214 Sentences With "arithmetic mean" | Random Sentence Generator

Quarterly growth in the arithmetic mean of the GDP and GDI averaged 1.6 percent over the 12 months ending in September.

But despite his claim of representing some harmless arithmetic mean of racism, he touched on many of the F.N.'s identitarian themes.

EPA promulgated the National Ambient Air Quality Standards (NAAQS) to establish basic air pollution control requirements across the U.S. The NAAQS sets standards on six main sources of pollutants, which include emissions of: ozone (0.12 ppm per 1 hour), carbon monoxide (35 ppm per 1 hour; primary standard), particulate matter (50g/m^3 at an annual arithmetic mean), sulfur dioxide (80g/m^3 at an annual arithmetic mean), nitrogen dioxide (100g/m^3 at an annual arithmetic mean), and lead emissions (1.5g/m^3 at an annual arithmetic mean).

If all the terms are equal: :x_1 = x_2 = \cdots = x_n, then their sum is , so their arithmetic mean is ; and their product is , so their geometric mean is ; therefore, the arithmetic mean and geometric mean are equal, as desired.

The design matrix for an arithmetic mean is a column vector of ones.

Loimu's audience arithmetic mean in the 2009–2010 season was 721 per game.

This measure too is valid only for data that are measured absolutely on a strictly positive scale. ; Weighted arithmetic mean: an arithmetic mean that incorporates weighting to certain data elements. ; Truncated mean or trimmed mean: the arithmetic mean of data values after a certain number or proportion of the highest and lowest data values have been discarded. ; Interquartile mean: a truncated mean based on data within the interquartile range.

The mid-range is the arithmetic mean of the highest and lowest values of a set.

The tetrahedron's center of mass computes as the arithmetic mean of its four vertices, see Centroid.

The arithmetic mean, the geometric mean and the harmonic mean are known collectively as the Pythagorean means.

The UPGMA (Unweighted Pair Group Method with Arithmetic mean) and WPGMA (Weighted Pair Group Method with Arithmetic mean) methods produce rooted trees and require a constant-rate assumption – that is, it assumes an ultrametric tree in which the distances from the root to every branch tip are equal.

The UPGMA (Unweighted Pair Group Method with Arithmetic mean) and WPGMA (Weighted Pair Group Method with Arithmetic mean) methods produce rooted trees and require a constant-rate assumption - that is, it assumes an ultrametric tree in which the distances from the root to every branch tip are equal.

In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f. It is also called Kolmogorov mean after Russian mathematician Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.

The geometric mean of two positive numbers is never bigger than the arithmetic mean (see inequality of arithmetic and geometric means). As a consequence, for , is an increasing sequence, is a decreasing sequence, and . These are strict inequalities if . is thus a number between the geometric and arithmetic mean of and ; it is also between and .

The geometric mean of a non-empty data set of (positive) numbers is always at most their arithmetic mean. Equality is only obtained when all numbers in the data set are equal; otherwise, the geometric mean is smaller. For example, the geometric mean of 242 and 288 equals 264, while their arithmetic mean is 265. In particular, this means that when a set of non-identical numbers is subjected to a mean-preserving spread — that is, the elements of the set are "spread apart" more from each other while leaving the arithmetic mean unchanged — their geometric mean decreases.

Since the arithmetic mean is not always appropriate for angles, the following method can be used to obtain both a mean value and measure for the variance of the angles: Convert all angles to corresponding points on the unit circle, e.g., \alpha to (\cos\alpha,\sin\alpha). That is, convert polar coordinates to Cartesian coordinates. Then compute the arithmetic mean of these points.

There are 65 ultra-prominent summits in Alaska. If an elevation or prominence is calculated as a range of values, the arithmetic mean is shown.

The weighted arithmetic mean is similar to an ordinary arithmetic mean (the most common type of average), except that instead of each of the data points contributing equally to the final average, some data points contribute more than others. The notion of weighted mean plays a role in descriptive statistics and also occurs in a more general form in several other areas of mathematics. If all the weights are equal, then the weighted mean is the same as the arithmetic mean. While weighted means generally behave in a similar fashion to arithmetic means, they do have a few counterintuitive properties, as captured for instance in Simpson's paradox.

Many important inequalities can be proved by the rearrangement inequality, such as the arithmetic mean – geometric mean inequality, the Cauchy–Schwarz inequality, and Chebyshev's sum inequality.

The geometric mean filter is most widely used to filter out Gaussian noise. In general it will help smooth the image with less data loss than an arithmetic mean filter.

An ultra-prominent summit is a summit with at least of topographic prominence. If an elevation or prominence is calculated as a range of values, the arithmetic mean is shown.

Dorsheim lies right near the Autobahn A 61 and has its own Autobahn interchange. The noise pollution rates an arithmetic mean according to measurements of roughly 63 dB/24 h.

The contraharmonic mean of a random variable is equal to the sum of the arithmetic mean and the variance divided by the arithmetic mean.Kingley MSC (1989) The distribution of hauled out ringed seals an interpretation of Taylor's law. Oecologia 79: 106-110 Since the variance is always ≥0 the contraharmonic mean is always greater than or equal to the arithmetic mean. The ratio of the variance and the mean was proposed as a test statistic by Clapham.

In signal processing, spectral flatness, a measure of how flat or spiky a spectrum is, is defined as the ratio of the geometric mean of the power spectrum to its arithmetic mean.

Then the 1 and 13 are removed to obtain the list 3, 7. Since there are two elements in this remaining list, the median is their arithmetic mean, (3 + 7)/2 = 5.

The harmonic and geometric means are concave symmetric functions of their arguments, and hence Schur- concave, while the arithmetic mean is a linear function of its arguments, so both concave and convex.

When the variance of the measurement error is only partially known, the reduced chi-squared may serve as a correction estimated a posteriori, see weighted arithmetic mean#Correcting for over- or under-dispersion.

In mathematics and statistics, the arithmetic mean (, stress on first and third syllables of "arithmetic"), or simply the mean or the average (when the context is clear), is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results of an experiment or an observational study, or frequently a set of results from a survey. The term "arithmetic mean" is preferred in some contexts in mathematics and statistics, because it helps distinguish it from other means, such as the geometric mean and the harmonic mean. In addition to mathematics and statistics, the arithmetic mean is used frequently in many diverse fields such as economics, anthropology and history, and it is used in almost every academic field to some extent.

The optical interpolation b, in the three stems a (thinnest), b (interpolation) and c (thickest), is set to the geometric mean of a and c, i.e. b² = ac (as opposed to the linear arithmetic mean).

As such, they represent the well-known fact that the harmonic mean is less than the geometric mean, which is less than the arithmetic mean, which is, in turn, less than the root mean square.

WPGMA (Weighted Pair Group Method with Arithmetic Mean) is a simple agglomerative (bottom-up) hierarchical clustering method, generally attributed to Sokal and Michener. The WPGMA method is similar to its unweighted variant, the UPGMA method.

In data sets containing real-numbered measurements, the suspected outliers are the measured values that appear to lie outside the cluster of most of the other data values. The outliers would greatly change the estimate of location if the arithmetic average were to be used as a summary statistic of location. The problem is that the arithmetic mean is very sensitive to the inclusion of any outliers; in statistical terminology, the arithmetic mean is not robust. In the presence of outliers, the statistician has two options.

A post-stack attribute that computes the arithmetic mean of the amplitudes of a trace within a specified window. This can be used to observe the trace bias which could indicate the presence of a bright spot.

Gauss showed that the arithmetic mean is indeed the best estimate of the location parameter by changing both the probability density and the method of estimation. He then turned the problem around by asking what form the density should have and what method of estimation should be used to get the arithmetic mean as estimate of the location parameter. In this attempt, he invented the normal distribution. An early demonstration of the strength of Gauss's method came when it was used to predict the future location of the newly discovered asteroid Ceres.

It remains to show that if not all the terms are equal, then the arithmetic mean is greater than the geometric mean. Clearly, this is only possible when . This case is significantly more complex, and we divide it into subcases.

An ultra-prominent summit is a summit with at least of topographic prominence. There are 126 ultra-prominent summits in the United States. If an elevation or prominence is calculated as a range of values, the arithmetic mean is shown.

For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic, meaning that it is greatly influenced by outliers (values that are very much larger or smaller than most of the values). Notably, for skewed distributions, such as the distribution of income for which a few people's incomes are substantially greater than most people's, the arithmetic mean may not coincide with one's notion of "middle", and robust statistics, such as the median, may provide better description of central tendency.

For example, Cesàro summation assigns Grandi's divergent series. :1 - 1 + 1 - 1 + \cdots the value . Cesàro summation is an averaging method, in that it relies on the arithmetic mean of the sequence of partial sums. Other methods involve analytic continuations of related series.

First, the statistician may remove the suspected outliers from the data set and then use the arithmetic mean to estimate the location parameter. Second, the statistician may use a robust statistic, such as the median statistic. Peirce's criterion is a statistical procedure for eliminating outliers.

Instead, either values for one measure are compared for a fixed level at the other measure (e.g. precision at a recall level of 0.75) or both are combined into a single measure. Examples of measures that are a combination of precision and recall are the F-measure (the weighted harmonic mean of precision and recall), or the Matthews correlation coefficient, which is a geometric mean of the chance- corrected variants: the regression coefficients Informedness (DeltaP') and Markedness (DeltaP). Accuracy is a weighted arithmetic mean of Precision and Inverse Precision (weighted by Bias) as well as a weighted arithmetic mean of Recall and Inverse Recall (weighted by Prevalence).

The geometric mean is more appropriate than the arithmetic mean for describing proportional growth, both exponential growth (constant proportional growth) and varying growth; in business the geometric mean of growth rates is known as the compound annual growth rate (CAGR). The geometric mean of growth over periods yields the equivalent constant growth rate that would yield the same final amount. Suppose an orange tree yields 100 oranges one year and then 180, 210 and 300 the following years, so the growth is 80%, 16.6666% and 42.8571% for each year respectively. Using the arithmetic mean calculates a (linear) average growth of 46.5079% (80% + 16.6666% + 42.8571%, that sum then divided by 3).

He had managed to complete Laplace's program of specifying a mathematical form of the probability density for the observations, depending on a finite number of unknown parameters, and define a method of estimation that minimises the error of estimation. Gauss showed that the arithmetic mean is indeed the best estimate of the location parameter by changing both the probability density and the method of estimation. He then turned the problem around by asking what form the density should have and what method of estimation should be used to get the arithmetic mean as estimate of the location parameter. In this attempt, he invented the normal distribution.

Similarly, if one wishes to estimate the density of an alloy given the densities of its constituent elements and their mass fractions (or, equivalently, percentages by mass), then the predicted density of the alloy (exclusive of typically minor volume changes due to atom packing effects) is the weighted harmonic mean of the individual densities, weighted by mass, rather than the weighted arithmetic mean as one might at first expect. To use the weighted arithmetic mean, the densities would have to be weighted by volume. Applying dimensional analysis to the problem while labelling the mass units by element and making sure that only like element-masses cancel, makes this clear.

The diagonal of a half square forms the basis for the geometrical construction of a golden rectangle. The golden ratio is the arithmetic mean of 1 and .Browne, Malcolm W. (July 30, 1985) New York Times Puzzling Crystals Plunge Scientists into Uncertainty. Section: C; Page 1.

The Edgeworth-Marshall index uses the arithmetic mean of mean characteristics of two periods t and t+1. A Walsh-type index uses the geometric average of two periods. And finally, the base quality index does not update characteristics (quality) and uses fixed base characteristics - \ z^0.

It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of the total number of values.

For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

Consider the sets R and S :R={1, 2, 3, 4} :S={5, 6, 7, 8, 9} The arithmetic mean of R is 2.5, and the arithmetic mean of S is 7. However, if 5 is moved from S to R, producing :R={1, 2, 3, 4, 5} :S={6, 7, 8, 9} then the arithmetic mean of R increases to 3, and the arithmetic mean of S increases to 7.5. Consider this more illustrative example :R={1,2} :S={99, 10,000, 20 000} with arithmetic means 1.5 for R and 10,033 for S. Moving 99 from S to R gives means 34 and 15,000. 99 is orders of magnitude above 1 and 2, and orders of magnitude below 10,000 and 20,000. It should come as no surprise that the transfer of 99 increases the mean of both R and S. The element which is moved does not have to be the very lowest of its set; it merely has to have a value that lies between the means of the two sets. Consider this example: :R={1, 3, 5, 7, 9, 11, 13} (mean = 7) :S={6, 8, 10, 12, 14, 16, 18} (mean = 12) Moving 10, which is larger than R's mean of 7 and smaller than S's mean of 12, from S to R will raise the mean of R from 7 to 7.375, and the mean of S from 12 to 12.333.

However, for some probability distributions, there is no guarantee that the least-squares solution is even possible given the observations; still, in such cases it is the best estimator that is both linear and unbiased. For example, it is easy to show that the arithmetic mean of a set of measurements of a quantity is the least-squares estimator of the value of that quantity. If the conditions of the Gauss–Markov theorem apply, the arithmetic mean is optimal, whatever the distribution of errors of the measurements might be. However, in the case that the experimental errors do belong to a normal distribution, the least-squares estimator is also a maximum likelihood estimator.

The "Unweighted Pairwise Group Method with Arithmetic-mean" (UPGMA) is a clustering technique which operates by repeatedly joining the two languages that have the smallest distance between them. It operates accurately with clock-like evolution but otherwise it can be in error. This is the method used in Swadesh's original lexicostatistics.

If the left and right derivatives are equal, then they have the same value as the usual ("bidirectional") derivative. One can also define a symmetric derivative, which equals the arithmetic mean of the left and right derivatives (when they both exist), so the symmetric derivative may exist when the usual derivative does not.

Data dependencies of a selected cell in the 2D array. To illustrate the formal definition, we'll have a look at how a two dimensional Jacobi iteration can be defined. The update function computes the arithmetic mean of a cell's four neighbors. In this case we set off with an initial solution of 0.

All elevations include an adjustment from the National Geodetic Vertical Datum of 1929 (NGVD 29) to the North American Vertical Datum of 1988 (NAVD 88). For further information, please see this United States National Geodetic Survey note. If an elevation or prominence is calculated as a range of values, the arithmetic mean is shown.

An important practical application in financial mathematics is to computing the rate of return: the annualized return, computed via the geometric mean, is less than the average annual return, computed by the arithmetic mean (or equal if all returns are equal). This is important in analyzing investments, as the average return overstates the cumulative effect.

Miwin's Dice are a set of nontransitive dice invented in 1975 by the physicist Michael Winkelmann. They consist of three different dice with faces bearing numbers from 1 to 9; opposite faces sum to 9, 10, or 11. The numbers on each die give the sum of 30 and have an arithmetic mean of 5.

All elevations include an adjustment from the National Geodetic Vertical Datum of 1929 (NGVD 29) to the North American Vertical Datum of 1988 (NAVD 88). For further information, please see this United States National Geodetic Survey note. If an elevation or prominence is calculated as a range of values, the arithmetic mean is shown.

For positive real numbers a and b, their arithmetic mean, geometric mean, and harmonic mean are the lengths of the sides of a right triangle if and only if that triangle is a Kepler triangle.Di Domenico, Angelo, "The golden ratio—the right triangle—and the arithmetic, geometric, and harmonic means," The Mathematical Gazette 89, 2005.

All elevations include an adjustment from the National Geodetic Vertical Datum of 1929 (NGVD 29) to the North American Vertical Datum of 1988 (NAVD 88). For further information, please see this United States National Geodetic Survey note. If an elevation or prominence is calculated as a range of values, the arithmetic mean is shown.

It is based on a non-linear function which, starting from the arithmetic mean of the normalized indicators, introduces a penalty for the units with unbalanced values of the indicators (De Muro et al., 2011De Muro, P., Mazziotta, M., Pareto, A. (2011). Composite Indices of Development and Poverty: An Application to MDGs. Social Indicators Research, 104, 1–18.).

Bakker, Arthur. "The early history of average values and implications for education." Journal of Statistics Education 11.1 (2003): 17-26. A possible precursor to the arithmetic mean is the mid-range (the mean of the two extreme values), used for example in Arabian astronomy of the ninth to eleventh centuries, but also in metallurgy and navigation.

In mathematics, there are different results that share the common name of the Ky Fan inequality. The Ky Fan inequality presented here is used in game theory to investigate the existence of an equilibrium. Another Ky Fan inequality is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval.

All elevations in this article include an elevation adjustment from the National Geodetic Vertical Datum of 1929 (NGVD 29) to the North American Vertical Datum of 1988 (NAVD 88). For further information, please see this United States National Geodetic Survey note. If an elevation or prominence is calculated as a range of values, the arithmetic mean is shown.

In geometry, a centroidal Voronoi tessellation (CVT) is a special type of Voronoi tessellation or Voronoi diagram. A Voronoi tessellation is called centroidal when the generating point of each Voronoi cell is also its centroid, i.e. the arithmetic mean or center of mass. It can be viewed as an optimal partition corresponding to an optimal distribution of generators.

See The Post-War History of the London Stock Market, The FT 30 index was calculated using the geometric mean.The Financial System Today, by Eric E. Rowley. (Manchester University Press, 1987) As Rowley states, this had the effect of understating movements in the index compared to using the arithmetic mean; and there are circumstances where this is undesirable.

For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 100 highest major summits of Washington by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

Even though comparison-sorting n items requires operations, selection algorithms can compute the th-smallest of items with only operations. This includes the median, which is the th order statistic (or for an even number of samples, the arithmetic mean of the two middle order statistics). Here: Section 3.6 "Order Statistics", p.97-99, in particular Algorithm 3.6 and Theorem 3.9.

In surveying, the repetition method is used to improve precision and accuracy of measurements of horizontal angles. The same angle is measured multiple times, with the survey instrument rotated so that systematic errors tend to cancel. The arithmetic mean of these observations gives true value of an angle. The precision of the measurement can exceed the least count of the instrument. used.

For projectiles that are launched by firearms and artillery, the nature of the gun's barrel is also important. Longer barrels allow more of the propellant's energy to be given to the projectile, yielding greater range. Rifling, while it may not increase the average (arithmetic mean) range of many shots from the same gun, will increase the accuracy and precision of the gun.

The arithmetic mean may be contrasted with the median. The median is defined such that no more than half the values are larger than, and no more than half are smaller than, the median. If elements in the data increase arithmetically, when placed in some order, then the median and arithmetic average are equal. For example, consider the data sample {1,2,3,4}.

The grading system used is the Romanian numerical grading system, with grades ranging from 1 to 10, 10 being the maximum; 5 is the graduating mark. At the end of each semester and at the end of the year, the final grade is obtained through the arithmetic mean of the grades received throughout the year. Where term papers apply (Romanian, Mathematics, Physics, Informatics), this arithmetic mean is multiplied by 3 added with the term paper’s result and divided by 4 to obtain the final grade. An American style ranking system is not offered but each group of students calculates its own rank and awards diplomas for the highest 3 final grades, and at the end of the 12th grade the school awards the distinction of Chief of graduates to the student with the highest final grade over the four years.

In mathematics and its applications, the mean square is defined as the arithmetic mean of the squares of a set of numbers or of a random variable, or as the arithmetic mean of the squares of the differences between a set of numbers and a given "origin" that may not be zero (e.g. may be a mean or an assumed mean of the data). When the mean square is calculated relative to a given "target" or "correct" value, or as the mean square of differences from a sequence of correct values, it is known as mean squared error. A typical estimate for the variance from a set of sample values x_i uses a divisor of one less then the number n of values, rather than a simple arithmetic average, and this is still called the mean square (e.g.

In statistics the assumed mean is a method for calculating the arithmetic mean and standard deviation of a data set. It simplifies calculating accurate values by hand. Its interest today is chiefly historical but it can be used to quickly estimate these statistics. There are other rapid calculation methods which are more suited for computers which also ensure more accurate results than the obvious methods.

Serum anti-aberrant-lactobacillus antibody titres (geometric mean and standard error) in 97 women vaccinated with SolcoTrichovac (Milovanović 1983). A booster dose was given 12 months after the first injection. Total secretory IgA concentration (arithmetic mean and standard error) of the vaginal secretions of 95 women vaccinated with SolcoTrichovac (Rüttgers 1988). Mucosal surfaces are a major portal of entry for pathogens into the body.

For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 100 highest major summits of the United States by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

Mean opinion score (MOS) is a measure used in the domain of Quality of Experience and telecommunications engineering, representing overall quality of a stimulus or system. It is the arithmetic mean over all individual "values on a predefined scale that a subject assigns to his opinion of the performance of a system quality".ITU-T Rec. P.10 (2006) Vocabulary for performance and quality of service.

The Value Line Arithmetic Composite Index was established on February 1, 1988, using the arithmetic mean to more closely mimic the change in the index if you held a portfolio of stocks in equal amounts. The daily price change of the Value Line Arithmetic Composite Index is calculated by adding the daily percent change of all the stocks, and then dividing by the total number of stocks.

UPGMA (unweighted pair group method with arithmetic mean) is a simple agglomerative (bottom-up) hierarchical clustering method. The method is generally attributed to Sokal and Michener. The UPGMA method is similar to its weighted variant, the WPGMA method. Note that the unweighted term indicates that all distances contribute equally to each average that is computed and does not refer to the math by which it is achieved.

The method is best explained with an example. Consider the following dataset: :5, 8, 4, 38, 8, 6, 9, 7, 7, 3, 1, 6 First sort the list from lowest-to-highest: :1, 3, 4, 5, 6, 6, 7, 7, 8, 8, 9, 38 There are 12 observations (datapoints) in the dataset, thus we have 4 quartiles of 3 numbers. Discard the lowest and the highest 3 values: : ~~1, 3, 4~~ , 5, 6, 6, 7, 7, 8, ~~8, 9, 38~~ We now have 6 of the 12 observations remaining; next, we calculate the arithmetic mean of these numbers: :xIQM = (5 + 6 + 6 + 7 + 7 + 8) / 6 = 6.5 This is the interquartile mean. For comparison, the arithmetic mean of the original dataset is :(5 + 8 + 4 + 38 + 8 + 6 + 9 + 7 + 7 + 3 + 1 + 6) / 12 = 8.5 due to the strong influence of the outlier, 38.

This is the only central tendency measure that can be used with nominal data, which have purely qualitative category assignments. ; Geometric mean: the nth root of the product of the data values, where there are n of these. This measure is valid only for data that are measured absolutely on a strictly positive scale. ; Harmonic mean: the reciprocal of the arithmetic mean of the reciprocals of the data values.

The midpoint of the distribution (a + b) / 2 is both the mean and the median of the uniform distribution. Although both the sample mean and the sample median are unbiased estimators of the midpoint, neither is as efficient as the sample mid-range, i.e. the arithmetic mean of the sample maximum and the sample minimum, which is the UMVU estimator of the midpoint (and also the maximum likelihood estimate).

An ultra-prominent summit is a summit with at least of topographic prominence. of the U.S. State of Alaska. The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown.

In many situations involving rates and ratios, the harmonic mean provides the correct average. For instance, if a vehicle travels a certain distance d outbound at a speed x (e.g. 60 km/h) and returns the same distance at a speed y (e.g. 20 km/h), then its average speed is the harmonic mean of x and y (30 km/h) - not the arithmetic mean (40 km/h).

The weighted harmonic mean is the preferable method for averaging multiples, such as the price–earnings ratio (P/E). If these ratios are averaged using a weighted arithmetic mean, high data points are given greater weights than low data points. The weighted harmonic mean, on the other hand, correctly weights each data point. The simple weighted arithmetic mean when applied to non-price normalized ratios such as the P/E is biased upwards and cannot be numerically justified, since it is based on equalized earnings; just as vehicles speeds cannot be averaged for a roundtrip journey (see above). For example, consider two firms, one with a market capitalization of $150 billion and earnings of $5 billion (P/E of 30) and one with a market capitalization of $1 billion and earnings of $1 million (P/E of 1000). Consider an index made of the two stocks, with 30% invested in the first and 70% invested in the second.

We use terms pattern and frequent sub-graph in this review interchangeably. There is an ensemble of random graphs corresponding to the null-model associated to . We should choose random graphs uniformly from and calculate the frequency for a particular frequent sub-graph in . If the frequency of in is higher than its arithmetic mean frequency in random graphs , where , we call this recurrent pattern significant and hence treat as a network motif for .

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit. The summit may be near its key col or quite far away.

Not all natal points are created equal. If a planet happens to be at the center of a lot of midpoints, then transits to that point will be more powerful. A midpoint refers to the arithmetic mean of the zodiac degrees of two planets. Say, in a particular chart, Venus is at 0° Aries, Mercury is at 0° Taurus, and Mars is at 0° Gemini, then Mercury is at the midpoint of Venus/Mars.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 55 highest major summits of Colorado by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

The geometric mean has from time to time been used to calculate financial indices (the averaging is over the components of the index). For example, in the past the FT 30 index used a geometric mean. It is also used in the recently introduced "RPIJ" measure of inflation in the United Kingdom and in the European Union. This has the effect of understating movements in the index compared to using the arithmetic mean.

In colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit. The summit may be near its key col or quite far away.

For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit. The summit may be near its key col or quite far away.

For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit. The summit may be near its key col or quite far away.

However, in practice and in the definition of MOS, it is considered acceptable to calculate the arithmetic mean. It has been shown that for categorical rating scales (such as ACR), the individual items are not perceived equidistant by subjects. For example, there may be a larger "gap" between Good and Fair than there is between Good and Excellent. The perceived distance may also depend on the language into which the scale is translated.

For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

In mathematics, the geometric–harmonic mean M(x, y) of two positive real numbers x and y is defined as follows: we form the geometric mean of g0 = x and h0 = y and call it g1, i.e. g1 is the square root of xy. We also form the harmonic mean of x and y and call it h1, i.e. h1 is the reciprocal of the arithmetic mean of the reciprocals of x and y.

For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

A Vernier scale on a caliper may have a least count of 0.1 mm while a micrometer may have a least count of 0.01 mm. The least count error occurs with both systematic and random errors. Instruments of higher precision can reduce the least count error. By repeating the observations and taking the arithmetic mean of the result, the mean value would be very close to the true value of the measured quantity.

Blind tasting was performed and the judges were asked to grade each wine out of 20 points. No specific grading framework was given, leaving the judges free to grade according to their own criteria. Rankings of the wines preferred by individual judges were based on the grades they individually attributed. An overall ranking of the wines preferred by the jury was also established in averaging the sum of each judge's individual grades (arithmetic mean).

For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 20 highest major summits of Arizona by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 40 highest major summits of Idaho by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 50 highest major summits of Montana by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 50 highest major summits of Nevada by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 30 highest major summits of Oregon by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 50 highest major summits of Utah by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 40 highest major summits of Wyoming by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit. The summit may be near its key col or quite far away.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 50 highest major summits of California by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 100 highest major summits of greater North America by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

Mean time between failures (MTBF) is the predicted elapsed time between inherent failures of a mechanical or electronic system, during normal system operation. MTBF can be calculated as the arithmetic mean (average) time between failures of a system. The term is used for repairable systems, while mean time to failure (MTTF) denotes the expected time to failure for a non- repairable system. The definition of MTBF depends on the definition of what is considered a failure.

In mathematics and statistics, the Fréchet mean is a generalization of centroids to metric spaces, giving a single representative point or central tendency for a cluster of points. It is named after Maurice Fréchet. Karcher mean is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove and Hermann Karcher... On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.

A computationally large safe prime is likely to be a cryptographically strong prime. Note that the criteria for determining if a pseudoprime is a strong pseudoprime is by congruences to powers of a base, not by inequality to the arithmetic mean of neighboring pseudoprimes. When a prime is equal to the mean of its neighboring primes, it's called a balanced prime. When it's less, it's called a weak prime (not to be confused with a weakly prime number).

Using the finite form of Jensen's inequality for the natural logarithm, we can prove the inequality between the weighted arithmetic mean and the weighted geometric mean stated above. Since an with weight has no influence on the inequality, we may assume in the following that all weights are positive. If all are equal, then equality holds. Therefore, it remains to prove strict inequality if they are not all equal, which we will assume in the following, too.

Because the standard deviation squares its differences, it tends to give more weight to larger differences and less weight to smaller differences compared to the mean absolute difference. When the arithmetic mean is finite, the mean absolute difference will also be finite, even when the standard deviation is infinite. See the examples for some specific comparisons. The recently introduced distance standard deviation plays similar role to the mean absolute difference but the distance standard deviation works with centered distances.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 100 highest major summits of greater North America by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

The F-score is also used for evaluating classification problems with more than two classes (Multiclass classification). In this setup, the final score is obtained by micro-averaging (biased by class frequency) or macro-averaging (taking all classes as equally important). For macro-averaging, two different formulas have been used by applicants: the F-score of (arithmetic) class-wise precision and recall means or the arithmetic mean of class-wise F-scores, where the latter exhibits more desirable properties.

This methodology is somewhat disputed amongst experts in quantitative methods used in educational and psychological measurement as it is basically only a linear scale transformation that cannot ensure or examine whether PISA and TIMSS scores are based on the same or at least comparable measurement constructs: The numerical values used to measure shoe size and intelligence can be transformed so that both have the same arithmetic mean and standard deviation, but they still represent two very different characteristics.

The mean center, or centroid, is the point on which a rigid, weightless map would balance perfectly, if the population members are represented as points of equal mass. Mathematically, the centroid is the point to which the population has the smallest possible sum of squared distances. It is easily found by taking the arithmetic mean of each coordinate. If defined in the three-dimensional space, the centroid of points on the Earth's surface is actually inside the Earth.

If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 30 highest major summits of New Mexico by elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.The topographic prominence of a summit is the topographic elevation difference between the summit and its highest or key col to a higher summit.

Mount Logan in the Saint Elias Mountains of Yukon is the highest peak of Canada. The following sortable table comprises the 150 most topographically prominent mountain peaks of Canada. The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown.

It is computed from the prices of selected stocks (typically a weighted arithmetic mean). Two of the primary criteria of an index are that it is investable and transparent: The method of its construction are specified. Investors can invest in a stock market index by buying an index fund, which are structured as either a mutual fund or an exchange-traded fund, and "track" an index. The difference between an index fund's performance and the index, if any, is called tracking error.

The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 13 major summits of Hawaii by topographic elevation. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

The calculation of scores for the five brand positioning variables is done by evaluating the arithmetic mean of the various sub-elements that a brand positioning variable includes. Thus, a set of five average scores, labeled as SB1-SB5, is obtained, which reflects the consumer preference for the various attributes. Similarly, a score is obtained for the ten sub-elements of ethics, and are labeled as SE1-SE10. A high score reflects greater liking, while a low score indicates dislike and insignificance.

An ultra-prominent summit is a summit with at least of topographic prominence. The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

Visual proof that . Taking square roots and dividing by two gives the AM–GM inequality. In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same. The simplest non-trivial case — i.e.

An ultra-prominent summit is a summit with at least of topographic prominence. The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

An ultra-prominent summit is a summit with at least of topographic prominence. The summit of a mountain or hill may be measured in three principal ways: # The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. # The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

An ultra-prominent summit is a summit with at least of topographic prominence. The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

For ordinal variables the median can be calculated as a measure of central tendency and the range (and variations of it) as a measure of dispersion. For interval level variables, the arithmetic mean (average) and standard deviation are added to the toolbox and, for ratio level variables, we add the geometric mean and harmonic mean as measures of central tendency and the coefficient of variation as a measure of dispersion. For interval and ratio level data, further descriptors include the variable's skewness and kurtosis.

All elevations in the 48 states of the contiguous United States include an elevation adjustment from the National Geodetic Vertical Datum of 1929 (NGVD 29) to the North American Vertical Datum of 1988 (NAVD 88). For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

All elevations in the 48 states of the contiguous United States include an elevation adjustment from the National Geodetic Vertical Datum of 1929 (NGVD 29) to the North American Vertical Datum of 1988 (NAVD 88). For further information, please see this United States National Geodetic Survey note. If a summit elevation or prominence has a range of values, the arithmetic mean is cited. Denali is one of only three summits on Earth (along with Mount Everest and Aconcagua) with more than of topographic prominence.

The site was designed to make the nature of review aggregation clear, opting for a simple arithmetic mean, in contrast to the hidden weights used by Metacritic. The site also highlights review authors' names and allows users to customize what reviews took priority. The site began development in 2014, and formally launched on September 30, 2015 with reviews from 75 publications. The site generally only supports video game reviews from its launch date forward and there are no plans to fully populate older games.

Each pair's result on a board is compared against a "datum" score which is the arithmetic mean of all the results (usually after exclusion of one or more of the top and bottom results), and the difference converted to IMPs. Sometimes, the median is used instead of the mean. ;Bye # A round of an event during which a team or pair is not scheduled to play. # A location ("bye-stand") such as a chair or table, where boards are kept when not in use during an event.

Wolff was so pleased with the solution that he sought to extend the arithmetic mean method to more divergent series such as . Briefly, if one expresses a partial sum of this series as a function of the penultimate term, one obtains either or . The mean of these values is , and assuming that at infinity yields as the value of the series. Leibniz's intuition prevented him from straining his solution this far, and he wrote back that Wolff's idea was interesting but invalid for several reasons.

Let v : M → R3 be a smooth immersion of a compact, orientable surface. Giving M the Riemannian metric induced by v, let H : M → R be the mean curvature (the arithmetic mean of the principal curvatures κ1 and κ2 at each point). In this notation, the Willmore energy W(M) of M is given by : W(M) = \int_M H^2 \, dA. It is not hard to prove that the Willmore energy satisfies W(M) ≥ 4π, with equality if and only if M is an embedded round sphere.

Winkel tripel projection of the world, 15° graticule The Winkel tripel projection with Tissot's indicatrix of deformation The Winkel tripel projection (Winkel III), a modified azimuthal map projection of the world, is one of three projections proposed by German cartographer Oswald Winkel (7 January 1874 – 18 July 1953) in 1921. The projection is the arithmetic mean of the equirectangular projection and the Aitoff projection: The name tripel (German for "triple") refers to Winkel's goal of minimizing three kinds of distortion: area, direction, and distance.

If a population is lognormally distributed then the harmonic mean of the population size (H) is related to the arithmetic mean (m) : H = m - am^{b - 1} Given that H must be > 0 for the population to persist then rearranging we have : m > a^{1/(2 - b)} is the minimum size of population for the species to persist. The assumption of a lognormal distribution appears to apply to about half of a sample of 544 species. suggesting that it is at least a plausible assumption.

A geometric construction of the quadratic mean and the Pythagorean means (of two numbers a and b). Harmonic mean denoted by H, geometric by G, arithmetic by A and quadratic mean (also known as root mean square) denoted by Q. Comparison of the arithmetic, geometric and harmonic means of a pair of numbers. The vertical dashed lines are asymptotes for the harmonic means. In mathematics, the three classical Pythagorean means are the arithmetic mean (AM), the geometric mean (GM), and the harmonic mean (HM).

This article defines Central America as the seven nations of Belize, Costa Rica, El Salvador, Guatemala, Honduras, Nicaragua, and Panamá. The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. The first table below ranks the 25 highest major summits of Central America by elevation.

For each trip segment i, the slowness si = 1/speedi. Then take the weighted arithmetic mean of the si's weighted by their respective distances (optionally with the weights normalized so they sum to 1 by dividing them by trip length). This gives the true average slowness (in time per kilometre). It turns out that this procedure, which can be done with no knowledge of the harmonic mean, amounts to the same mathematical operations as one would use in solving this problem by using the harmonic mean.

However, there are various older vague references to the use of the arithmetic mean (which are not as clear, but might reasonably have to do with our modern definition of the mean). In a text from the 4th century, it was written that (text in square brackets is a possible missing text that might clarify the meaning): : In the first place, we must set out in a row the sequence of numbers from the monad up to nine: 1, 2, 3, 4, 5, 6, 7, 8, 9. Then we must add up the amount of all of them together, and since the row contains nine terms, we must look for the ninth part of the total to see if it is already naturally present among the numbers in the row; and we will find that the property of being [one] ninth [of the sum] only belongs to the [arithmetic] mean itself... Even older potential references exist. There are records that from about 700 BC, merchants and shippers agreed that damage to the cargo and ship (their "contribution" in case of damage by the sea) should be shared equally among themselves.

The median can be used as a measure of location when one attaches reduced importance to extreme values, typically because a distribution is skewed, extreme values are not known, or outliers are untrustworthy, i.e., may be measurement/transcription errors. For example, consider the multiset : 1, 2, 2, 2, 3, 14. The median is 2 in this case, (as is the mode), and it might be seen as a better indication of the center than the arithmetic mean of 4, which is larger than all-but-one of the values.

For a finite population, the population mean of a property is equal to the arithmetic mean of the given property, while considering every member of the population. For example, the population mean height is equal to the sum of the heights of every individual—divided by the total number of individuals. The sample mean may differ from the population mean, especially for small samples. The law of large numbers states that the larger the size of the sample, the more likely it is that the sample mean will be close to the population mean.

Comparison of the arithmetic mean, median and mode of two skewed (log-normal) distributions. Geometric visualization of the mode, median and mean of an arbitrary probability density function. In descriptive statistics, the mean may be confused with the median, mode or mid-range, as any of these may be called an "average" (more formally, a measure of central tendency). The mean of a set of observations is the arithmetic average of the values; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or the most likely value (mode).

For a particular cluster of documents, we can calculate the centroid by finding the arithmetic mean of all the document vectors. If an entry in the centroid vector has a high value, then the corresponding term occurs frequently within the cluster. These terms can be used as a label for the cluster. One downside to using centroid labeling is that it can pick up words like "place" and "word" that have a high frequency in written text, but have little relevance to the contents of the particular cluster.

A more general method for defining an average takes any function g(x1, x2, ..., xn) of a list of arguments that is continuous, strictly increasing in each argument, and symmetric (invariant under permutation of the arguments). The average y is then the value that, when replacing each member of the list, results in the same function value: . This most general definition still captures the important property of all averages that the average of a list of identical elements is that element itself. The function provides the arithmetic mean.

Given a time series such as daily stock market prices or yearly temperatures people often want to create a smoother series. This helps to show underlying trends or perhaps periodic behavior. An easy way to do this is the moving average: one chooses a number n and creates a new series by taking the arithmetic mean of the first n values, then moving forward one place by dropping the oldest value and introducing a new value at the other end of the list, and so on. This is the simplest form of moving average.

149 is the 35th prime number, and with the next prime number, 151, is a twin prime, thus 149 is a Chen prime. 149 is an emirp, since the number 941 is also prime. 149 is a strong prime in the sense that it is more than the arithmetic mean of its two neighboring primes. 149 is an irregular prime since it divides the numerator of the Bernoulli number B130. 149 is an Eisenstein prime with no imaginary part and a real part of the form 3n - 1.

Gann maintained two formulas for calculating planetary averages. The first one is called the Mean of Five (MOF), which is arithmetic mean of the geocentric and heliocentric longitudes of the outer five planets: Jupiter, Saturn, Uranus, Neptune and Pluto. The other is called the Cycle of Eight (COE), which include all eight planets in its calculation, using both geocentric and heliocentric longitudes (the Sun replaces the Earth in the geocentric calculation). He thinks that when these averaged values make an aspect to the price, it will create support or resistance.

As one adjunct to data, the IMDb offers a rating scale that allows users to rate films on a scale of one to ten. IMDb indicates that submitted ratings are filtered and weighted in various ways in order to produce a weighted mean that is displayed for each film, series, and so on. It states that filters are used to avoid ballot stuffing; the method is not described in detail to avoid attempts to circumvent it. In fact, it sometimes produces an extreme difference between the weighted average and the arithmetic mean.

The frequency axis of this symbolic diagram may be linearly or logarithmically scaled. Except in special cases, the peak response will not align precisely with the center frequency. In electrical engineering and telecommunications, the center frequency of a filter or channel is a measure of a central frequency between the upper and lower cutoff frequencies. It is usually defined as either the arithmetic mean or the geometric mean of the lower cutoff frequency and the upper cutoff frequency of a band-pass system or a band-stop system.

The summit of a mountain or hill may be measured in three principal ways: #topographic elevation: the height of the summit above a geodetic sea level.All elevations in the 48 states of the contiguous United States include an elevation adjustment from the National Geodetic Vertical Datum of 1929 (NGVD 29) to the North American Vertical Datum of 1988 (NAVD 88). For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown.

This article defines Central America as the seven nations of Belize, Costa Rica, El Salvador, Guatemala, Honduras, Nicaragua, and Panamá. The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

Prior to the introduction of the euro, TALIBOR or the Tallinn Interbank Offered Rate was a daily reference rate based on the interest rates at which banks offer to lend unsecured funds to other banks in the Estonian wholesale money market (or interbank market in Estonian kroons. TALIBOR was published daily by the Bank of Estonia, together with TALIBID (Tallinn Interbank Bid Rate). TALIBOR was calculated based on the quotes for different maturities provided by reference banks at about 11.00 am each business day by disregarding highest and lowest quotation and calculating arithmetic mean of the quotations.

The mean absolute difference (univariate) is a measure of statistical dispersion equal to the average absolute difference of two independent values drawn from a probability distribution. A related statistic is the relative mean absolute difference, which is the mean absolute difference divided by the arithmetic mean, and equal to twice the Gini coefficient. The mean absolute difference is also known as the absolute mean difference (not to be confused with the absolute value of the mean signed difference) and the Gini mean difference (GMD). The mean absolute difference is sometimes denoted by Δ or as MD.

The number of rows of the Macaulay matrix is less than (ed)^n, where is the usual mathematical constant, and is the arithmetic mean of the degrees of the P_i. It follows that all solutions of a system of polynomial equations with a finite number of projective zeros can be determined in time d^{O(n)}. Although this bound is large, it is nearly optimal in the following sense: if all input degrees are equal, then the time complexity of the procedure is polynomial in the expected number of solutions (Bézout's theorem). This computation may be practically viable when , and are not large.

Dice may have numbers that do not form a counting sequence starting at one. One variation on the standard die is known as the "average" die. These are six-sided dice with sides numbered `2, 3, 3, 4, 4, 5`, which have the same arithmetic mean as a standard die (3.5 for a single die, 7 for a pair of dice), but have a narrower range of possible values (2 through 5 for one, 4 through 10 for a pair). They are used in some table-top wargames, where a narrower range of numbers is required.

Mount Logan in the Saint Elias Mountains of Yukon is the highest summit of Canada. This article comprises four sortable tables of mountain summits of Canada that are the higher than any other point north or south of their latitude or east or west their longitude in Canada. The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown.

This article comprises four sortable tables of mountain summits of Central America that are the higher than any other point north or south of their latitude or east or west their longitude in the region. The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown. #The topographic prominence of a summit is a measure of how high the summit rises above its surroundings.

Group nesting through associations is especially prevalent and varied in P. annularis, with an arithmetic mean of between 3.82 to 4.93 and a full range from one to 28. Variation from year to year explains only 2% of the variance in the size of associations. Only 5% of queens run a nest without any co-foundresses, while about three-quarters of foundresses become subordinate to a queen on a cooperative nest. The largest aggregations of foundresses are seen when females reuse the nest in which they were born, typically by constructing a new nest on the old one.

The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.All elevations in the 48 states of the contiguous United States include an elevation adjustment from the National Geodetic Vertical Datum of 1929 (NGVD 29) to the North American Vertical Datum of 1988 (NAVD 88). For further information, please see this United States National Geodetic Survey note.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown.

For the arithmetic mean, the speed of each portion of the trip is weighted by the duration of that portion, while for the harmonic mean, the corresponding weight is the distance. In both cases, the resulting formula reduces to dividing the total distance by the total time.) However one may avoid the use of the harmonic mean for the case of "weighting by distance". Pose the problem as finding "slowness" of the trip where "slowness" (in hours per kilometre) is the inverse of speed. When trip slowness is found, invert it so as to find the "true" average trip speed.

However, if one connects the resistors in series, then the average resistance is the arithmetic mean of x and y (with total resistance equal to the sum of x and y). This principle applies to capacitors in parallel or to inductors in series. As with the previous example, the same principle applies when more than two resistors, capacitors or inductors are connected, provided that all are in parallel or all are in series. The "conductivity effective mass" of a semiconductor is also defined as the harmonic mean of the effective masses along the three crystallographic directions.

As a quantitative measure, the "forecast bias" can be specified as a probabilistic or statistical property of the forecast error. A typical measure of bias of forecasting procedure is the arithmetic mean or expected value of the forecast errors, but other measures of bias are possible. For example, a median- unbiased forecast would be one where half of the forecasts are too low and half too high: see Bias of an estimator. In contexts where forecasts are being produced on a repetitive basis, the performance of the forecasting system may be monitored using a tracking signal, which provides an automatically maintained summary of the forecasts produced up to any given time.

In mathematics and its applications, the root mean square (RMS or rms) is defined as the square root of the mean square (the arithmetic mean of the squares of a set of numbers). The RMS is also known as the quadratic mean and is a particular case of the generalized mean with exponent 2\. RMS can also be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values during a cycle. For alternating electric current, RMS is equal to the value of the direct current that would produce the same average power dissipation in a resistive load.

The median is the middle number of the group when they are ranked in order. (If there are an even number of numbers, the mean of the middle two is taken.) Thus to find the median, order the list according to its elements' magnitude and then repeatedly remove the pair consisting of the highest and lowest values until either one or two values are left. If exactly one value is left, it is the median; if two values, the median is the arithmetic mean of these two. This method takes the list 1, 7, 3, 13 and orders it to read 1, 3, 7, 13.

In mathematics, there are two different results that share the common name of the Ky Fan inequality. One is an inequality involving the geometric mean and arithmetic mean of two sets of real numbers of the unit interval. The result was published on page 5 of the book Inequalities by Edwin F. Beckenbach and Richard E. Bellman (1961), who refer to an unpublished result of Ky Fan. They mention the result in connection with the inequality of arithmetic and geometric means and Augustin Louis Cauchy's proof of this inequality by forward-backward-induction; a method which can also be used to prove the Ky Fan inequality.

A semicircle with arithmetic and geometric means of a and b A semicircle can be used to construct the arithmetic and geometric means of two lengths using straight-edge and compass. For a semicircle with a diameter of a + b, the length of its radius is the arithmetic mean of a and b (since the radius is half of the diameter). The geometric mean can be found by dividing the diameter into two segments of lengths a and b, and then connecting their common endpoint to the semicircle with a segment perpendicular to the diameter. The length of the resulting segment is the geometric mean.

For Centrist Orthodoxy, moderation "is the result neither of guile nor of indifference nor of prudence, it is a matter of sacred principle ... it is not the mindless application of the arithmetic mean... [rather] it is the earnest sober and intelligent assessment of each situation... [Thus], moderation issues from a broad weltanschauung rather than from tunnel vision." This moderation, "seeking what is allowed rather than forbidden", is manifest in three ways. Firstly, along with the Haredi community, the ideology demands adherence to the halakha; however it is not insistent that strictures (chumras) are normative, rather, these are a matter of personal choiceEdah.org (see 3.1 and 4.1 under Modern Orthodox Judaism).

The mode, median, and arithmetic mean are allowed to measure central tendency of interval variables, while measures of statistical dispersion include range and standard deviation. Since one can only divide by differences, one cannot define measures that require some ratios, such as the coefficient of variation. More subtly, while one can define moments about the origin, only central moments are meaningful, since the choice of origin is arbitrary. One can define standardized moments, since ratios of differences are meaningful, but one cannot define the coefficient of variation, since the mean is a moment about the origin, unlike the standard deviation, which is (the square root of) a central moment.

Pico de Orizaba (Citlaltépetl), a stratovolcano on the boundary between the states of Puebla and Veracruz, is the highest mountain peak of Mexico. This article comprises four sortable tables of mountain summits of Mexico that are higher than any other point north or south of their latitude or east or west their longitude in Mexico. The summit of a mountain or hill may be measured in three principal ways: #The topographic elevation of a summit measures the height of the summit above a geodetic sea level.If the elevation or prominence of a summit is calculated as a range of values, the arithmetic mean is shown.

Relative atomic mass (symbol: A) or atomic weight is a dimensionless physical quantity defined as the ratio of the average mass of atoms of a chemical element in a given sample to the atomic mass constant. The atomic mass constant (symbol: m) is defined as being of the mass of a carbon-12 atom. Since both quantities in the ratio are masses, the resulting value is dimensionless; hence the value is said to be relative. For a single given sample, the relative atomic mass of a given element is the weighted arithmetic mean of the masses of the individual atoms (including their isotopes) that are present in the sample.

In the political science of the United States Congress, slurge is the arithmetic mean of retirement slump and sophomore surge. The term was coined by John Alford and David R. Brady in a 1988 academic paper. The slurge is one of the more accurate means of measuring incumbency advantage in congressional elections. If the "retirement slump" is the difference in percentage of the vote share between a retiring incumbent and a new candidate, and the "sophomore surge" is the difference in percentage between a new candidate's first campaign and second campaign as an incumbent, then the slurge, being an average of the two, indicates a higher incumbency advantage when higher.

According to a survey determined in 2016, students in Darmstadt paid an arithmetic mean of 348 euros a month for rent, heat and utilities. With the German average being 323 euros at the time, this made Darmstadt the ninth most expensive city for students in Germany. This value only includes students who live alone, are not married and are pursuing their first degree. In 2016, on national average, approximately 20% lived with their parents, 12% lived in a hall of residence, 1% were lodgers, 30% were sharing a flat with others, 17% were living alone and 21% were sharing a flat with their partner.

On a molecular level, amino acid residues at different locations of a protein may evolve non-independently because they have a direct physicochemical interaction, or indirectly by their interactions with a common substrate or through long-range interactions in the protein structure. Conversely, the folded structure of a protein could potentially be inferred from the distribution of residue interactions. One of the earliest applications of ancestral reconstruction, to predict the three- dimensional structure of a protein through residue contacts, was published by Shindyalov and colleagues. Phylogenies relating 67 different protein families were generated by a distance-based clustering method (unweighted pair group method with arithmetic mean, UPGMA), and ancestral sequences were reconstructed by parsimony.

In mathematics, a mean of circular quantities is a mean which is sometimes better-suited for quantities like angles, daytimes, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on circular quantities. For example, the arithmetic mean of 0° and 360° is 180°, which is misleading because for most purposes 360° is the same thing as 0°.Christopher M. Bishop: Pattern Recognition and Machine Learning (Information Science and Statistics), As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day.

The classification is done through dividing things into large (or called the head) and small (or called the tail) things around the arithmetic mean or average, and then recursively going on for the division process for the large things or the head until the notion of far more small things than large ones is no longer valid, or with more or less similar things left only.Jiang, Bin (2013). "Head/tail breaks: A new classification scheme for data with a heavy-tailed distribution", The Professional Geographer, 65 (3), 482 – 494. Head/tail breaks is not just for classification, but also for visualization of big data by keeping the head, since the head is self-similar to the whole.

642, in : Al-Biruni's methods resembled the modern scientific method, particularly in his emphasis on repeated experimentation. Biruni was concerned with how to conceptualize and prevent both systematic errors and observational biases, such as "errors caused by the use of small instruments and errors made by human observers." He argued that if instruments produce errors because of their imperfections or idiosyncratic qualities, then multiple observations must be taken, analyzed qualitatively, and on this basis, arrive at a "common-sense single value for the constant sought", whether an arithmetic mean or a "reliable estimate." In his scientific method, "universals came out of practical, experimental work" and "theories are formulated after discoveries", as with inductivism.

For example, courage is worthy, for too little of it makes one defenseless. But too much courage can result in foolhardiness in the face of danger. To be clear, Aristotle emphasizes that the moderate state is not an arithmetic mean, but one relative to the situation: sometimes the mean course is to be angry at, say, injustice or mistreatment, at other times anger is wholly inappropriate. Additionally, because people are different, the mean for one person may be bravery, but for another it is recklessness. For Aristotle, the key to finding this balance is to enjoy and recognize the value of developing one’s rational powers, and then using this recognition to determine which actions are appropriate in which circumstances.

The mean absolute difference is twice the L-scale (the second L-moment), while the standard deviation is the square root of the variance about the mean (the second conventional central moment). The differences between L-moments and conventional moments are first seen in comparing the mean absolute difference and the standard deviation (the first L-moment and first conventional moment are both the mean). Both the standard deviation and the mean absolute difference measure dispersion—how spread out are the values of a population or the probabilities of a distribution. The mean absolute difference is not defined in terms of a specific measure of central tendency, whereas the standard deviation is defined in terms of the deviation from the arithmetic mean.

In the first three months of 2016, the average total compensation per employee was $72,931 for the three months, 44% below the same period in 2015. The average total compensation per employee for the full year of 2006 was $622,000. The average compensation in the first three months of 2013 was $135,594. However, these numbers represent the arithmetic mean of total compensation and are highly skewed upwards as several hundred of the top recipients command the majority of the bonus pools, leaving the median that most employees receive well below this number. In Business Week's September 2008 release of the Best Places to Launch a Career, Goldman Sachs was ranked No. 4 out of 119 total companies on the list.

In statistics, the term higher-order statistics (HOS) refers to functions which use the third or higher power of a sample, as opposed to more conventional techniques of lower-order statistics, which use constant, linear, and quadratic terms (zeroth, first, and second powers). The third and higher moments, as used in the skewness and kurtosis, are examples of HOS, whereas the first and second moments, as used in the arithmetic mean (first), and variance (second) are examples of low-order statistics. HOS are particularly used in estimation of shape parameters, such as skewness and kurtosis, as when measuring the deviation of a distribution from the normal distribution. On the other hand, due to the higher powers, HOS are significantly less robust than lower-order statistics.

It is the reciprocal dual of the arithmetic mean for positive inputs: :1/H(1/x_1 \ldots 1/x_n) = A(x_1 \ldots x_n) The harmonic mean is a Schur-concave function, and dominated by the minimum of its arguments, in the sense that for any positive set of arguments, \min(x_1 \ldots x_n) \le H(x_1 \ldots x_n) \le n \min(x_1 \ldots x_n). Thus, the harmonic mean cannot be made arbitrarily large by changing some values to bigger ones (while having at least one value unchanged). The harmonic mean is also concave, which is an even stronger property than Schur-concavity. One has to take care to only use positive numbers though, since the mean fails to be concave if negative values are used.

However, this approach of sequencing after cloning was rarely done for the ITS sequences that make up the reference libraries used for DNA barcode-aided identification, thus potentially giving an underestimate of the existing ITS sequence variation in many samples. The weighted arithmetic mean of the intraspecific (within-species) ITS variability among fungi is 2.51%. This variability, however, can range from 0% for example in Serpula lacrymans (n=93 samples) over 0.19% in Tuber melanosporum (n=179) up to 15.72% in Rhizoctonia solani (n=608), or even 24.75% in Pisolithus tinctorius (n=113). In cases of high intraspecific ITS variability, the application of a threshold of 3% sequence variability - a canonical upper value for intraspecific variation - will therefore lead to a higher estimate of operational taxonomic units (OTUs), i.e.

Each valid solution to the puzzle arranges the blocks in an approximate grid of blocks, with the sides of the blocks all parallel to the sides of the outer cube, and with one block of each width along each axis-parallel line of three blocks. Counting reflections and rotations as being the same solution as each other, the puzzle has 21 combinatorially distinct solutions. The total volume of the pieces, , is less than the volume of the cube that they pack into. If one takes the cube root of both volumes, and divides by three, then the number obtained in this way from the total volume of the pieces is the geometric mean of , , and , while the number obtained in the same way from the volume of the cube is their arithmetic mean.

Typically, supercritical fluids are completely miscible with each other, so that a binary mixture forms a single gaseous phase if the critical point of the mixture is exceeded. However, exceptions are known in systems where one component is much more volatile than the other, which in some cases form two immiscible gas phases at high pressure and temperatures above the component critical points. This behavior has been found for example in the systems N2-NH3, NH3-CH4, SO2-N2 and n-butane-H2O. The critical point of a binary mixture can be estimated as the arithmetic mean of the critical temperatures and pressures of the two components, For greater accuracy, the critical point can be calculated using equations of state, such as the Peng-Robinson, or group-contribution methods.

The fourth chapter of this treatise includes an exposition of the method of least squares, a remarkable testimony to Laplace's command over the processes of analysis. In 1805 Legendre had published the method of least squares, making no attempt to tie it to the theory of probability. In 1809 Gauss had derived the normal distribution from the principle that the arithmetic mean of observations gives the most probable value for the quantity measured; then, turning this argument back upon itself, he showed that, if the errors of observation are normally distributed, the least squares estimates give the most probable values for the coefficients in regression situations. These two works seem to have spurred Laplace to complete work toward a treatise on probability he had contemplated as early as 1783.

The Color Cell Compression algorithm processes an image in eight steps, although one of the steps (step #6) is optional. It is assumed here that the input is a 24 bits/pixel image, as assumed in the original journal article, although other bit depths could be used. ::# For each 8-bit RGB octet triple contained in each 24-bit color value in the input image, the NTSC luminance y is computed using the following formula: y = 0.30 \times red + 0.59 \times green + 0.11 \times blue ::# The image is now subdivided into 4-pixel by 4-pixel blocks, and, the arithmetic mean of the luminance of each pixel in the block is used to select a representative luminance value. ::# Each block of pixels is then divided into two groups.

The granularity-related inconsistency of means (GRIM) test is a simple statistical test used to identify inconsistencies in the analysis of data sets. The test relies on the fact that, given a dataset containing N integer values, the arithmetic mean (commonly called simply the average) is restricted to a few possible values: it must always be expressible as a fraction with an integer numerator and a denominator N. If the reported mean does not fit this description, there must be an error somewhere; the preferred term for such errors is "inconsistencies", to emphasise that their origin is, on first discovery, typically unknown. GRIM inconsistencies can result from inadvertent data-entry or typographical errors or from scientific fraud. The GRIM test is most useful in fields such as psychology where researchers typically use small groups and measurements are often integers.

In common usage, elevations are often cited in height above sea level, although what “sea level” actually means is a more complex issue than might at first be thought: the height of the sea surface at any one place and time is a result of numerous effects, including waves, wind and currents, atmospheric pressure, tides, topography, and even differences in the strength of gravity due to the presence of mountains etc. For the purpose of measuring the height of objects on land, the usual datum used is mean sea level (MSL). This is a tidal datum which is described as the arithmetic mean of the hourly water elevation taken over a specific 19 years cycle. This definition averages out tidal highs and lows (caused by the gravitational effects of the sun and the moon) and short term variations.

OpenCritic collects links to external websites for video game reviews, providing a landing page for the reader. Reviews include both those that are scored (and thus entered into their aggregate score) and unscored reviews, including reviews that come from popular YouTube reviewers. Reviews are summarized in three ways: an arithmetic mean of all scored reviews, a percent recommended which represents the percent of all reviews that recommend the game (including unscored reviews), and a percentile rank which indicates where a game's aggregate score falls in the distribution of all games on the site. A user of the site is able to mark any of these review sources as trusted publications which will be reflected in how the reviews are presented to the user and tailor the content to them, as well as generating a personalized aggregate score for that user.

In number theory, a strong prime is a prime number that is greater than the arithmetic mean of the nearest prime above and below (in other words, it's closer to the following than to the preceding prime). Or to put it algebraically, given a prime number p, where n is its index in the ordered set of prime numbers, . For example, 17 is the seventh prime: the sixth and eighth primes, 13 and 19, add up to 32, and half that is 16; 17 is greater than 16, and so 17 is a strong prime. The first few strong primes are :11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439, 457, 461, 479, 487, 499 .

In a twin prime pair (p, p + 2) with p > 5, p is always a strong prime, since 3 must divide p − 2, which cannot be prime. It is possible for a prime to be a strong prime both in the cryptographic sense and the number theoretic sense. For the sake of illustration, 439351292910452432574786963588089477522344331 is a strong prime in the number theoretic sense because the arithmetic mean of its two neighboring primes is 62 less. Without the aid of a computer, this number would be a strong prime in the cryptographic sense because 439351292910452432574786963588089477522344330 has the large prime factor 1747822896920092227343 (and in turn the number one less than that has the large prime factor 1683837087591611009), 439351292910452432574786963588089477522344332 has the large prime factor 864608136454559457049 (and in turn the number one less than that has the large prime factor 105646155480762397).

In computer science, specifically information retrieval and machine learning, the harmonic mean of the precision (true positives per predicted positive) and the recall (true positives per real positive) is often used as an aggregated performance score for the evaluation of algorithms and systems: the F-score (or F-measure). This is used in information retrieval because only the positive class is of relevance, while number of negatives, in general, is large and unknown. It is thus a trade-off as to whether the correct positive predictions should be measured in relation to the number of predicted positives or the number of real positives, so it is measured versus a putative number of positives that is an arithmetic mean of the two possible denominators. A consequence arises from basic algebra in problems where people or systems work together.

The Dow Jones Industrial Average (DJIA), Dow Jones, or simply the Dow (), is a stock market index that measures the stock performance of 30 large companies listed on stock exchanges in the United States. Although it is one of the most commonly followed equity indices, many consider the Dow to be an inadequate representation of the overall U.S. stock market compared to broader market indices such as the S&P; 500 Index or Russell 3000 because it only includes 30 large cap companies, is not weighted by market capitalization, and does not use a weighted arithmetic mean. The value of the index is the sum of the stock prices of the companies included in the index, divided by a factor which is currently () approximately 0.152. The factor is changed whenever a constituent company undergoes a stock split so that the value of the index is unaffected by the stock split.

Demonstration, with Cuisenaire rods, of the perfection of the number 6 For any integer M, as Ore observed, the product of the harmonic mean and arithmetic mean of its divisors equals M itself, as can be seen from the definitions. Therefore, M is harmonic, with harmonic mean of divisors k, if and only if the average of its divisors is the product of M with a unit fraction 1/k. Ore showed that every perfect number is harmonic. To see this, observe that the sum of the divisors of a perfect number M is exactly 2M; therefore, the average of the divisors is M(2/τ(M)), where τ(M) denotes the number of divisors of M. For any M, τ(M) is odd if and only if M is a square number, for otherwise each divisor d of M can be paired with a different divisor M/d.

Stigler, 1975 In two important papers in 1810 and 1811, Laplace first developed the characteristic function as a tool for large-sample theory and proved the first general central limit theorem. Then in a supplement to his 1810 paper written after he had seen Gauss's work, he showed that the central limit theorem provided a Bayesian justification for least squares: if one were combining observations, each one of which was itself the mean of a large number of independent observations, then the least squares estimates would not only maximise the likelihood function, considered as a posterior distribution, but also minimise the expected posterior error, all this without any assumption as to the error distribution or a circular appeal to the principle of the arithmetic mean. In 1811 Laplace took a different non-Bayesian tack. Considering a linear regression problem, he restricted his attention to linear unbiased estimators of the linear coefficients.

For a railway line as a whole (in one direction), one would use the average linear density along the whole line, where most points on the line have no trains on them and thus have zero density there. One would also need to use the average velocity which turns out to be the weighted harmonic mean where the weights are the lengths of the various segments of the train run (with the speed approximately constant on each segment). Another way to find the average velocity is to simply find the weighted arithmetic mean where the weights is this case are the times on each segment. For a segment of a train run where the speed is very slow, the train thus spends a long time traversing this segment and thus the weight for this segment is quite high (since it is weighted by time).

The second advantage is that the IVhet model maintains the inverse variance weights of individual studies, unlike the RE model which gives small studies more weight (and therefore larger studies less) with increasing heterogeneity. When heterogeneity becomes large, the individual study weights under the RE model become equal and thus the RE model returns an arithmetic mean rather than a weighted average. This side-effect of the RE model does not occur with the IVhet model which thus differs from the RE model estimate in two perspectives: Pooled estimates will favor larger trials (as opposed to penalizing larger trials in the RE model) and will have a confidence interval that remains within the nominal coverage under uncertainty (heterogeneity). Doi & Barendregt suggest that while the RE model provides an alternative method of pooling the study data, their simulation results demonstrate that using a more specified probability model with untenable assumptions, as with the RE model, does not necessarily provide better results.

Theaetetus classified the known irrational numbers into three types, based on analogies to the geometric mean, arithmetic mean, and harmonic mean, and this classification was then greatly extended by Eudoxus of Cnidus; Knorr speculates that this extension stemmed out of Eudoxus' studies of the golden section.Review of The Evolution of the Euclidean Elements by Sabetai Unguru (1977), Isis 68: 314–316, .. Although published as a regular paper, this is an extended review of The Evolution of the Euclidean Elements, for which Unguru's review in Isis is a precis. :Along with this history of irrational numbers, Knorr reaches several conclusions about the history of Euclid's Elements and of other related mathematical documents; in particular, he ascribes the origin of the material in Books 1, 3, and 6 of the Elements to the time of Hippocrates of Chios, and of the material in books 2, 4, 10, and 13 to the later period of Theodorus, Theaetetus, and Eudoxos.

Likewise with CATV, although many broadcast TV installations and CATV headends use 300 Ω folded dipole antennas to receive off-the-air signals, 75 Ω coax makes a convenient 4:1 balun transformer for these as well as possessing low attenuation. The arithmetic mean between 30 Ω and 77 Ω is 53.5 Ω; the geometric mean is 48 Ω. The selection of 50 Ω as a compromise between power- handling capability and attenuation is in general cited as the reason for the number. 50 Ω also works out tolerably well because it corresponds approximately to the feedpoint impedance of a half-wave dipole, mounted approximately a half-wave above "normal" ground (ideally 73 Ω, but reduced for low-hanging horizontal wires). RG-62 is a 93 Ω coaxial cable originally used in mainframe computer networks in the 1970s and early 1980s (it was the cable used to connect IBM 3270 terminals to IBM 3274/3174 terminal cluster controllers).

Sea of Galilee Historically, Israel had about 1,780 million cubic meters of conventional freshwater and brackish water resources at its disposal in an average year. More than half of these resources consisted of inflows from the Golan Heights (275 million), Lebanon (310 million) and the West Bank (345 million). 92 percent of the conventional water resources were considered economically exploitable, the rest being excess floodwaters. The usable amount consisted of about 1.1 billion cubic meters of groundwater and springs and 0.6 billion from surface water. About 80% of the water resources are located in the North of the country and only 20% in the South. However, average rainfall has declined, possibly as a result of climate change, so that conventional water resources are now estimated at less than 1,336 million cubic meters in an average year (arithmetic mean) over the period 1975–2011. The median was even lower at 1,202 million cubic meters per year. The security of these resources is impacted by riparian conflicts.