Average

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In mathematics, there are numerous methods for calculating the average or central tendency of a set of n numbers. The most common method, and the one generally referred to simply as the average, is the arithmetic mean. Please see the table of mathematical symbols for explanations of the symbols used.

Table of contents

1 Arithmetic Mean
2 Median
3 Mode
4 Geometric Mean
5 Harmonic Mean
6 Generalized Mean
7 Weighted Mean
8 Truncated mean
9 Interquartile mean
10 Weblink
11 Further reading

Arithmetic Mean

The arithmetic mean is the "standard" average, often simply called the "mean". It is used for many purposes and may be abused by using it to describe skewed distributions, with highly misleading results. A classic example is average income. The arithmetic mean may be used to imply that most people's incomes are higher than is in fact the case. When presented with an "average" one may be led to believe that most people's incomes are near this number. This "average" (arithmetic mean) income is higher than most people's incomes, because high income outliers skew the result higher (in contrast, the median income "resists" such skew). However, this "average" says nothing about the number of people near the median income (nor does it say anything about the income that most people are near). Nevertheless, because one might carelessly relate "average" and "most people" one might incorrectly assume that most people's incomes would be higher (nearer this inflated "average") than they are. Consider the scores {1, 2, 2, 2, 3, 9}. The arithmetic mean is 3.17, but five out of six scores are below this!

Median

The median is the value below which 50% of the scores fall, or the middle score. Where there is an even number of scores, the median is the mean of the two centermost scores. It is primarily used for skewed distributions, which it represents more accurately than the arithmetic mean. (Consider {1, 2, 2, 2, 3, 9} again: the median is 2, in this case, a much better indication of central tendency than the arithmetic mean of 3.16.)

Mode

The Mode is simply the most frequent score. It is most useful where the scores are not numeric: for example, while the mode {1, 2, 2, 2, 3, 9} is 2, the mode of {apple, apple, banana, orange, orange, orange, peach} is orange.

Geometric Mean

The geometric mean is an average which is useful for sets of numbers which are interpreted according to their product and not their sum (as is the case with the arithmetic mean). For example rates of growth.

Harmonic Mean

The harmonic mean is an average which is useful for sets of numbers which are defined in relation to some unit, for example speed (distance per unit of time).

Generalized Mean

The generalized mean is an abstraction of the Arithmetic, Geometric and Harmonic Means.

By choosing the appropriate value for the parameter m we can get the arithmetic mean (m = 1), the geometric mean (m -> 0) or the harmonic mean (m = -1). A further abstraction would give

for a suitable invertible function f(x).

Weighted Mean

The weighted mean is used, if one wants to combine average values from samples of the same population with different sample sizes:

The weights represent the bounds of the partial sample. In other applications they represent a measure for the reliability of the influence upon the mean by respective values.

Truncated mean

Sometimes a set of numbers (the data) might be contaminated by inaccurate outliers, i.e. values which are much too low or much too high. In this case one can use a truncated mean. 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 total number of values.

Interquartile mean

The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values.

assuming the values have been ordered.

Weblink

Calculations and comparisons between arithmetic and geometric mean between two values