Average

In mathematics, there are numerous methods for calculating the average or central tendency of a list 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.

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 modal 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!

In general, given n numbers, $$x_1, x_2, \dots, x_n,$$ their arithmetic mean is computed by the formula
 * $$ \bar{x} = {1 \over n} \sum_{i=1}^n{x_i}. $$

Which means: Divide the sum of the numbers in the list by the count of numbers in the list.

For example:
 * If you have 3 numbers then add them and divide them by 3. $$[(x_1 + x_2 + x_3)/3]$$
 * If you have 4 numbers add them and divide by 4. $$[(x_1 + x_2 + x_3 + x_4)/4]$$

Median
The median is the value below which 50% of the scores fall, or the middle score ( 1/2 of the population will have values less than the median and 1/2 of the population will have values greater than the median ). 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. Also note that 1/2 of the scores, namely{1,2,2}, have values less than the median and the other half, namely {2,3,9}, have values greater than the median.)

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.

Other averages
The geometric mean, harmonic mean, generalized mean, weighted mean, truncated mean, and interquartile mean are described in their own articles and in the Mean article.

Other more sophisticated averages, usually more representative of the whole dataset are: trimean, trimedian, and normalised mean, to name a few. One can create one's own average metric using the generic formula y = f -1((f(x1)+f(x2)+...+f(xn))/n) where f is any invertible function. For example, expmean (exponential mean) is a mean using the function f(x) = e^x and due to its nature, it is biased towards the higher values.

The only significant reason why the arithmetic mean (classical average) is generally used in scientific papers is that there are various (statistical) tests which can be applied to test the statistical significance of the results, as well as the correlations that are explored through these metrics.

Etymology
Average is derived from Romance avaria, a 'tax'; this is in turn connected with the Arabic or Turkish avaria or avania of the same meaning. The OED considers it uncertain which is older. If the Italian and related forms are not from the Arabic, they may be from Italian avere, 'goods'.