Concepts

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#### Description

The arithmetic mean is what is commonly called the average: When the word "mean" is used without a modifier, it can be assumed that it refers to the arithmetic mean. The mean is the sum of all the scores divided by the number of scores. The formula in summation notation is: μ = ΣX/N where μ is the population mean and N is the number of scores. If the scores are from a sample, then the symbol M refers to the mean and N refers to the sample size. The formula for M is the same as the formula for μ.

The mean is a good measure of central tendency for roughly symmetric distributions but can be misleading in skewed distributions since it can be greatly influenced by extreme scores. Therefore, other statistics such as the median may be more informative for distributions such as reaction time or family income that are frequently very skewed.

The sum of squared deviations of scores from their mean is lower than their squared deviations from any other number.

For normal distributions, the mean is the most efficient and therefore the least subject to sample fluctuations of all measures of central tendency.

The formal definition of the arithmetic mean is µ = E[X] where μ is the population mean of the variable X and E[X] is the expected value of X.

#### Discussion

For a data set, the mean is just the sum of all the observations divided by the number of observations. Once we have chosen this method of describing the communality of a data set, we usually use the standard deviation to describe how the observations differ. The standard deviation is the square root of the average of squared deviations from the mean.

The mean is the unique value about which the sum of squared deviations is a minimum. If you calculate the sum of squared deviations from any other measure of central tendency, it will be larger than for the mean. This explains why the standard deviation and the mean are usually cited together in statistical reports.

The mean may often be confused with the median or mode. The mean is the arithmetic average of a set of values, or distribution; however, for skewed distributions, the mean is not necessarily the same as the middle value (median), or most likely (mode). For example, mean income is skewed upwards by a small number of people with very large incomes, so that the majority have an income lower than the mean. By contrast, the median income is the level at which half the population is below and half is above. The mode income is the most likely income, and favors the larger number of people with lower incomes. The median or mode are often more intuitive measures of such data.

That said, many skewed distributions are best described by their mean - such as the Exponential and Poisson distributions.

#### An amusing example

Most people have an above average number of legs - think about it. The mean number of legs is going to be less than 2 (because there are people with one leg and people with no legs). The mean is probably 1.999997 or somesuch figure. So since most people have two legs, they have an above average number!

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