Arithmetic mean
The arithmetic mean is defined as the sum of all the members of a set of data divided by the number of items in the set. If the set used is a population, then we speak of the "population mean". When the set is a sample, we call the resulting statistic a "sample mean". The arithmetic mean is often used to express the "average" of a set of numbers.
We denote the set of data by X = {x1, x2, ..., xn}. The symbol µ (Greek: mu) is used to denote the arithmetic mean of a population. We use the name of the variable, X, with a horizontal bar over it as the symbol ("X bar") for a sample mean. Both are computed in the same way:
- Arithmetic mean = ( x1 + x2 + ... + xn ) / n
The arithmetic mean is greatly influenced by outliers. For instance, reporting the "average" annual income in the U.S. as the arithmetic mean of all annual incomes would yield a surprisingly high number because of Bill Gates. It is therefore often prudent to accompany the arithmetic mean by the median which is much less affected by outliers.
In certain situations, the arithmetic mean is the wrong concept of "average" altogether. For example, if a stock rose 10% in the first year, 30% in the second year and fell 10% in the third year, then it would be incorrect to report its "average" increase per year over this three year period as the arithmetic mean (10% + 30% + (-10%))/3 = 10%; the correct average in this case is the geometric mean which yields an average increase per year of only 8.8%.
If X is a random variable, then the expected value of X can be seen as the long-term arithmetic mean that occurs on repeated measurements of X. This is the content of the law of large numbers. As a result, the sample mean is used to estimate unknown expected values.
See also: generalized mean, geometric mean, harmonic mean, arithmetic geometric mean, weighted mean, variance, central tendency
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