Arithmetic mean

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In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. If one particular number occurs more times than others in the list, it is called a mode. The arithmetic mean is what students are taught very early to call the "average". If the list is a statistical population, then the mean of that population is called a population mean. If the list is a statistical sample, we call the resulting statistic a sample mean.

When the mean is not an accurate estimate of the median, the list of numbers, or frequency distribution, is said to be skewed.

1 Introduction
- 1.1 Examples
2 Problems with the mean
3 See also
4 External links

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Introduction

The symbol μ (Greek: mu) is used to denote the arithmetic mean of a population.

If we denote a list of data by X = (x₁, x₂, ..., x_n), then the sample mean is typically denoted with a horizontal bar over the variable (x̅, generally enunciated "x bar").

In practice, the difference between μ and x̅ is that μ is typically unobservable because one observes only a sample rather than the whole population, and if the sample is drawn randomly, then one may treat x̅, but not μ, as a random variable, attributing a probability distribution to it (the sampling distribution of the mean).

Both are computed in the same way:

<math>\bar{x} = \mathrm{mean} = \frac1n\sum_{i=1}^n x_i = \frac1n (x_1+\cdots+x_n).</math>

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.

Note that several other "means" have been defined, including the generalized mean, the generalized f-mean, the harmonic mean, the arithmetic-geometric mean, and various weighted means.

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Examples

If you have 3 numbers then add them and divide them by 3. <math>\frac{x_1 + x_2 + x_3}{3}</math>
If you have 4 numbers add them and divide by 4. <math>\frac{x_1 + x_2 + x_3 + x_4}{4}</math>

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Problems with the mean

While the mean is often used to report central tendency, it may not be appropriate for describing skewed distributions, because it is easily misinterpreted. The arithmetic mean is greatly influenced by outliers. These distortions can occur when the mean is different from the median. When this happens the median may be a better description of central tendency.

A classic example is average income. The arithmetic mean may be misinterpreted 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. For instance, reporting the "average" net worth in Redmond, Washington as the arithmetic mean of all annual net worths would yield a surprisingly high number because of Bill Gates. 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 certain situations, the arithmetic mean is the wrong measure of central tendency altogether. For example, if a stock fell 10% in the first year, and rose 30% in the second year, then it would be incorrect to report its "average" increase per year over this two year period as the arithmetic mean (−10% + 30%)/2 = 10%; the correct average in this case is the geometric mean which yields an average increase per year of only 8.1%. The reason for this is that each of those percents have different starting points. If the stock starts at $30 and falls 10%, it is now at $27. If the stock then rises 30%, it is now $35.1. The arithmetic mean of those rises is 10%, but since the stock rose by $5.1 in 2 years, an average of 8.1% would result in the final $35.1 figure [$30(1-10%)(1+30%) = $30(1+8.1%)(1+8.1%) = $35.1]. If one used the arithmetic mean 10% in the same way, you would not get the actual increase [$30(1+10%)(1+10%) = $36.3].

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