As well as statistics, means are often used in geometry and analysis; a wide range of means have been developed for these purposes, which are not much used in statistics. See the section below for a list of means.
Sample mean is often used as an Estimator of the Central tendency such as the population mean. However, other estimators are also used. For example, the Median is a more Robust estimator of the central tendency than the sample mean.
For a real-valued Random variable X, the mean is the expectation of X. If the expectation does not exist, then the random variable has no mean.
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.
An alternative measure of dispersion is the mean deviation, equivalent to the average Absolute deviation from the mean. It is less sensitive to outliers, but less tractable when combining data sets.
Note that not every Probability distribution has a defined mean or Variance - see the Cauchy distribution for an example.
The following is a summary of some of the multiple methods for calculating the mean of a set of n numbers. See the Table of mathematical symbols for explanations of the symbols used.
| –– x | = | 1 –– n | n Σ i = 1 | {x i } |
Do not confuse the mean 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 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.
| –– x | = | n √ | ––––––––––––––––––––––––– Π i = 1 -> n {x i } |
| –– x | = | n ––––––––––––––––––––––– Σ i = 1 -> n 1/x i |
| –– x | (m) = | m √ | –––––––––––––––––––––––––––––– 1/n Σ i = 1 -> n {x i m } |
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)
This could be generalised further as
| –– x | = f -1 | ( | 1 –– n | n Σ i = 1 | {f (x i )}) |
and again a suitable choice of an invertible f(x) will give the arithmetic mean with f(x)=x, the geometric mean with f(x)=log(x), and the harmonic mean with f(x)=1/x.
| –– x | = | Σ i = 1 -> n {w i • x i } ––––––––––––––––––––––––––––––––––– Σ i = 1 -> n {w i } |
The weights
| –– x | = | 2 –– n | 3 n / 4 Σ i = ( n / 4 ) + 1 | {x i } |
| –– f | = | 1 ––––– b-a | ∫ | b a | f (x)dx. |
| –– f | = | 1 –––––––––––– { | ∫ | U | f |