Mean Mean From Wikipedia, the free encyclopedia. In statistics, mean has two related meanings: the average in ordinary English, which is more correctly called the arithmetic mean, to distinguish it from geometric mean or harmonic mean. The average is also called sample mean. the expected value of a random variable, which is also called the population mean. 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 'Other Means' 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. The mean value of a function, <math>f(x)<math>, on an interval, <math>a 0) or the harmonic mean (m = -1) This could be generalised further as <math> ar{x} = f^{-1}left({frac{1}{n}sum_{i=1}^n{f(x_i)}} ight) <math> 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. Weighted mean The weighted mean is used, if one wants to combine average values from samples of the same population with different sample sizes: <math> ar{x} = frac{sum_{i=1}^n{w_i cdot x_i}}{sum_{i=1}^n {w_i}} <math> The weights <math>w_i<math> represent the bounds of the partial sample. In other applications they represent a measure for the reliability of the influence upon the mean by respective values. Truncated mean Sometimes a set of numbers (the data) might be contaminated by inaccurate outliers, i.e. values which are much too low or much too high. In this case one can use a truncated mean. It involves discarding given parts of the data at the top or the bottom end, typically an equal amount at each end, and then taking the arithmetic mean of the remaining data. The number of values removed is indicated as a percentage of total number of values. Interquartile mean The interquartile mean is a specific example of a truncated mean. It is simply the arithmetic mean after removing the lowest and the highest quarter of values. <math> ar{x} = {2 over n} sum_{i=(n/4)+1}^{3n/4}{x_i} <math> assuming the values have been ordered. Other means Arithmetic mean Arithmetic-geometric mean Arithmetic-harmonic mean Cesàro mean Chisini mean Generalized mean (also known as power mean) Geometric mean Geometric-harmonic mean Harmonic mean Heronian mean Hölder mean Identric mean Lehmer mean Quadratic mean root mean square Stolarsky mean weighted mean weighted geometric mean weighted harmonic mean Rényi's entropy (a generalized f-mean) See also Central tendency Descriptive statistics Kurtosis Median Mode (statistics) Summary statistics Quelle "Mean" : Wikipedia.org Main page for "Mean"