Mean (arithmetic)

The arithmetic average of a set of data values (also known as the sample mean where the data are a sample from a larger population). Note that if the set {f_i} are regarded as weights rather than frequencies the result is known as the weighted mean. Other mean values include the geometric and harmonic mean. The population mean is often denoted by the symbol μ. In many instances the sample mean is the best (unbiased) estimate of the population mean and is sometimes denoted by μ with a ^ symbol above it) or as a variable such as x with a bar above it.

Mean (harmonic)

The harmonic mean, H, is the mean of the reciprocals of the data values, which is then adjusted by taking the reciprocal of the result. The harmonic mean is less than or equal to the geometric mean, which is less than or equal to the arithmetic mean

Mean (geometric)

The geometric mean, G, is the mean defined by taking the products of the data values and then adjusting the value by taking the n^th root of the result. The geometric mean is greater than or equal to the harmonic mean and is less than or equal to the arithmetic mean

, hence

Mean (power)

The general (limit) expression for mean values. Values for p give the following means: p=1 arithmetic; p=2 root mean square; p=‑1 harmonic. Limit values for p (i.e. as p tends to these values) give the following means: p=0 geometric; p=‑∞ minimum; p=∞ maximum

Trim-mean, TM, t, Olympic mean

The mean value computed with a specified percentage (proportion), t/2, of values removed from each tail to eliminate the highest and lowest outliers and extreme values. For small samples a specific number of observations (e.g. 1) rather than a percentage, may be ignored. In general an equal number, k, of high and low values should be removed and the number of observations summed should equal n(1‑t) expressed as an integer. This variant is sometimes described as the Olympic mean, as is used in scoring Olympic gymnastics for example

, t[0,1]

Mode

The most common or frequently occurring value in a set. Where a set has one dominant value or range of values it is said to be unimodal; if there are several commonly occurring values or ranges it is described as multi-modal. Note that arithmetic mean‑mode≈3 (arithmetic mean‑median) for many unimodal distributions

Median, Med

The middle value in an ordered set of data if the set contains an odd number of values, or the average of the two middle values if the set contains an even number of values. For a continuous distribution the median is the 50% point (0.5) obtained from the cumulative distribution of the values or function

Med{x_i}=X_(n+1)/2 ; n odd

Med{x_i}=iX_n/2+X_n/2+1)/2; n even

Mid-range, MR

The middle value of the Range

MR{x_i}=Range/2

Root mean square (RMS)

The root of the mean of squared data values. Squaring removes negative values