Standard deviation

ERDC TN-DOER-C15

July 2000

Mean The mean is a measure of central tendency of the data, that is, the central value about which

the data are grouped. There are several different types of means, including the arithmetic mean, the

harmonic mean, and the geometric mean. The arithmetic mean x is most commonly used, but the

harmonic mean H and geometric mean G are important when the data set includes a few very high

or low values that may influence the arithmetic mean (McBean and Rovers 1998). The arithmetic

mean is defined as follows, and has the same units as the individual data observations (e.g., mg/kg):

∑ xi

x = i -1

(I1)

where

xi = individual observations (values)

n = the number of observations

The relationship between the different means is as follows:

H≤G≤ x

(I2)

Median The median is also a measure of central tendency, and may be more reflective of the

center of gravity of a skewed distribution than the mean (Figure I2). The median is found by ranking

the n measurements from smallest to largest. If n is odd, the median is the value with rank (n + 1)/2;

if n is even, the median is the value halfway between the measurements with rank n/2 and n/2 + 1,

that is, the average of the two middle values (Mendenhall and Beaver 1994).

Mode The mode is the most frequently occurring value of the measured variable.

Standard deviation The standard deviation s is a measure of the scatter of the data (how closely

the data are grouped around the central value, the mean). A small standard deviation indicates

closely grouped data with little variability. A large standard deviation indicates data that are widely

variable. The number of standard deviations a value is away from the mean x is an indication of its

probability of occurrence. For example, by the empirical rule, for a data distribution that is

approximately bell shaped (normally distributed, or nearly so), 68 percent of the values will be

within one standard deviation of the mean (x plus n minus s), 95 percent of the values will be within

two standard deviations of the mean, and most or all of the values will be within three standard

deviations of the mean (Figure I3) (Mendenhall and Beaver 1994; McBean and Rovers 1998). A

value falling more than three standard deviations from the mean has a very small probability of

occurrence. Values within one to two standard deviations would be reasonably expected to occur.

This is the predictive value of statistical application to sampling; it allows the data to be extended

to speculate on values expected in unsampled areas. Data with low variability increase the level of

confidence in determining how likely a certain value is to occur, or threshold to be exceeded.