Standard deviation

Standard deviation 

In statistics, the standard deviation (SD, also represented by the Greek letter sigma σ or the Latin letter s) is a measure that is used to quantify the amount of variation or dispersion of a set of data values.

Video: How to Calculate Standard Deviation

A low standard deviation indicates that the data points tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the data points are spread out over a wider range of values.

The standard deviation of a random variable, statistical population, data set, or probability distribution is the square root of its variance. It is algebraically simpler, though in practice less robust, than the average absolute deviation. A useful property of the standard deviation is that, unlike the variance, it is expressed in the same units as the data.

Basic examples

Sample standard deviation of metabolic rate of Northern Fulmars

Logan gives the following example. Furness and Bryant measured the resting metabolic rate for 8 male and 6 female breeding Northern fulmars. The table shows the Furness data set.

Furness data set on metabolic rates of Northern fulmars
Sex Metabolic rate Sex Metabolic rate
Male 525.8 Female 727.7
Male 605.7 Female 1086.5
Male 843.3 Female 1091.0
Male 1195.5 Female 1361.3
Male 1945.6 Female 1490.5
Male 2135.6 Female 1956.1
Male 2308.7    
Male 2950.0    

The graph shows the metabolic rate for males and females. By visual inspection, it appears that the variability of the metabolic rate is greater for males than for females.

Graph of metabolic rates for northern fulmars

The graph shows the metabolic rate for males and females. By visual inspection, it appears that the variability of the metabolic rate is greater for males than for females.

Graph of metabolic rates for northern fulmars

The sample standard deviation of the metabolic rate for the female fulmars is calculated as follows. The formula for the sample standard deviation is

 

where are the observed values of the sample items,  is the mean value of these observations, and N is the number of observations in the sample.

In the sample standard deviation formula, for this example, the numerator is the sum of the squared deviation of each individual animal's metabolic rate from the mean metabolic rate. The table below shows the calculation of this sum of squared deviations for the female fulmars. For females, the sum of squared deviations is 886047.09, as shown in the table.

Sum of squares calculation for female fulmars
Animal Sex Metabolic rate Mean Difference from mean Squared difference from mean
1 Female 727.7 1285.5 -557.8 311140.84
2 Female 1086.5 1285.5 -199.0 39601.00
3 Female 1091.0 1285.5 -194.5 37830.25
4 Female 1361.3 1285.5 75.8 5745.64
5 Female 1490.5 1285.5 205.0 42025.00
6 Female 1956.1 1285.5 670.6 .......

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