Standard deviation
Standard deviation
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.
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.
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.
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.
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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