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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.



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


 * $$s = \sqrt{\frac{\sum_{i=1}^N (x_i - \overline{x})^2}{N-1} }.$$

where $$\scriptstyle\{x_1,\,x_2,\,\ldots,\,x_N\}$$ are the observed values of the sample items, $$\scriptstyle\overline{x}$$ 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.



The denominator in the sample standard deviation formula is N – 1, where N is the number of animals. In this example, there are N = 6 females, so the denominator is 6 – 1 = 5. The sample standard deviation for the female fulmars is therefore


 * $$s = \sqrt{\frac{\sum_{i=1}^N (x_i - \overline{x})^2}{N-1} } = \sqrt{\frac{886047.09}{5}} = 420.96.$$

For the male fulmars, a similar calculation gives a sample standard deviation of 894.37, approximately twice as large as the standard deviation for the females. The graph shows the metabolic rate data, the means (red dots), and the standard deviations (red lines) for females and males.



Use of the sample standard deviation implies that these 14 fulmars are a sample from a larger population of fulmars. If these 14 fulmars comprised the entire population (perhaps the last 14 surviving fulmars), then instead of the sample standard deviation, the calculation would use the population standard deviation. In the population standard deviation formula, the denominator is N instead of N-1. It is rare that measurements can be taken for an entire population, so, by default, statistical software packages calculate the sample standard deviation. Similarly, journal articles report the sample standard deviation unless otherwise specified.

Population standard deviation of grades of eight students
Suppose that the entire population of interest was eight students in a particular class. For a finite set of numbers, the population standard deviation is found by taking the square root of the average of the squared deviations of the values from their average value. The marks of a class of eight students (that is, a statistical population) are the following eight values:

Standard deviation of average height for adult men
If the population of interest is approximately normally distributed, the standard deviation provides information on the proportion of observations above or below certain values.