Sample Standard Deviation - Basic Examples

Basic Examples

For a finite set of numbers, the standard deviation is found by taking the square root of the average of the squared differences of the values from their average value. For example, consider a population consisting of the following eight values:

These eight data points have the mean (average) of 5:

To calculate the population standard deviation, first compute the difference of each data point from the mean, and square the result of each:

Next, compute the average of these values, and take the square root:

This quantity is the population standard deviation, and is equal to the square root of the variance. The formula is valid only if the eight values we began with form the complete population. If the values instead were a random sample drawn from some larger parent population, then we would have divided by 7 (which is n−1) instead of 8 (which is n) in the denominator of the last formula, and then the quantity thus obtained would be called the sample standard deviation. Dividing by n−1 gives a better estimate of the population standard deviation than dividing by n.

As a slightly more complicated real-life example, the average height for adult men in the United States is about 70 in, with a standard deviation of around 3 in. This means that most men (about 68 percent, assuming a normal distribution) have a height within 3 in of the mean (67–73 in) – one standard deviation – and almost all men (about 95%) have a height within 6 in of the mean (64–76 in) – two standard deviations. If the standard deviation were zero, then all men would be exactly 70 in tall. If the standard deviation were 20 in, then men would have much more variable heights, with a typical range of about 50–90 in. Three standard deviations account for 99.7 percent of the sample population being studied, assuming the distribution is normal (bell-shaped).