What is the standard deviation formula?

The standard deviation formula is the square root of the variance, which measures data spread around the mean.

To calculate the standard deviation, you first need to find the mean (average) of your data set. Add up all the numbers in your data set and then divide by the number of data points. Next, subtract the mean from each data point to find the deviation of each point from the mean. Square each of these deviations to make them positive and then add all these squared deviations together.

For a population, divide this sum by the number of data points (N) to find the variance. For a sample, divide by the number of data points minus one (N-1) to get the sample variance. Finally, take the square root of the variance to get the standard deviation.

In formula terms, for a population, the standard deviation (σ) is:
\[ \sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}} \]
where \( x_i \) represents each data point, \( \mu \) is the mean, and \( N \) is the number of data points.

For a sample, the standard deviation (s) is:
\[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{N-1}} \]
where \( \bar{x} \) is the sample mean and \( N-1 \) accounts for the sample size.

Understanding standard deviation helps you analyse how spread out your data is. A small standard deviation means the data points are close to the mean, while a large standard deviation indicates they are spread out over a wider range.