4.3:
Standard Deviation
Consider the goals scored by two teams with identical mean scores. To find out the better performing team, one can use standard deviation, another measure of variation, and compare the spread of all values from the mean.
The standard deviation formula depends on whether the data was drawn from a sample or from an entire population. If it's sample data, the standard deviation is denoted by s. For population data, it is represented by sigma. Note that the denominator is population size N for population data, instead of n minus 1 as in sample data.
If one plots the data from the example, the graph on the left shows more spread, hence, greater standard deviation. In contrast, the one on the right shows less spread and a smaller standard deviation. So, Team 2 is more consistent than Team 1.
Standard deviation values are usually positive and are zero only if all the dataset values are equal. The standard deviation and the dataset share the same units: here, it is the number of goals.
4.3:
Standard Deviation
The most commonly used measure of variation is the standard deviation. It is a numerical value measuring how far data values are from their mean. The standard deviation value is small when the data are concentrated close to the mean, exhibiting slight variation or spread. The standard deviation value is never negative, it is either positive or zero. The standard deviation is larger when the data values are more spread out from the mean, which means the data values are exhibiting more variation.
Consider the waiting time for customers at the checkout at two supermarkets, X and Y. The average waiting time at both supermarkets is five minutes. At supermarket X, the standard deviation for the wait time is two minutes; at supermarket Y, the standard deviation for the waiting time is four minutes. As supermarket Y has a higher standard deviation, there is more variation in the wait time at supermarket Y. Overall, wait times at supermarket Y are more spread out or show more deviations from the average. In contrast, wait times at supermarket X are more concentrated near the average.
The lowercase letter s signifies the sample standard deviation, while the Greek letter σ (sigma, lowercase) represents the population standard deviation.
The sample standard deviation is given by the formula
The population standard deviation is given by the following formula:
This text is adapted from 2.7 Measures of the Spread of the Data – Introductory Statistics | OpenStax