Standard Deviation and Variance
Measure spread by how far the values sit from the mean on average, using squared deviations so that distances on both sides count. Compare data sets by mean and standard deviation together.
What a learner can do afterwards
- Calculate the standard deviation of a small data set by hand
- Compute it from Σx and Σx² for a frequency table
- Compare two sets with the same mean but different spreads
1 · Read
Variance and standard deviation measure spread around the mean. For 2, 4, 4, 4, 5, 5, 7, 9, the values add to 40 and there are 8 of them, so the mean is 5. The gaps from 5 are negative 3, negative 1, negative 1, negative 1, 0, 0, 2 and 4. Square each gap to get 9, 1, 1, 1, 0, 0, 4 and 16, which add to 32. Divide by 8 for the population variance 4, and take the root for the standard deviation 2, back in the original units.
With big tables, totals do the same job faster. For those 8 values the total is 40 and the total of squares is 232. The mean of squares is 232 over 8, which is 29, and the mean squared is 25. Variance is the mean of squares minus the mean squared: 29 minus 25 is 4.
Compare sets by mean and spread together. Sets A with 4, 5, 6 and B with 1, 5, 9 share the mean 5. A strays at most 1 from the mean while B strays up to 4, so B has the larger standard deviation. Stretching values out pushes the number up.
Standard deviation is the root of the variance, so it always carries the original units. Check the means match before you compare spreads, or the wider number may just follow a bigger mean.
Square the gaps from the mean, average them for variance, and root it for standard deviation.
2 · Watch
Take it off screen
Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.