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. Take 3, 4, 7, 10 and 11. They add to 35 and there are 5 of them, so the mean is 7. The gaps from 7 are negative 4, negative 3, 0, 3 and 4. Square each gap, and a negative times a negative is positive: 16, 9, 0, 9 and 16, which add to 50. Divide by 5 for the population variance, 10. The standard deviation is its square root, about 3.16 on a calculator, back in the original units.
Totals give the same answer faster. For 3, 4, 7, 10 and 11, the total is 35 and the total of squares is 9 + 16 + 49 + 100 + 121 = 295. The mean of squares is 295 over 5, which is 59, and the mean squared is 7 × 7 = 49. Variance is the mean of squares minus the mean squared: 59 minus 49 is 10.
As when finding a mean from a frequency table, count each value f times. Σ means add them all up. Values 1, 2 and 3 with frequencies 2, 5 and 3 make n = 10. Σfx = 2 + 10 + 9 = 21, so the mean is 2.1. Σfx² = 2 × 1 + 5 × 4 + 3 × 9 = 49, so the mean of squares is 4.9. Variance is 4.9 minus 2.1 squared, which is 4.9 minus 4.41 = 0.49. The standard deviation is 0.7.
Compare sets by mean and spread together. C with 6, 7, 8 and D with 2, 7, 12 share the mean 7. C strays at most 1 from the mean while D strays up to 5, so D has the larger standard deviation. Stretching values away from the mean pushes the number up.
Standard deviation is the root of the variance, so it carries the original units. Adding the same amount to every value moves the mean but leaves the standard deviation unchanged.
Square the gaps from the mean, average them for variance, and take the square root for standard deviation.
The rewritten readings were written for this dataset. Adapted reading summaries: Khan Academy, "Center, spread, and shape of distributions | Lesson". Questions, flashcards and "Dividing by n or by n - 1" were written for this dataset. Khan Academy publishes the retained videos "Variance of a population" and "Measures of spread: range, variance & standard deviation". Source licenses are not specified.
2 · Watch
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