The variance is the average of the squared distances from the mean, and the standard deviation is its square root. Both measure how far values sit from the mean.
This lesson is part of dispersion of ungrouped data. It follows calculating range and interquartile range.
What are the two formulas?
| Method | Formula | Best when |
|---|---|---|
| Deviations | σ² = Σ(x − μ)² ÷ N | The mean is a whole number |
| Σx² | σ² = Σx² ÷ N − μ² | The values are large |
The standard deviation is σ = √(σ²). Both methods give the same variance, which makes a good cross-check.
Worked example: 4, 6, 8, 10, 12
The mean is (4 + 6 + 8 + 10 + 12) ÷ 5 = 40 ÷ 5 = 8.
Deviation method. The deviations are −4, −2, 0, 2, 4. Squared: 16, 4, 0, 4, 16, which add to 40. The variance is 40 ÷ 5 = 8.
Σx² method. The squares are 16, 36, 64, 100, 144, which add to 360. Then 360 ÷ 5 − 8² = 72 − 64 = 8. The two methods agree.
The standard deviation is √8 = 2.83 (to two decimal places).
The slip that costs marks
The common slip is to subtract the mean instead of its square in the second method: 72 − 8 = 64. That gives a variance of 64, which is 8 times too large.
The check is quick. A variance of 64 would mean the typical distance from the mean is 8, but every value is at most 4 away. A standard deviation cannot exceed the largest distance from the mean, so 8 is impossible.
Reading the answer
A standard deviation of 2.83 means the values sit, in a typical sense, about 2.83 away from the mean of 8. A set with a smaller standard deviation is more tightly packed.
Standard deviation is never negative. If your working gives a negative variance, a sign slipped in the Σx² method, because the mean squared cannot exceed the mean of the squares.
Check yourself
Find the variance and standard deviation of 2, 3, 5, 6, 9.
Answer
Mean = 25 ÷ 5 = 5. Deviations: −3, −2, 0, 1, 4. Squared: 9, 4, 0, 1, 16, which add to 30. Variance = 30 ÷ 5 = 6.
Check with Σx²: 4 + 9 + 25 + 36 + 81 = 155, so 155 ÷ 5 − 25 = 31 − 25 = 6. Standard deviation = √6 = 2.45.
What to study next
Continue with comparing data sets using centre and spread. If squaring and roots still cause slips, try the algebra step repair trainer.
For a teacher to watch your working, see online one-to-one Mathematics tuition.