Math 1 · Unit 12 · Lesson 3 · part 3 of 4
The standard deviation: square, add, square root
The standard deviation is a typical distance from the mean, found by squaring each distance instead of...
The standard deviation: square, add, square root
The standard deviation is a typical distance from the mean, found by squaring each distance instead of dropping its sign, averaging the squares, and then taking the square root at the end to get back to the original units. Square, add, square root: the same move as the distance formula in Unit 11, with the square root undoing the square, the inverse idea from Unit 1. Work it once by hand on 2, 4, 6, 8, 10.
| value x | distance from the mean, x − 6 | squared |
|---|---|---|
| 2 | −4 | 16 |
| 4 | −2 | 4 |
| 6 | 0 | 0 |
| 8 | 2 | 4 |
| 10 | 4 | 16 |
| add the squares | 40 |
- 1Divide by n.405 = 8. This is the average squared distance.
- 2Square root.√8 ≈ 2.83. Back in the original units: a typical value is about 2.83 from the mean.
- 3Which symbol?Dividing by n gives σ, the Greek letter sigma, the population standard deviation, for when the data is the whole group.
Sample or population?If the data is only a sample of a bigger group, divide by n − 1 instead of n. That gives s, the sample standard deviation: 404 = 10, and √10 ≈ 3.16. The reason, kept simple: a sample tends to underestimate how spread out the whole group is, so dividing by a slightly smaller number makes up for it. Use σ when you have everyone; use s when you have a sample.
σ = √Σ(x − μ)²n s = √Σ(x − x̄)²n − 1
The Σ, sigma again but capital, means "add them all up". After this one example, the tool does the arithmetic. Fill in the table one row at a time.
What the number means. A bigger standard deviation means the values are more spread out from the mean. A standard deviation of 0 means every value is the same. The tight set above has σ ≈ 1.41; the wide set has σ ≈ 3.69. Same mean, and the wide set is about twice as spread out.