Sample vs Population Standard Deviation
Population standard deviation σ uses n in the denominator, while sample standard deviation s uses (n-1) — a critical statistical distinction in enVision Algebra 1 Chapter 11 for Grade 11. σ = √[Σ(x-μ)²/n] is used when you have data for every member of the group. s = √[Σ(x-x̄)²/(n-1)] is used when your data is a sample from a larger population; the (n-1) denominator (Bessel's correction) makes the estimate less biased. For 5 test scores from a class of 5 (complete population), use n = 5. For heights from 20 students sampled from a school of 500, use n-1 = 19.
Key Concepts
Population standard deviation: $\sigma = \sqrt{\frac{\sum(x \mu)^2}{n}}$.
Sample standard deviation: $s = \sqrt{\frac{\sum(x \bar{x})^2}{n 1}}$.
Common Questions
When do you use population standard deviation vs sample standard deviation?
Use population standard deviation (σ, denominator n) when you have data for every member of the group. Use sample standard deviation (s, denominator n-1) when your data is a subset drawn from a larger group.
Why does sample standard deviation use n-1 instead of n?
Dividing by n-1 (Bessel's correction) corrects for the bias introduced by using the sample mean x̄ instead of the true population mean μ. It makes the sample variance an unbiased estimator.
For 5 test scores from a class of exactly 5 students, which formula applies?
Population standard deviation (n = 5 in denominator) because you have data for every member of the group — not a sample.
For heights from 20 students randomly selected from 500, which applies?
Sample standard deviation (n-1 = 19 in denominator) because the 20 students are a sample from the larger population of 500.
On a calculator, which button gives sample vs population standard deviation?
Typically Sx (or s) gives sample standard deviation (n-1) and σx (or σ) gives population standard deviation (n). Check your calculator's statistics mode.