Sample Standard Deviation

s = \frac{\sum ( x _{i} - x ˉ ) ^{2}}{n - 1}

s estimates the population σ from a sample of n observations. The n−1 denominator (Bessel's correction) makes s² unbiased for σ². Population σ uses n in the denominator. s has the same units as the data.

$x_{i}$: Observation i
$x ˉ$: Sample mean
$n$: Sample size
$s$: Sample standard deviation

Step-by-step solved example

Find s for the sample 2, 4, 6, 8.

s estimates σ of the normal that generated the sample.

1. Mean and squares
x̄ = 5. Deviations −3, −1, 1, 3. Squares sum to 9+1+1+9 = 20.
$\sum (x_{i} - x ˉ)^{2} = 20$
2. Divide and root
s = √(20 / 3) = √6.667 = 2.58.
$s = 2.58$

Answer: s = 2.58

10 practice questions

0/10 correct

1.Sample s divides the sum of squared deviations by

2.For the data 5, 5, 5, 5, s equals

3.n = 5, Σ(x_i − x̄)² = 16. Then s =

4.Population σ uses the denominator

5.Mean of 2, 4, 6, 8 is

6.For the two points 1 and 3, s equals

7.Sample variance equals

8.The n−1 correction makes s²

9.The units of s are

10.n = 10, Σ(x_i − x̄)² = 36. s =