1. Definition
- The sample standard deviation (s) measures how much the values in a sample spread out around the sample mean.
- It is the square root of the sample variance.
- It tells us, on average, how far each data point is from the sample mean.
2. Formula
For a sample of size $n$:
$s = \sqrt{\frac{\sum_{i=1}^{n} (x_i – \bar{x})^2}{n – 1}}$
Where:
- $x_i$ = each observation
- $\bar{x}$ = sample mean
- $n$ = sample size
- Denominator = $n-1$ (Bessel’s correction, to make $s$ an unbiased estimator of the population standard deviation σ).
3. Step-by-Step Example
Sample Data: {5, 7, 9}
- Find the mean:
$\bar{x} = \frac{5 + 7 + 9}{3} = \frac{21}{3} = 7$
- Subtract the mean (deviations):
- (5 – 7) = –2
- (7 – 7) = 0
- (9 – 7) = +2
- Square deviations:
- (–2)² = 4
- 0² = 0
- (2)² = 4
- Sum squared deviations:
$4 + 0 + 4 = 8$
- Divide by (n – 1):
$\frac{8}{3 – 1} = \frac{8}{2} = 4$
- Take square root:
$s = \sqrt{4} = 2$
Sample standard deviation = 2
4. Interpretation
- Small $s$: Data points are close to the mean (low variability).
- Large $s$: Data points are spread out from the mean (high variability).
- If $s = 0$: All values are identical.
5. Sample vs Population Standard Deviation
| Concept | Formula | Denominator | Used for |
|---|---|---|---|
| Population standard deviation (σ) | $\sqrt{\frac{\sum (x_i – μ)^2}{N}}$ | $N$ | Whole population |
| Sample standard deviation (s) | $\sqrt{\frac{\sum (x_i – \bar{x})^2}{n-1}}$ | $n-1$ | A sample (estimate of σ) |
n – 1 instead of n corrects for bias → otherwise, sample variance would underestimate σ.
6. When It’s Used
- To measure spread/variability in sample data.
- In t-tests, ANOVA, regression, etc. (standard error calculations use sss).
- To build confidence intervals around sample means.
In short:
The sample standard deviation (s) is the square root of the average squared deviation from the sample mean, using denominator n−1n-1n−1. It measures how spread out the sample data is, and it’s the best unbiased estimator of the population standard deviation (σ).