Standard deviation Formula
The standard deviation measures how spread out values are from the mean.
Standard Deviation Formula
Population Standard Deviation (σ)
Used when you have all values in a population:
$$\sigma = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \mu)^2}{N}}$$
Where:
- $x_i$ = each individual value
- $\mu$ = population mean
- $N$ = total number of values in the population
Sample Standard Deviation (s)
Used when you have a sample (subset) of a population:
$$s = \sqrt{\frac{\sum_{i=1}^{N}(x_i - \bar{x})^2}{n - 1}}$$
Where:
- $x_i$ = each individual value
- $\bar{x}$ = sample mean
- $n$ = number of values in the sample
- $n - 1$ = Bessel's correction (reduces bias in estimating population variance)
Step-by-Step Calculation
- Find the mean - add all values, divide by count
- Subtract the mean from each value - these are the deviations
- Square each deviation - eliminates negatives
- Sum the squared deviations
- Divide by N (population) or n-1 (sample)
- Take the square root - this converts variance back to original units
Quick Example
Data: 2, 4, 4, 4, 5, 5, 7, 9
| Step | Value |
|---|---|
| Mean ($\bar{x}$) | (2+4+4+4+5+5+7+9) / 8 = 5 |
| Squared deviations | 9, 1, 1, 1, 0, 0, 4, 16 |
| Sum of squares | 32 |
| Variance (sample) | 32 / 7 = 4.57 |
| Std Dev (sample) | √4.57 = 2.14 |
Key Notes
- SD = 0 means all values are identical
- A small SD means values cluster tightly around the mean
- A large SD means values are widely spread
- Standard deviation is the square root of variance
- Always use sample SD (n-1) when working with a sample drawn from a larger population
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