What is Standard Deviation?
Standard deviation (σ or s) is a measure of how spread out numbers are from the mean. A low standard deviation indicates that values tend to be close to the mean, while a high standard deviation indicates that values are spread out over a wider range.
Population vs Sample
There are two types of standard deviation depending on your data:
- Population (σ): Use when you have data for the entire population. Divide by N (total count).
- Sample (s):Use when you have a sample from a larger population. Divide by (N-1) to correct for bias (Bessel's correction).
Formulas
Population Standard Deviation (σ)
σ = √[Σ(xᵢ - μ)² / N]
Sample Standard Deviation (s)
s = √[Σ(xᵢ - x̄)² / (n-1)]
Variance
Variance is the square of standard deviation. It represents the average of squared deviations from the mean. While mathematically useful, standard deviation is easier to interpret because it's in the same units as the original data.
Variance = (Standard Deviation)²
Coefficient of Variation (CV)
The coefficient of variation is the ratio of standard deviation to mean, expressed as a percentage. It allows comparison of variability between datasets with different units or scales.
CV = (σ / μ) × 100%
Interpreting Standard Deviation
For normally distributed data (bell curve):
- ~68% of values fall within 1 standard deviation of the mean
- ~95% of values fall within 2 standard deviations
- ~99.7% of values fall within 3 standard deviations