How to Calculate Standard Deviation
Standard Deviation is a key statistical measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that data points tend to be close to the mean, while a high standard deviation indicates that the data is spread out over a wider range.
Sample vs. Population Standard Deviation
When using this calculator, it is crucial to choose the correct method based on your data source:
- Population Standard Deviation (σ): Use this when your data set includes the entire population you are studying. The formula divides by N (total count).
- Sample Standard Deviation (s): Use this when your data is a sample subset of a larger population. This is the most common method in statistics. The formula divides by N-1 (Bessel's correction) to provide an unbiased estimate.
Standard Deviation Formulas
Population: σ = √ [ Σ(xi - μ)2 / N ]
Sample: s = √ [ Σ(xi - x̄)2 / (N - 1) ]
Why is Variance Important?
Variance is the square of the standard deviation. While standard deviation gives you a measure of spread in the same units as the original data, variance is expressed in squared units. It is fundamental in risk assessment, finance, and quality control.
Example Calculation
For a data set of {1, 2, 3, 4, 5}:
- Mean: 3
- Differences: (-2, -1, 0, 1, 2)
- Squared Differences: (4, 1, 0, 1, 4) → Sum = 10
- Population Variance: 10 / 5 = 2
- Population Std Dev: √2 ≈ 1.4142