How to Use the Standard Deviation Calculator

Choose population or sample mode, enter your data values separated by commas or spaces, and click Calculate. For grouped data, enter midpoint:frequency pairs.

Calculator Modes

Population Standard Deviation (σ)

Use when your data represents the entire population you're studying. The formula divides by N (total count):

Formula: σ = √(Σ(x - μ)² / N)
Where μ is the population mean and N is the population size

Sample Standard Deviation (s)

Use when your data is a sample from a larger population. Uses Bessel's correction (n-1) for an unbiased estimate:

Formula: s = √(Σ(x - x̄)² / (n-1))
Where x̄ is the sample mean and n is the sample size

Grouped Data

For frequency distribution data where you have class midpoints and frequencies. Enter data as midpoint:frequency pairs (e.g., "5:10, 15:20, 25:15").

Compare Datasets

Compare two datasets and calculate pooled statistics. Useful for:

Pooled Standard Deviation: Common variance estimate for two groups
Combined Statistics: Statistics if datasets were merged

Understanding Standard Deviation

Standard deviation measures the spread or dispersion of data from the mean. A low standard deviation means data points are close to the mean; a high value means data is spread out over a wider range.

Related Statistics

Variance: The square of standard deviation (σ² or s²)
Coefficient of Variation (CV): Standard deviation as a percentage of the mean
Range: Difference between maximum and minimum values

Common Applications

Quality control and manufacturing
Financial risk assessment
Scientific research and experiments
Academic grading and assessment
Weather forecasting variability
Sports performance analysis

Interpretation Guide

~68% of data falls within 1 standard deviation of the mean
~95% of data falls within 2 standard deviations
~99.7% of data falls within 3 standard deviations (Empirical Rule)

Standard Deviation Calculator

σStandard Deviation Calculator