This calculator computes the mean, variance, population standard deviation, and sample standard deviation for a set of numbers. Standard deviation measures how spread out the values in a dataset are relative to their average.
Formula Used
Population SD (σ) = √[Σ(x − μ)² ÷ N]
Sample SD (s) = √[Σ(x − x̄)² ÷ (n − 1)]
x
— each individual value in the dataset
μ / x̄
— the arithmetic mean (average) of the values
N / n
— the total number of values in the dataset (N for population, n for sample)
Example Calculation
For a dataset [10, 12, 23, 23, 16, 23, 21, 16] (mean = 18): Variance = 24. Population Standard Deviation = √24 = 4.8990. Sample Standard Deviation (dividing by n-1 = 7) = 5.2372.
Frequently Asked Questions
Standard deviation is a statistical metric that quantifies the amount of variation or dispersion in a set of data values. A low SD indicates values are close to the mean, while a high SD indicates wider dispersion.
Population SD is used when you have the complete dataset of all members. Sample SD is used when the data represents a sample of a larger population, using Bessel's correction (n-1) to adjust for bias.
Variance is the average of the squared differences from the mean. Standard deviation is simply the square root of the variance.
Squaring the differences prevents positive and negative differences from cancelling each other out when summed, and gives extra mathematical weight to outliers.
It is widely used in finance to measure asset volatility and risk, in manufacturing quality control to ensure consistency, and in weather forecasting.