Statistics Calculator

Standard Deviation Calculator

Calculate variance and standard deviation for population or sample data with detailed breakdown.

Enter your data (comma or space separated)

Data Type

Sample (n-1)

Population (N)

Learning Guide

What is Standard Deviation?

Standard deviation measures the spread or dispersion of data points from the mean. A low standard deviation indicates data points are close to the mean, while a high standard deviation indicates data is spread out.

Population vs Sample

There are two types of standard deviation:

Population (σ): When you have data for the entire population. Divide by N.
Sample (s): When you have a sample from a larger population. Divide by (n-1) for unbiased estimate.

Formula

σ = √[Σ(xᵢ - μ)² / N] (Population)

s = √[Σ(xᵢ - x̄)² / (n-1)] (Sample)

Where μ (or x̄) is the mean, xᵢ is each data point, and N (or n) is the count.

Variance

Variance is the square of standard deviation. It represents the average squared deviation from the mean. Standard deviation is preferred because it's in the same units as the original data.

The Empirical Rule (68-95-99.7)

For normally distributed data:

~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

Interpreting Results

CV (Coefficient of Variation): (SD/Mean) × 100%. Useful for comparing variability across datasets with different means.
Low SD: Data points cluster closely around the mean
High SD: Data points are spread out widely

Real-World Applications

Finance: Measuring stock volatility and risk
Quality Control: Manufacturing tolerance limits
Science: Experimental measurement uncertainty
Education: Test score distribution analysis