Standard Deviation Calculator
Standard deviation measures how spread out numbers are around their mean. A low SD means values cluster tightly; a high SD means they are widely dispersed. It is the most common measure of statistical variability in science, finance, and engineering.
Population vs. Sample Standard Deviation
The distinction matters depending on your dataset:
- Population (σ): Use when your data is the entire population. Divide by n.
- Sample (s): Use when your data is a sample drawn from a larger population. Divide by n − 1 (Bessel's correction) to correct for sampling bias.
σ = √[ Σ(xᵢ − μ)² / N ]
s = √[ Σ(xᵢ − x̄)² / (n − 1) ]
s = √[ Σ(xᵢ − x̄)² / (n − 1) ]
Other Statistics Explained
- Mean: The arithmetic average — sum divided by count.
- Median: The middle value when sorted. Less sensitive to outliers than the mean.
- Mode: The most frequently occurring value. A dataset can have multiple modes.
- Range: Max minus min. A basic measure of spread.
- Variance: The square of standard deviation. Used in many statistical tests.
When to Use Standard Deviation
- Finance: Volatility of returns — a higher SD means a riskier investment.
- Quality control: Detecting how much a manufacturing process deviates from specification.
- Science: Reporting measurement uncertainty and replicability.
- Grades/scores: Understanding how a score compares to the class average.
Frequently Asked Questions
Which should I use — population or sample SD?
If your data is the entire group (e.g., all 30 students in a class), use population SD. If your data is a sample meant to represent a larger group (e.g., 30 students selected from a school of 1,000), use sample SD.
Can standard deviation be negative?
No. It is the square root of variance, which is a sum of squared deviations — always non-negative. An SD of zero means all values in the dataset are identical.