How to use Standard Deviation Calculator
The standard deviation calculator computes the complete descriptive statistics for a list of numbers: count, sum, mean, variance, standard deviation, min, max, and range. Toggle between population standard deviation (divides by n) and sample standard deviation (divides by n−1) depending on whether your data represents an entire population or a sample drawn from a larger group. Results update immediately after you click Calculate, and you can copy the full summary with one click.
- Paste or type your numbers in the input box, separated by spaces, commas, or newlines.
- Choose Population (÷n) if your data covers the entire group, or Sample (÷n−1) if it is a subset.
- Click Calculate to see all statistics in the results table.
- Copy the full results using the Copy button, or use Copy Result to share.
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Population vs. sample standard deviation
Population standard deviation uses n as the divisor and applies when you have data for every member of the group (e.g., all test scores in a class). Sample standard deviation uses n−1 (Bessel's correction) and applies when your data is a random sample from a larger population. Using n−1 corrects for the tendency of a sample to underestimate the true population spread. For large samples the difference is negligible, but for small samples (fewer than 30 values) the choice matters significantly.
| Statistic | Formula | Description |
|---|---|---|
| Mean (μ) | Σx / n | Sum of all values divided by count |
| Population variance (σ²) | Σ(x−μ)² / n | Average squared deviation from the mean |
| Sample variance (s²) | Σ(x−μ)² / (n−1) | Corrected variance for sample data |
| Population SD (σ) | √σ² | Square root of population variance |
| Sample SD (s) | √s² | Square root of sample variance |
| Range | max − min | Difference between largest and smallest values |
Interpreting standard deviation in practice
In a normal distribution, about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three (the 68-95-99.7 rule). A low standard deviation means values cluster tightly around the mean; a high SD means they are spread out. In finance, SD is used as a measure of investment volatility — a stock with a high SD is considered more risky. In quality control, a process is considered under control when measurements stay within ±3σ of the target (Six Sigma principle). In education, SD helps compare score distributions across different tests.
Worked examples
Population standard deviation
Inputs: 2, 4, 6
Result: μ = 4 · variance 2.67 · σ ≈ 1.63
Sample standard deviation
Inputs: 2, 4, 6 (÷ n−1)
Result: s = 2.00
Wider spread
Inputs: 10, 20, 30, 40, 50
Result: μ = 30 · σ ≈ 14.14
The 68-95-99.7 rule (normal distribution)
| Range from mean | Values within | Interpretation |
|---|---|---|
| ±1σ | ~68% | Most values cluster here |
| ±2σ | ~95% | Outside this is uncommon |
| ±3σ | ~99.7% | Beyond this is rare (Six Sigma) |
Glossary
- Standard deviation
- A measure of how spread out values are around the mean; the square root of variance.
- Variance
- The average of the squared differences from the mean; the square of the standard deviation.
- Bessel's correction
- Using n−1 instead of n when computing sample variance, to correct for the bias in estimating the population variance from a sample.
- Normal distribution
- A symmetric, bell-shaped probability distribution characterised by its mean and standard deviation.
Related reading
Frequently Asked Questions
Why use Standard Deviation Calculator?
- Computes population or sample standard deviation with the complete formula working shown
- Calculates six statistics simultaneously: count, sum, mean, variance, SD, and range
- Toggles between σ (÷n) and s (÷n−1) with a single click for the correct statistical context
- Accepts numbers separated by spaces, commas, or newlines for flexible data entry
Common use cases
- Find the spread of a dataset for a statistics assignment that requires full working
- Calculate sample standard deviation from survey responses for quantitative research
- Measure monthly investment-return volatility across a series of percentage figures
- Assess test-score consistency across a class to understand grade distribution width
- Run quality-control variance checks on repeated measurements from a production batch
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