Standard Deviation Calculator
Calculate mean, variance, and standard deviation (population and sample) from a data set.
By Konstantin Iakovlev · Updated September 2026 · Source: Khan Academy
Population SD
4.8990
Sample SD
5.2372
Mean
18.0000
Statistics
| Count (n) | 8 |
| Sum | 144.00 |
| Mean | 18.0000 |
| Population Variance (σ²) | 24.0000 |
| Population Std Dev (σ) | 4.8990 |
| Sample Variance (s²) | 27.4286 |
| Sample Std Dev (s) | 5.2372 |
| Minimum | 10.00 |
| Maximum | 23.00 |
| Range | 13.00 |
Sorted Data
10.00, 12.00, 16.00, 16.00, 21.00, 23.00, 23.00, 23.00
Use the Standard Deviation Calculator above to calculate your results. Enter your values and see instant results — all calculations run in your browser.
Disclaimer: This calculator is for informational purposes only. Results are estimates based on the information you provide and the assumptions described on this page.
How It Works
Feed in a data set and this calculator returns its mean, variance, and both the population and sample standard deviation in a single pass. Those figures describe how tightly or loosely values cluster, which is why they turn up wherever spread matters—gauging stock market volatility, tracking temperature swings, or checking whether a manufacturing line stays within tolerance.
Calculation starts with the mean, the simple average of your values. Variance comes next, averaging the squared distance of each point from that mean to capture overall spread. Taking the square root of the variance yields the standard deviation, expressed in the same units as your data; the population version divides by N, while the sample version divides by N-1 to correct for the bias that creeps in when you estimate from a subset.
Choosing the right version is not optional—applying the population formula to sample data, or vice versa, skews your conclusions, and the distortion grows sharper as data sets shrink. Confirm that every entry is numerical and that nothing was mistyped. One frequent slip is treating variance and standard deviation as interchangeable; only the standard deviation carries the original units, which is what makes it directly interpretable.
Example: Spread in Quarterly Sales
- 1 A small business had quarterly sales of $120, $135, $110 and $145 thousand over four quarters. Enter them as 120, 135, 110, 145: separate the values with commas or line breaks, and leave out dollar signs and thousands separators, which the calculator cannot read.
- 2 Mean: (120 + 135 + 110 + 145) / 4 = 510 / 4 = 127.5. The deviations from the mean are −7.5, 7.5, −17.5 and 17.5; their squares, 56.25, 56.25, 306.25 and 306.25, sum to 725.
- 3 Population variance = 725 / 4 = 181.25, so the population SD is √181.25 = 13.4629. Sample variance = 725 / 3 = 241.6667, so the sample SD is √241.6667 = 15.5456. The calculator shows both, with the mean as 127.5000.
- 4 Treating the four quarters as the whole population of interest, sales varied by about $13,463 around the $127,500 average. The SD is a root-mean-square distance, so it is larger than the plain average distance from the mean, $12,500. If the four quarters stand in for a longer run, the sample SD, about $15,546, is the better estimate of that spread.
Source: Khan Academy · Last updated: September 2026
Frequently Asked Questions
What is the difference between population and sample standard deviation?
What does standard deviation tell you?
What is considered a high standard deviation?
You might also need
Percentage Calculator
Calculate percentages three ways: what is X% of Y, X is what % of Y, and percentage change from X to Y.
Fraction Calculator
Add, subtract, multiply, and divide fractions. Results shown as fraction, mixed number, and decimal.
Probability Calculator
Calculate probability from favorable and total outcomes. Find combined probability for independent or mutually exclusive events.