Math Tools
Enter numbers to calculate standard deviation, variance, and other key statistical measures.
Sample Standard Deviation
5.24
Based on sample data (N - 1 divisor)
Standard deviation is a statistic that measures how far individual numbers are from the average. A small standard deviation means data points cluster near the mean; a large one means they are scattered. It has two forms: population SD (for the entire group) and sample SD (for a subset). Variance is the square of standard deviation.
The calculator accepts any list of numbers (separated by commas, spaces, or line breaks). It computes the mean, then calculates the average of squared differences from that mean. The square root of that average is the standard deviation. For sample data, it divides by N−1 instead of N to account for estimation bias.
Population SD = √[(Σ(x − mean)²) / N] | Sample SD = √[(Σ(x − mean)²) / (N − 1)]Find the mean (average) of all numbers. For each number, calculate how far it is from the mean and square that difference. Sum all squared differences. Divide by N (population) or N−1 (sample). Take the square root. The result is standard deviation; its square is variance.
| Input | Result | Notes |
|---|---|---|
| 10, 12, 23, 23, 16, 23, 21, 16 | 5.61 | Sample SD for 8 test scores: spread is moderate, data varies about 5.6 points from mean (17.5) |
| 100, 100, 100, 100 | 0.00 | All values identical: zero standard deviation, no variation |
| 5, 15, 25, 35, 45 | 15.81 | Evenly spaced values: high SD (≈15.8), indicating wide spread across the range |
Population SD uses N (all data). Sample SD uses N−1 (a subset of a larger population). Use sample SD when analyzing a sample; it's an unbiased estimator. Use population SD only when you have the complete dataset.
Dividing by N−1 instead of N makes the result slightly larger. This correction (Bessel's correction) accounts for estimation uncertainty when sampling, giving a more accurate estimate of the true population SD.
On average, data points deviate about 5 units from the mean. If your mean is 100, expect most values near 95–105. SD is in the same units as your original data.
Yes. Negative numbers are treated normally. Squaring them removes the sign, so the result depends only on distance from the mean, not direction.
Variance is the average of squared deviations (SD²). SD is the square root of variance. SD is more intuitive because it's in the same units as your data; variance is used in more advanced statistics.
At least 2 for sample SD. With only 1 value, there's no spread to measure. More data generally gives a more reliable SD.