Standard Deviation Calculator

Standard deviation measures how spread out values are from the mean (average) of a data set. A low standard deviation means values cluster tightly around the mean, while a high standard deviation indicates wide dispersion. It is one of the most commonly used statistics in science, business and social research.

The calculator computes both population standard deviation (σ) and sample standard deviation (s). The sample version divides by n−1 (Bessel's correction) to give an unbiased estimate when your data is a subset of a larger population. It also shows variance, mean, range and sum.

Enter your numbers separated by commas or spaces. The calculator handles data sets of any size and displays each step of the calculation: mean, deviations from mean, squared deviations, sum of squares and the final standard deviation.

What does standard deviation measure? Spread of data around the mean. Higher SD = data spread out widely; lower SD = data clustered near mean. Same mean, different SD: critical insight for understanding variability. Example: two classes both averaging 60% in maths exam. Class A SD = 5% → most scores 55-65%, predictable. Class B SD = 20% → some scores 20%, others 95% — same average, vastly different teaching implication. SD is in the same units as the data (£, kg, scores).

How to calculate standard deviation. Step 1: Calculate mean (sum ÷ count). Step 2: Subtract mean from each value (gives deviations). Step 3: Square each deviation (removes negatives). Step 4: Sum all squared deviations. Step 5: Divide by n (population SD) or n-1 (sample SD — Bessel's correction). Step 6: Square root the result. Example: scores [4, 8, 6, 10, 7]. Mean = 7. Deviations: -3, 1, -1, 3, 0. Squared: 9, 1, 1, 9, 0 (sum 20). Variance = 20/5 = 4. SD = √4 = 2.

Population vs sample SD — which to use? Population SD (divide by n): you have all the data — entire population. E.g. heights of every Premier League player this season. Sample SD (divide by n-1): you're estimating from a sample. E.g. height of 50 randomly-selected UK adults to estimate UK population. n-1 correction (Bessel's): compensates for the fact that sample mean is closer to sample values than population mean would be — without correction, sample SD underestimates true population SD. Scientific research almost always uses sample SD.

Normal distribution — the 68/95/99.7 rule. For normally-distributed data: 68% of values within ±1 SD of mean; 95% within ±2 SD; 99.7% within ±3 SD. IQ example: mean 100, SD 15. 68% have IQ 85-115; 95% have IQ 70-130; 99.7% have IQ 55-145. Mensa eligibility (top 2%): IQ 132+ (~2 SD above mean). UK university A-level grades: A 90%+, A* 95%+ — top 5-10% (1.3-1.5 SD above mean). Used in: quality control (Six Sigma = 6 SD = 3.4 defects per million), finance (Value at Risk), psychology, medicine.

Variance vs Standard Deviation — what's the difference? Variance = mean of squared deviations (SD before taking square root). Same information, different units. If data in £: variance in £², SD in £. SD usually preferred for reporting because units match data. Variance preferred for mathematical operations (variances of independent variables add; SDs don't). Coefficient of variation (CV) = SD ÷ mean × 100. CV allows comparison of variability across different scales. Sample CV: hospital wait times in UK 30 days mean ± 15 SD = CV 50% (high variability).