Standard Deviation Calculator

Use sample SD (n − 1 denominator) when your data represents a sample from a larger population — almost always the case in research, surveys, and quality testing. Use population SD (n denominator) when you have the complete population — all 50 states, all employees, every quarter. Excel STDEV.S, R sd(), and Python statistics.stdev() all default to sample SD.

It quantifies the spread or dispersion of your data around the mean. A low SD means values cluster tightly around the mean; a high SD means values are spread out. Two datasets can have the same mean but very different SDs — comparing both reveals how variable each is.

For normally-distributed data, 68% of values fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This is the basis for confidence intervals, hypothesis testing p-values, and quality control limits. Real-world data is often approximately normal but rarely exactly so — always check with a histogram or QQ plot.

It's called Bessel's correction. When estimating population SD from a sample, dividing by n underestimates true variability because the sample mean is closer to sample values than the population mean would be. Dividing by n − 1 corrects this bias. The effect is small for large samples but meaningful for small ones.

Compute monthly SD of returns, then multiply by √12 (about 3.46). So a 5% monthly SD becomes ~17.3% annualized volatility. This assumes returns are independent across months, which is approximately true for liquid assets. Sharpe ratio uses annualized SD as the denominator.

Variance is the average squared deviation from the mean (units²). Standard deviation is the square root of variance (same units as the data). SD is more interpretable because it's in the same units as the original data — a SAT score SD of 200 points is more meaningful than a variance of 40,000 squared points.

Six Sigma quality control aims to keep process output within ±6 SDs of the target mean — equivalent to 3.4 defects per million opportunities (DPMO). Cp and Cpk indices measure process capability relative to specification limits, both expressed in SDs. The "sigma level" of a process is how many SDs fit between the mean and the spec limit.

Outliers can inflate SD dramatically — one extreme value can double the SD. For outlier-prone data (income, home prices, web traffic), prefer robust alternatives: median absolute deviation (MAD), interquartile range (IQR), or trimmed SD (remove top/bottom 5% before calculating). Always plot a histogram or boxplot before using SD on real-world data.