📊 Standard Deviation Calculator

What Standard Deviation Actually Measures

Standard deviation tells you how spread out the values in a dataset are relative to the mean. A small standard deviation means most values cluster tightly around the average. A large one means values are widely dispersed. If 10 students scored an average of 70% on a test with a standard deviation of 2, nearly everyone scored between 66–74%. A standard deviation of 15 means scores ranged widely — some students failed while others aced it.

Population vs. Sample Standard Deviation

Use population standard deviation (σ) when you have data for every member of the group you care about. Use sample standard deviation (s) when your data is a subset of a larger group — the formula divides by n−1 instead of n (Bessel's correction), which corrects for the tendency of small samples to underestimate true spread. In most real-world analysis, use the sample version unless you have census-level data.

The 68-95-99.7 Rule

For normally distributed data, approximately 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. This is the foundation of statistical quality control, IQ scoring, and clinical reference ranges. If a test has mean 100 and SD 15 (as many IQ tests do), then 95% of scores fall between 70 and 130. Values more than three SDs from the mean occur less than 0.3% of the time — they're genuinely unusual.