Standard Deviation: What It Measures and How to Interpret It

Standard deviation (SD) measures how spread out a dataset is around its mean. A small SD means most values cluster close to the average; a large SD means values are widely scattered. It is one of the most important statistics in science, finance, and quality control.

How it is calculated

Step 1: Find the mean. Step 2: Subtract the mean from each value and square the result. Step 3: Average those squared differences (divide by n for a population, by n−1 for a sample). Step 4: Take the square root. The calculator does this automatically for any list of numbers you enter.

The 68-95-99.7 rule

In a normal (bell-curve) distribution, roughly 68% of values fall within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This rule makes SD immediately interpretable: if exam scores have a mean of 72 and SD of 8, about 68% of students scored between 64 and 80.

Population vs. sample SD

Use population SD when you have data on every member of a group. Use sample SD (divides by n−1 instead of n) when your data is a sample drawn from a larger group — which is nearly always the case in research and quality control. The calculator offers both options.