Standard Deviation Calculator
Population & Sample Spread Tool
Understanding Standard Deviation
Standard deviation measures how spread out a set of numbers is from its average. A small standard deviation means the values cluster tightly around the mean; a large one means they're scattered widely.
Use this when your data set is the entire group you're studying — for example, the test scores of every student in a single class. It divides the sum of squared deviations by n.
Use this when your data set is a sample drawn from a larger population — for example, 30 students out of an entire school. It divides by n − 1 to correct for sampling bias.
The Formulas
Population Standard Deviation
Sample Standard Deviation
Step by Step
The calculator works in three stages: first it finds the mean of your list, then it subtracts the mean from every value and squares each result to get the sum of squared deviations, and finally it divides that sum by either n (population) or n − 1 (sample) and takes the square root to arrive at the standard deviation. Variance is simply that division result before the square root is applied.
Population standard deviation divides by the full count.
Sample standard deviation divides by one less than the count.
Standard deviation squared — the same spread, in squared units.
Key Takeaways
- Choose population vs. sample based on your data. Use population statistics only when your list is the entire group of interest.
- Sample standard deviation is always slightly larger than population standard deviation for the same data, since it divides by a smaller number.
- A single value has no sample standard deviation — you need at least two numbers to estimate spread from a sample.
- Need mean, median, mode, or quartiles too? Try the Statistics Calculator for a complete summary in one pass.
Frequently Asked Questions
- Type a number into each row.
- Click "Add Number" to add more rows, or the icon to remove a row.
- Click "Calculate" to see the mean, the sum of squared deviations, and both population and sample variance/standard deviation.
Population standard deviation is used when your list contains every member of the group you care about — you divide the sum of squared deviations by n (the count). Sample standard deviation is used when your list is only a sample drawn from a larger population — you divide by n − 1 (called Bessel's correction) instead, which slightly inflates the result to correct for the tendency of a sample to underestimate the true population spread.
It's the numerator shared by both variance formulas: for every number in your list, subtract the mean, square the result, then add up all those squares — Σ(x − mean)². This calculator shows that running total explicitly so you can see exactly what goes into the variance and standard deviation before it gets divided.
Sample standard deviation divides by n − 1. With only one value, n − 1 = 0, which makes the division undefined — there's no spread to measure from a single data point. The calculator shows "N/A" for the sample statistics in that case, while population standard deviation (which is always 0 for a single value) still computes normally.
Variance is the standard deviation squared. Standard deviation is usually more useful for interpretation because it's expressed in the same units as your original data (variance is in squared units), but variance is the more fundamental quantity used in further statistical calculations like ANOVA and regression.
Yes. Every row accepts any real number, including negative values and decimals — useful for measuring the spread of things like temperature changes, investment returns, or lab measurements.