Standard Deviation Calculator
Paste a list of numbers and get the sample (s) and population (σ) standard deviation, with the full working laid out the way your teacher wants to see it.
The standard deviation formulas
s = √( Σ(x − x̄)² ÷ (n − 1) ) σ = √( Σ(x − μ)² ÷ N )
Both formulas do the same four things: find the mean, measure each value’s distance from it, square those distances so negatives don’t cancel positives, then average and take a square root to get back to the original units. The only difference is the divisor.
How to calculate standard deviation by hand
Take the scores 4, 8, 6, 5, 3, 7, 9, 5. They add to 47, so the mean is 47 ÷ 8 = 5.875. The deviations are −1.875, 2.125, 0.125, −0.875, −2.875, 1.125, 3.125 and −0.875. Squared and added, they give 30.875. Divide by n − 1 = 7 to get a sample variance of 4.411, and the square root gives s ≈ 2.100. If those eight numbers were the whole population you would divide by 8 instead: σ ≈ 1.965.
A quick sanity check: the standard deviation is usually between a quarter and a sixth of the range for moderate samples. Here the range is 6, and 2.1 fits comfortably.
Reading the result
A small standard deviation means the values huddle close to the mean; a large one means they are scattered. Because it shares the data’s units, you can say things like “scores averaged 81 points, give or take about 9.” To locate a single value relative to that spread, turn it into a z-score. To describe how precisely a sample estimates the mean, use the standard error, which is s ÷ √n.
Frequently asked questions
Should I use sample or population standard deviation?
Use the sample standard deviation (s, divide by n − 1) when your numbers are a subset drawn from a bigger group you want to describe, which is almost always the case in coursework and research. Use the population standard deviation (σ, divide by n) only when you have measured every member of the group you care about, such as every student in one class.
Why does the sample formula divide by n − 1?
Deviations are measured from the sample mean, which sits closer to the sample values than the true population mean does. That makes the sum of squares slightly too small. Dividing by n − 1 instead of n (Bessel’s correction) removes that bias from the variance.
What is a “good” standard deviation?
There is no universal good value. Standard deviation is in the same units as the data, so judge it against the mean and the context. A 2 cm spread in adult heights is tiny; a 2 cm spread in screw lengths is huge. The coefficient of variation (SD ÷ mean) helps compare spreads across scales.
Can the standard deviation be negative or zero?
It can never be negative, because it is the square root of an average of squared numbers. It is zero only when every value is identical.
How do I calculate standard deviation in Excel or Google Sheets?
Use =STDEV.S(range) for the sample standard deviation and =STDEV.P(range) for the population standard deviation. The older STDEV function equals STDEV.S. On a TI-84, Sx is the sample value and σx the population value.
What does standard deviation tell me about a normal distribution?
For roughly bell-shaped data, about 68% of values fall within one standard deviation of the mean, about 95% within two and about 99.7% within three. This is the empirical rule, and it only holds when the data really are close to normal.