Standard Deviation
Sample or population standard deviation with every step shown.
About this calculator
Paste your numbers, choose sample or population, and read the standard deviation with all four steps of the computation written out โ mean, sum of squared differences, variance, square root. It's the working a stats teacher wants to see, done honestly.
What the number means, and nโ1
Standard deviation is spread measured in the data's own units โ roughly, how far a typical value sits from the mean. In bell-shaped data the famous rule applies: about 68% of values fall within one deviation of the mean, 95% within two, 99.7% within three; a test with mean 70 and deviation 8 has most of the class between 62 and 78.
The nโ1 question, answered plainly: when your numbers are a sample standing in for a bigger population, the sample's own mean sits slightly too close to its values, understating the spread โ dividing by nโ1 instead of n (Bessel's correction) compensates. Since almost all real data is a sample of something, sample is the default here; divide by n only when your data is literally the entire population (every student in the class, every unit produced). For big n the two answers converge anyway. Mean, median and mode are the natural companions.
Frequently asked questions
Should I use sample or population standard deviation?
Sample (รท nโ1) almost always โ it corrects for the fact that a sample understates its parent population's spread. Population (รท n) is only right when your data is literally everyone, not a subset.
What does a standard deviation of 8 mean?
Typical values sit about 8 units from the mean. If the data is roughly bell-shaped, ~68% of values fall within 8 of the mean and ~95% within 16.
What's the difference between variance and standard deviation?
Variance is the average squared distance from the mean; standard deviation is its square root, which puts the answer back in the data's own units โ metres rather than metres-squared โ and is why it's the quoted number.
Can standard deviation be zero?
Only when every value is identical. It can never be negative โ it's built from squared distances.