Why does sample standard deviation divide by n - 1?
A sample uses its own mean to estimate a larger population, which tends to understate spread unless the denominator is adjusted.
Standard Deviation Calculator
Measure how tightly a data list clusters around its mean, compare population and sample standard deviation, inspect variance, and understand outlier-driven spread.
Spread is distance from the mean
Every value contributes by how far it sits above or below the average. Standard deviation summarizes those distances in the original data unit, while variance keeps the squared version.
Read a data list in four passes
- Enter the values.
- Find the mean.
- Inspect deviations from the mean.
- Compare variance and standard deviation.
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Population, sample, and variance formulas
Choose population when the list is the whole group; choose sample when it estimates a larger group.
Population variance
Population standard deviation
Sample variance
Sample standard deviation
Variance is squared spread; standard deviation returns to the original unit
Variance can feel abstract because deviations are squared. Taking the square root gives standard deviation, which is easier to compare with the original values.
Outliers and interpretation checks
- One large value can inflate spread.
- Compare with and without a suspected outlier when appropriate.
- Small standard deviation means values cluster near the mean.
- Large standard deviation means values are dispersed.
FAQ
What does a high standard deviation mean?
It means values are farther from the mean on average; check the data list because one outlier may be responsible for much of that spread.
