The formula
How to calculate standard deviation
Standard deviation measures how spread out a set of values is around its mean. This calculator works from summary statistics — count, sum and sum of squares — which is how the figure is computed inside every statistics package.
The computational form above avoids calculating each deviation separately. Sample standard deviation divides by n − 1 rather than n, a correction called Bessel's, because a sample mean sits slightly closer to its own data than the true population mean does.
The calculator asks for:
- Number of values
- Sum of the values
- Sum of the squared values — square each value, then add them up
No submit button: type and the answer moves. Your inputs end up in the link, so the page can be shared already filled in.
Units matter more here than the arithmetic itself: the formula assumes a specific set of units for each input, stated next to the field, and converting into those units first is usually the difference between a correct result and one that is wrong by a clean power of ten.
Why standard deviation matters
The formula behind standard deviation is standard and appears in the same form across textbooks and reference material; what a calculator adds is speed and the ability to see instantly how the result responds to a change in any one of the inputs, which is far slower to do by hand.
Beyond a single check, the same calculation is worth rerunning whenever a measured input changes — a new reading, a corrected value, an updated assumption — since the result here always reflects exactly what is currently in the fields above rather than a value calculated once and then left stale.
A formula like this one is rarely the last step in a piece of work — the figure it produces usually feeds into a further calculation, a comparison against a published value, or a write-up that needs to state both the result and how confident it is. Getting this step right the first time, rather than propagating a small arithmetic slip through several more steps, is the main practical reason to check a manual calculation against a tool like this one before building on top of it.
It is worth keeping a note of which inputs were used to produce a given result, particularly where the figure is going into a report or a further calculation — reproducing a result later, or explaining how it was reached, is far easier with the original inputs to hand than by trying to reverse-engineer them from the output alone.
Worked example
Here is the calculation with the starting values:
- Number of values: 25
- Sum of the values: 1,850
- Sum of the squared values: 141,500
That gives:
- Sample standard deviation: 13.8444
- Population standard deviation: 13.5647
- Mean: 74
The figures above are the calculator's own default values, shown purely so the working is visible rather than hidden — the same steps apply exactly to your own numbers, entered in the fields at the top of this page.
Reading the result
For roughly normal data, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. That is what makes the figure interpretable rather than merely descriptive.
Where this goes wrong. Standard deviation is in the same units as the data and is not comparable across different scales. Use the coefficient of variation to compare spread between datasets measured in different units.
A result that is wrong by an exact factor of ten, a hundred or a similar round number is almost always a units error rather than a mistake in the formula itself — checking each input against the unit stated beside it is the fastest way to track it down.
Use the sample version — dividing by n − 1 — whenever your data is a sample from a larger group, which is almost always. The population version applies only when you have measured every member.
It depends entirely on the scale. A standard deviation of 5 is enormous for exam marks out of 10 and negligible for house prices. Compare it against the mean.
It returns sample standard deviation. With 25 number of values, 1,850 sum of the values and 141,500 sum of the squared values, that comes to 13.8444. Change any field and the figure moves with it.
Generally, no more than the least precise input justifies — a result reported to six decimal places from inputs measured to two significant figures is implying a precision the calculation does not actually have. The calculator shows full precision so you can round appropriately for your own use.
Yes — the equation shown in the formula section above is the standard form used in textbooks and reference material for this calculation, not a simplified or approximate version.
Yes, in the sense that it applies the correct standard formula and returns an accurate result for the inputs given — but check your own course or publication's requirements for how results should be rounded, presented and referenced, since those conventions vary and are not something a calculator can know on your behalf.