About the Standard Deviation Calculator
Standard deviation measures how spread out a dataset is — the typical distance of values from their mean. It is the most widely used measure of variability, appearing in finance as volatility, in manufacturing as process control, and in research as the basis for significance testing.
The formula
Population: σ = √(Σ(x−μ)²/N) Sample: s = √(Σ(x−x̄)²/(n−1))The only difference is the denominator. Dividing by N treats your data as the whole population; dividing by n − 1 corrects for the bias that arises when estimating a population from a sample.
How to use this calculator
- 1Enter your Value 1. The field starts at
10, which you can overwrite. - 2Enter your Value 2. The field starts at
20, which you can overwrite. - 3Enter your Value 3. The field starts at
30, which you can overwrite. - 4Enter your Value 4. The field starts at
40, which you can overwrite. - 5Enter your Value 5. The field starts at
50, which you can overwrite. - 6Enter your Value 6 (optional). The field starts at
60, which you can overwrite. - 7Read the result straight away — it recalculates as you type, so there is no button to press. Use Share to copy a link that reopens the page with your exact numbers filled in.
Worked example
| Input | Value |
|---|---|
| Value 1 | 10 |
| Value 2 | 20 |
| Value 3 | 30 |
| Value 4 | 40 |
| Value 5 | 50 |
| Value 6 (optional) | 60 |
Result
Population σ: 17.0783
Sample s: 18.7083
Mean: 35.0000
Variance (pop): 291.6667
Those are the values the page loads with, so you can reproduce this result yourself and then change one field at a time to see what drives the outcome.
Understanding your result
The n − 1 correction, called Bessel's correction, exists because a sample mean is always closer to its own data than the true population mean would be. That systematically understates the spread, and dividing by a smaller number compensates. With large samples the difference is negligible; with ten data points it matters.
The empirical rule makes standard deviation interpretable for roughly normal data: about 68 percent of values fall within one standard deviation of the mean, 95 percent within two, and 99.7 percent within three. This is why a result more than two standard deviations from expectation is treated as noteworthy and three as exceptional.
Things worth knowing
- Use the sample formula unless you genuinely have every member of the population. Most real data is a sample.
- Standard deviation shares the units of your data, which is why it is reported rather than variance.
- It is sensitive to outliers, because deviations are squared before averaging.
- In finance, standard deviation of returns is the standard measure of volatility and therefore of risk.
- To compare spread across different units, use the coefficient of variation instead.
Frequently asked questions
What is the difference between population and sample standard deviation?+
The population version divides by N and applies when you have all the data. The sample version divides by n − 1 to correct for bias when estimating a population from a subset, giving a slightly larger and more honest figure.
What does a high standard deviation mean?+
That values are widely dispersed around the mean. In investment returns that means volatility and risk; in manufacturing it means inconsistency; in test scores it means a wide range of performance.
Why divide by n − 1 for a sample?+
Because the sample mean sits closer to the sample data than the true population mean does, which understates variability. Dividing by n − 1 rather than n corrects that systematic underestimate.
Can standard deviation be negative?+
No. It is a square root of squared deviations, so it is always zero or positive. A standard deviation of zero means every value in the dataset is identical.