Standard Deviation Calculator
Enter the values the standard deviation form asks for and the result is computed here, in the page, with no sign-up and no upload. Change one field at a time to see how sensitive the standard deviation figure is to that assumption.
Enter Data Values
Standard Deviation Formulas
| Measure | Formula | Description |
|---|---|---|
| Population Mean (μ) | μ = (Σxᵢ) / N | Sum of all values divided by count |
| Population Variance (σ²) | σ² = Σ(xᵢ - μ)² / N | Average squared deviation from mean |
| Population SD (σ) | σ = √(σ²) | Square root of population variance |
| Sample Mean (x̄) | x̄ = (Σxᵢ) / n | Same as population mean formula |
| Sample Variance (s²) | s² = Σ(xᵢ - x̄)² / (n-1) | Uses Bessel's correction (n-1) |
| Sample SD (s) | s = √(s²) | Square root of sample variance |
When to Use Each Type?
Population Standard Deviation (σ)
Use when you have data for the entire population. For example:
- All students in a class
- All products manufactured in a day
- Complete census data
Sample Standard Deviation (s)
Use when you have a sample from a larger population. For example:
- Survey responses from 100 people
- Quality control testing of random samples
- Research studies with limited participants
Note: Sample SD uses (n-1) instead of n to provide an unbiased estimate of the population variance.
Interpretation Guide
- Low Standard Deviation: Data points are close to the mean (less spread)
- High Standard Deviation: Data points are spread out over a wider range
- SD = 0: All values are identical
- 68-95-99.7 Rule: In normal distribution, ~68% of data falls within ±1 SD, ~95% within ±2 SD, ~99.7% within ±3 SD
Use the scientific calculator with steps
Population or sample standard deviation
Enter comma-separated values and choose whether the data is a full population or a sample. The page calculates the mean and standard deviation from the values entered. Choose the interpretation before comparing results with a textbook or spreadsheet.
Formula
Population variance divides the sum of squared deviations by N. Sample variance divides by N minus 1 to estimate the population variance from a sample. Standard deviation is the square root of the selected variance.
Worked example
For 2, 4, 4, 6, and 8, the mean is 4.8. The population and sample results differ because their divisors differ.
Frequently Asked Questions about Standard Deviation Calculator
How do I sanity-check the standard deviation result?
Rerun the calculator with a known example and compare it against the figure you expect; then swap back to your own numbers.
Which standard deviation inputs change the outcome most?
Adjust one field at a time. The figure that shifts the result hardest is the assumption worth double-checking before you rely on it.
Which option should I choose?
Choose population when the entered values are the complete group of interest. Choose sample when they represent a sample from a larger group.
Can a data set have zero standard deviation?
Yes. If every value is identical, every deviation from the mean is zero.
Is this a statistical test?
No. It is a descriptive summary of the values entered.