What is the difference between sample and population standard deviation?
Use sample standard deviation (dividing by n - 1) when your data represents a sample drawn from a larger group. Use population standard deviation (dividing by N) only when you have measured every single member of the entire population.
Why does sample standard deviation divide by (n - 1) instead of n?
This is Bessel's correction. Because the sample mean is used instead of the true unknown population mean, dividing by n underestimates true variance. Dividing by (n - 1) produces an unbiased estimator.
What does the Empirical Rule (68-95-99.7 Rule) state?
For a normal distribution: roughly 68% of data falls within 1 standard deviation of the mean (μ ± 1σ), 95% falls within 2 standard deviations (μ ± 2σ), and 99.7% falls within 3 standard deviations (μ ± 3σ).
Can standard deviation be negative?
No. Because it is calculated by taking the principal square root of non-negative squared deviations, standard deviation is always ≥ 0.
What does a standard deviation of 0 mean?
A standard deviation of 0 means all values in the dataset are identical, indicating zero variability.
How accurate is the result?
Results use JavaScript's 64-bit floating point — accurate to 15+ significant digits for most calculations.
Can I enter decimal numbers?
Yes, decimals are fully supported.
Is it free?
Yes, completely free.
Does it show the formula?
Yes, the formula is displayed on the page.
Can I use it on mobile?
Yes, the calculator is fully responsive.
Does it save my inputs?
No data is stored — all calculations are done locally in your browser.
What if I get an error?
Check for invalid inputs such as division by zero or negative square roots.