Standard Deviation Calculator
Standard Deviation Calculator
Enter a list of numbers and choose whether it's a sample or the full population. This calculates the standard deviation — the most widely used measure of how spread out the values are around the mean.
The sample formula divides by n−1 (known as Bessel's correction) and should be used whenever your data is a sample drawn from a larger population — which describes most real-world datasets, from a survey of some customers to a batch of test measurements. The population formula divides by n instead, and is only correct when your dataset genuinely contains every member of the group you're studying, with nothing left out. Using the wrong one is a common source of small but real errors in statistical reporting.
- Sample formula: Standard Deviation = √(Σ(x − mean)² ÷ (n − 1)) — divides by n−1 (Bessel's correction), used when your data is a sample drawn from a larger population.
- Population formula: Standard Deviation = √(Σ(x − mean)² ÷ n) — divides by n, used only when your dataset IS the entire population you care about.
- Interpreting the result: a small standard deviation means values cluster tightly around the mean; a large one means they're spread widely — it's in the same units as your original data.
Should I use sample or population standard deviation?
Use sample (n−1) almost always in practice — it's the correct choice whenever your data represents a subset of a larger group you're trying to describe, which is the overwhelmingly common case; use population (n) only when your dataset genuinely contains every member of the group.
Why does the sample formula divide by n−1 instead of n?
Dividing by n tends to underestimate the true population variance when working from a sample — Bessel's correction (n−1) adjusts for this bias, giving a more accurate estimate.
Standard Deviation Calculator
Enter a list of numbers and choose whether it's a sample or the full population. This calculates the standard deviation — the most widely used measure of how spread out the values are around the mean.
The sample formula divides by n−1 (known as Bessel's correction) and should be used whenever your data is a sample drawn from a larger population — which describes most real-world datasets, from a survey of some customers to a batch of test measurements. The population formula divides by n instead, and is only correct when your dataset genuinely contains every member of the group you're studying, with nothing left out. Using the wrong one is a common source of small but real errors in statistical reporting.
- Sample formula: Standard Deviation = √(Σ(x − mean)² ÷ (n − 1)) — divides by n−1 (Bessel's correction), used when your data is a sample drawn from a larger population.
- Population formula: Standard Deviation = √(Σ(x − mean)² ÷ n) — divides by n, used only when your dataset IS the entire population you care about.
- Interpreting the result: a small standard deviation means values cluster tightly around the mean; a large one means they're spread widely — it's in the same units as your original data.
Should I use sample or population standard deviation?
Use sample (n−1) almost always in practice — it's the correct choice whenever your data represents a subset of a larger group you're trying to describe, which is the overwhelmingly common case; use population (n) only when your dataset genuinely contains every member of the group.
Why does the sample formula divide by n−1 instead of n?
Dividing by n tends to underestimate the true population variance when working from a sample — Bessel's correction (n−1) adjusts for this bias, giving a more accurate estimate.