This standard deviation calculator measures how spread out a list of numbers is. Enter comma separated values and choose sample or population mode. Sample mode divides by n-1. Population mode divides by n.
Students and analysts use it for labs, quality checks, and basic stats homework. Pair it with average style summaries when you need center and spread together, such as a mean median mode range calculator when available.
How the math works
Find the mean. Sum (each value – mean)². Divide by n-1 (sample) or n (population). Take the square root. Larger results mean wider spread around the mean.
Worked example
Values 10, 12, 15, 18, 20. Mean = 15. Sample standard deviation ≈ 4.12. Population standard deviation ≈ 3.69.
| Input | Value |
|---|---|
| Values | 10, 12, 15, 18, 20 |
| Mode | Sample |
| Standard deviation | ~4.12 |
How to use the fields
- Values should be plain numbers separated by commas or spaces.
- Mode selects sample (n-1) or population (n).
Sample versus population
Sample SD is the usual classroom choice when your data are a subset. Population SD fits when you truly have every member of the group you care about. Switching modes on a small list changes the result noticeably.
Reading the result
Standard deviation shares the unit of the data. If values are test scores, SD is in score points. A small SD means values cluster near the mean. A large SD means wider scatter.
Common mistakes
- Using population mode on a small sample by accident
- Including text or empty tokens in the list
- Comparing SDs from differently scaled variables
- Treating SD as a percent without converting
Practical tips
- Remove clear data entry errors before you compute.
- Report n beside SD so readers know sample size.
- Use charts when outliers dominate the square terms.
Outliers inflate SD because deviations are squared. If one bad reading drives the result, investigate the point before you rewrite a process.
Worked population contrast
On the default list 10, 12, 15, 18, 20, sample SD is about 4.12 while population SD is about 3.69. The same squared deviations produce a smaller population value because the divisor is larger. State the mode whenever you report SD.
Classroom reporting
Include units, n, and mode in one sentence with the number. Example style: sample SD was 4.12 score points for n = 5. That sentence prevents later misreading as a percent or as a population value.
Outliers and data cleaning
Because deviations are squared, one extreme value can dominate. Investigate outliers for entry errors before deleting them. If the point is real, consider reporting SD with and without it and explaining why.
Do not confuse standard deviation with standard error. Standard error involves sample size in a different way and answers a different question about mean precision.
Process control glimpse
Shops tracking a measurement over time watch whether SD shrinks after a process change. Keep the measurement method fixed so SD changes reflect the process, not the meter.
For tiny n, SD is unstable. Collect more points before making expensive decisions from a two or three value list.
Comparing two groups
When two classes have similar means but different SD values, the wider class has more score spread. Report both mean and SD so readers do not assume identical distributions.
Do not average two SD values from different studies without a proper statistical method. Recalculate from raw values when you can.
Spreadsheet hygiene
Paste the same list into this tool and into your spreadsheet SD function and confirm the mode matches (sample versus population). Function names differ across sheet apps and silently switch divisors.
Interpretation limits
SD assumes you care about spread around the mean. Skewed data can share an SD with a symmetric set and still look very different on a histogram. Plot when stakes are high.
For grades, a high SD can mean a wide mix of preparation levels. For manufacturing, a high SD can mean a process that needs tighter control. Context decides whether high SD is a problem.
Rounding
Round SD in a way that matches measurement precision. Reporting many digits from this calculator does not increase the precision of a coarse meter.
Limitations
The tool does not compute confidence intervals, z scores, or hypothesis tests. It returns standard deviation for the list and mode you enter.