Measure Data Spread with Standard Deviation
Standard deviation tells you how far your values sit from the mean. This calculator works out both sample (s) and population (σ) at the same time and shows the full step-by-step working, so you can check every stage of the math. Everything runs in your browser.
Where Standard Deviation Is Used
Lab & Research
Finance & Risk
Quality Control
How to Calculate Standard Deviation
Enter Your Numbers
Type or paste your values into the input, separated by commas, spaces, semicolons, tabs, or new lines. A live counter shows how many numbers were read.
Pick Sample or Population
Choose Sample (n − 1) when your data is a subset of a larger group, or Population (n) when you have every data point. Sample is selected by default.
Show the Steps
Click Show Steps to reveal the mean, each squared difference, the variance, and the final square root that produces the answer.
Read the Summary
The summary panel lists 16 related statistics at once — mean, median, mode, variance, quartiles, range, min, max, and more.
The Standard Deviation Formulas
| Dataset | Population (σ) | Sample (s) |
|---|---|---|
2, 4, 4, 4, 5, 5, 7, 9 | σ = 2 | s ≈ 2.14 |
10, 10, 10, 10 | σ = 0 | s = 0 |
1, 100 | σ = 49.5 | s = 70 |
Features
Sample & Population
Reports both s (n − 1) and σ (n) so you always have the right measure at hand.
Step-by-Step Solution
Shows the mean, every squared difference, the variance, and the final square root.
Flexible Input
Accepts numbers separated by commas, spaces, semicolons, tabs, or new lines.
Adjustable Precision
Round the result to 2, 4, 6, or 8 decimal places (4 by default).
Full Statistics Summary
16 statistics computed at once — mean, median, mode, variance, quartiles, range, and more.
Private by Design
All math runs locally in your browser — your data never leaves your device.
Frequently Asked Questions
What is the difference between sample and population standard deviation?
Population standard deviation (σ) divides by n and describes a group you have measured in full. Sample standard deviation (s) divides by n − 1 and estimates the spread of a larger population from a subset of it. This tool shows both.
When should I use n − 1 instead of n?
Use n − 1 (sample) whenever your data is a subset used to generalize beyond it — for example, surveying 50 students from a school of 2,000. Use n (population) only when the dataset already includes every member of the group you are describing.
Why does sample standard deviation divide by n − 1?
This is Bessel's correction. A sample's values naturally cluster closer to their own mean than to the true population mean, so dividing by n would underestimate the real spread. Dividing by n − 1 corrects that bias and gives an unbiased estimate.
How is standard deviation related to variance?
Standard deviation is the square root of the variance. Variance is expressed in squared units, while taking the square root returns the value to the original units of the data, which makes it easier to interpret. The summary reports both.
What does a standard deviation of 0 mean?
A standard deviation of 0 means there is no spread at all — every value in the dataset is identical, so each one equals the mean. Both σ and s are 0 in that case.
How do I enter my data?
Type or paste your numbers into the input box, separated by commas, spaces, semicolons, tabs, or new lines. You can mix separators freely, and a live counter confirms how many values were read before you calculate.
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