Standard Deviation (SD) Calculator
Enter your numbers below (separated by commas, spaces, or new lines), choose population or sample SD, and click calculate.
What does SD mean?
SD stands for standard deviation. It measures how spread out a group of numbers is around the mean (average). A small SD means values are close to the average. A large SD means values are more spread out.
How do you calculate SD? (The basic formula)
There are two common formulas, depending on your data:
- Population standard deviation (σ): √[ Σ(x - μ)2 / N ]
- Sample standard deviation (s): √[ Σ(x - x̄)2 / (n - 1) ]
Use population SD when you have every value in the full group. Use sample SD when your data is just a sample from a larger population.
Step-by-step: calculate SD by hand
Suppose your data is: 2, 4, 4, 4, 5, 5, 7, 9
1) Find the mean
Add all values and divide by the count: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5.
2) Subtract the mean from each value
Deviations: -3, -1, -1, -1, 0, 0, 2, 4
3) Square each deviation
Squared deviations: 9, 1, 1, 1, 0, 0, 4, 16
4) Add the squared deviations
9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32
5) Divide and take the square root
- Population variance: 32 / 8 = 4, so population SD = √4 = 2
- Sample variance: 32 / 7 ≈ 4.5714, so sample SD ≈ 2.1381
Quick checklist for any SD problem
- Find the mean.
- Find each deviation from the mean.
- Square each deviation.
- Add all squared deviations.
- Divide by N (population) or n - 1 (sample).
- Take the square root.
Common mistakes to avoid
- Using n instead of n - 1 for sample SD.
- Forgetting to square the deviations.
- Rounding too early (round only near the end).
- Mixing up variance and SD (SD is the square root of variance).
When should you use SD?
Standard deviation is widely used in:
- Test score analysis
- Quality control in manufacturing
- Financial risk and return
- Scientific and social science research
Final takeaway
If you remember one thing, remember this: SD is the typical distance from the mean. Compute the mean, find squared differences, average them (with the right denominator), and take the square root. Use the calculator above to get instant results and check your manual work.