Standard Deviation Calculator
Find the standard deviation and variance of any data set. Choose population or sample, enter your values, and see the mean, variance and every calculation step.
Standard Deviation Calculator
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Step-by-Step Solution
What Is Standard Deviation?
Standard deviation measures how spread out the numbers in a data set are. A small standard deviation means the values are clustered close to the mean. A large standard deviation means the values are spread far from the mean.
It is written in the same units as your data, which makes it easier to interpret than variance. For example, if you measure heights in centimeters, the standard deviation is also in centimeters.
To see how precisely a sample mean estimates the true mean, divide the standard deviation by √n. Our Standard Error Calculator does this for you.
Our calculator applies the correct formula for you and shows the work behind the answer.
How to Calculate Standard Deviation
You can calculate standard deviation by hand in five steps.
1. Find the mean
Add all the values and divide by how many there are.
2. Subtract the mean from each value
This gives the deviation of every value from the mean.
3. Square each deviation
Squaring makes every number positive, so deviations above and below the mean do not cancel out.
4. Find the variance
Add up the squared deviations. Divide by N for a population, or by n − 1 for a sample.
5. Take the square root
The square root of the variance is the standard deviation.
Standard Deviation Calculator Example
Find the standard deviation of these eight numbers:
The sum is 40 and there are 8 values, so the mean is 40 ÷ 8 = 5.
Now square each value's distance from the mean:
Sum of squares = 32
Population: variance = 32 ÷ 8 = 4, so the standard deviation is √4 = 2.
Sample: variance = 32 ÷ 7 ≈ 4.5714, so the standard deviation is √4.5714 ≈ 2.1381.
Population vs. Sample Standard Deviation
Use the population formula when your data contains every member of the group you care about, such as the scores of all students in one class. Divide by N.
Use the sample formula when your data is only a part of a larger group, such as a survey of 100 people from a whole city. Divide by n − 1. This small change corrects for the fact that a sample usually underestimates the spread of the full group.
If you are not sure which to choose, the sample option is the usual choice for most homework and real-world data.
More Standard Deviation Examples
These extra worked examples cover a very common homework set, a full step-by-step calculation, data with no spread, negative numbers, two data sets with the same mean, and what a standard deviation tells you in practice. Remember that the sample formula divides by n − 1 and the population formula divides by N.
Example 2: Standard deviation of 1, 2, 3, 4, 5
Answer: the sample standard deviation is about 1.5811, and the population standard deviation is about 1.4142.
The mean is (1 + 2 + 3 + 4 + 5) ÷ 5 = 3. The squared distances from the mean are 4, 1, 0, 1 and 4, which add up to 10. For a sample, the variance is 10 ÷ 4 = 2.5, and the standard deviation is √2.5 ≈ 1.5811. For a population, the variance is 10 ÷ 5 = 2, and the standard deviation is √2 ≈ 1.4142.
Example 3: Step-by-step calculation for 5, 8, 12, 15, 20
Answer: the sample standard deviation is about 5.8737, and the population standard deviation is about 5.2536.
Step 1: find the mean. The sum is 5 + 8 + 12 + 15 + 20 = 60, and there are 5 values, so the mean is 60 ÷ 5 = 12.
Step 2: subtract the mean from each value. The deviations are −7, −4, 0, 3 and 8.
Step 3: square each deviation. The squares are 49, 16, 0, 9 and 64, and their sum is 138.
Step 4: find the variance. Sample: 138 ÷ 4 = 34.5. Population: 138 ÷ 5 = 27.6.
Step 5: take the square root. Sample: √34.5 ≈ 5.8737. Population: √27.6 ≈ 5.2536.
Example 4: What if every value is the same?
Find the standard deviation of 7, 7, 7, 7.
Answer: the standard deviation is 0.
The mean is 7, and every value is exactly 7, so every deviation is 0. The sum of squares is 0, so the variance is 0 and the standard deviation is 0. A standard deviation of 0 means there is no spread at all.
Example 5: Negative numbers
Find the standard deviation of −2, 0, 2.
Answer: the sample standard deviation is 2, and the population standard deviation is about 1.6330.
The mean is (−2 + 0 + 2) ÷ 3 = 0. The squared deviations are 4, 0 and 4, which add up to 8. Sample: the variance is 8 ÷ 2 = 4, so the standard deviation is 2. Population: the variance is 8 ÷ 3 ≈ 2.6667, so the standard deviation is √2.6667 ≈ 1.6330. Negative values work exactly like positive ones, because every deviation is squared.
Example 6: Same mean, different spread
Set A is 48, 49, 50, 51, 52. Set B is 30, 40, 50, 60, 70. Both have a mean of 50. Which one is more spread out?
Answer: Set B is much more spread out.
For Set A, the squared deviations are 4, 1, 0, 1 and 4, which add up to 10. The sample variance is 10 ÷ 4 = 2.5, so the standard deviation is about 1.5811. For Set B, the squared deviations are 400, 100, 0, 100 and 400, which add up to 1,000. The sample variance is 1,000 ÷ 4 = 250, so the standard deviation is about 15.8114. The averages are equal, but the standard deviation shows that Set B is about ten times more spread out.
Example 7: What does a standard deviation tell you in practice?
A test has a mean score of 100 and a standard deviation of 15. For data that is roughly bell-shaped, about 68% of scores fall within one standard deviation of the mean, which is 85 to 115. About 95% fall within two standard deviations, which is 70 to 130. About 99.7% fall within three standard deviations, which is 55 to 145. To see how far one value is from the mean in standard deviations, use our Z Score Calculator.
Common mistakes when finding standard deviation
A common mistake is dividing by n when your data is a sample, instead of n − 1. Another is stopping at the variance and forgetting to take the square root. Also be careful not to mix up the sample and population formulas in the same problem. If you only need the variance, try our Variance Calculator.
Created by Sajid Ali, SmartStatCalc · Last updated: 7 November 2026
Found a mistake or have a suggestion? Contact us.
How to Use Our Standard Deviation Calculator
- Type or paste your numbers into the Data Values box.
- Choose Sample or Population in the Data Type menu.
- Click the Calculate Standard Deviation button.
- Read the count, mean, variance and standard deviation.
- Use the step-by-step solution to check each stage of the calculation.
Click Reset at any time to clear your data and start again. If you only need the mean, try our Average Calculator.
Why Use SmartStatCalc?
SmartStatCalc is built to make everyday statistics quick and easy to understand. Instead of squaring and adding every value by hand, you can enter your data and see a clear result right away.
Who Can Use a Standard Deviation Calculator?
Students use standard deviation in statistics, science and economics courses. Teachers use it to see how spread out class scores are. It is also used to describe the consistency of measurements, test results, prices, or any numbers where you want to know how much values vary around the average.
Frequently Asked Questions
What does a high standard deviation mean?
It means the values are spread far from the mean. A low standard deviation means the values are close to the mean.
What is the difference between variance and standard deviation?
Variance is the average of the squared deviations. Standard deviation is the square root of the variance, so it is in the same units as your data.
Should I use sample or population?
Use population if your data is the whole group. Use sample if it is only part of a larger group. When unsure, sample is the usual choice.
Why do we divide by n − 1 for a sample?
A sample tends to show less spread than the full group, so dividing by n − 1 instead of n corrects for this.
Can the standard deviation be negative?
No. It is a square root, so it is always zero or positive. It equals zero only when every value is the same.
How many numbers do I need?
Population needs at least one number. Sample needs at least two, because it divides by n − 1.
Is this Standard Deviation Calculator free?
Yes. SmartStatCalc's Standard Deviation Calculator is free to use online.
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SmartStatCalc. (2026). Standard Deviation Calculator. https://smartstatcalc.com/standard-deviation-calculator/