Standard Deviation Calculator
Calculate the mean, variance, and standard deviation of any list of numbers.
Separate values with commas, spaces, or new lines.
Enter a list of numbers to see the mean and standard deviation.
How to use the standard deviation calculator
Paste or type your numbers, separated by commas, spaces, or new lines. The calculator reports the count, the mean, the sum, and both the sample and population standard deviation as you type.
It shows the variance too, since the standard deviation is just its square root. Seeing both makes the steps clearer when you are checking homework or a spreadsheet.
Sample vs population standard deviation
The two formulas differ in the divisor. Population standard deviation divides by n, the full count. Sample standard deviation divides by n minus 1, which corrects for measuring only part of a larger group.
Use the population formula when your list is the entire group. Use the sample formula when it is a sample meant to represent something bigger. The sample value is always a little larger.
A worked example
Take the set 2, 4, 4, 4, 5, 5, 7, 9. The mean is 5. The squared distances from the mean add up to 32.
Divide 32 by 8 and the population variance is 4, so the population standard deviation is 2. Divide instead by 7 and the sample variance is about 4.57, giving a sample standard deviation near 2.14.
Frequently asked questions
- What is standard deviation?
- It measures how spread out numbers are from their mean. A small value means the data clusters near the average; a large one means it is widely scattered.
- Should I use sample or population standard deviation?
- Use population when your numbers are the whole group. Use sample when they are a subset of something larger. The sample formula divides by n minus 1 and gives a slightly larger result.
- What is the difference between variance and standard deviation?
- Variance is the average of the squared distances from the mean. Standard deviation is its square root, which brings the figure back to the original units.
- How many numbers do I need?
- At least two. A single value has no spread, and the sample formula divides by n minus 1, which would be zero for one number. Two or more gives a meaningful result.