Standard Deviation Calculator
Sample or population, with variance, mean and working
| # | Value x | x − mean | (x − mean)² |
|---|---|---|---|
| 1 | 12 | -12.7 | 161.29 |
| 2 | 15 | -9.7 | 94.09 |
| 3 | 18 | -6.7 | 44.89 |
| 4 | 22 | -2.7 | 7.29 |
| 5 | 24 | -0.7 | 0.49 |
| 6 | 25 | 0.3 | 0.09 |
| 7 | 29 | 4.3 | 18.49 |
| 8 | 30 | 5.3 | 28.09 |
| 9 | 34 | 9.3 | 86.49 |
| 10 | 38 | 13.3 | 176.89 |
| Total | 247 | 0 | 618.1 |
Standard Deviation Calculator: Sample and Population Statistics
This free standard deviation calculator finds the standard deviation, variance, mean, median, range and more for any list of numbers. Paste your data separated by commas, spaces or new lines, choose sample or population, and the answer appears as you type. Under the result you get the working step by step, a table of each value's distance from the mean, a chart that shows the spread, and plain-language notes about what the numbers mean, including possible outliers.
Quick answer: for the values 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the population standard deviation is 2 and the sample standard deviation is 2.138. Use the population figure when your data covers everyone or everything in the group, and the sample figure when it is only part of a larger group.
What Is Standard Deviation?
Standard deviation tells you how far values typically sit from their average. If every value is close to the mean, the standard deviation is small, and if they are spread widely, it is large. Because it is in the same units as your data, it is easy to read: a standard deviation of 3 cm means measurements usually differ from the mean by about 3 cm. Test scores, product measurements, daily temperatures and investment returns are all described this way.
Standard Deviation Formulas
Here x̄ is the mean, n is how many values you have, and Σ means add them all up. The calculator shows each of these steps with your own numbers.
Sample or Population?
Use population when your list contains every member of the group you care about, such as the scores of all 30 students in one class. Use sample when your list is only part of a larger group, such as 30 people surveyed out of a whole city. The sample version divides by n − 1 instead of n, which corrects for the fact that a sample usually understates the true spread. When in doubt, sample is the usual choice in research and coursework.
Examples
| Data set | Values (n) | Mean | Population σ | Sample s |
|---|---|---|---|---|
| Classic 8 values | 8 | 5 | 2 | 2.14 |
| Exam scores | 10 | 24.7 | 7.86 | 8.29 |
| Measurements | 10 | 5.16 | 0.57 | 0.6 |
| Even steps | 11 | 125 | 15.81 | 16.58 |
| Temperatures | 10 | 21.66 | 2 | 2.11 |
For the exam scores, the mean is 24.7, the sample variance is 68.68 and the sample standard deviation is 8.29. The two columns are close here because there are ten values, but the gap grows for small data sets.
How to Read the Results
In a bell-shaped distribution, about 68% of values fall within one standard deviation of the mean, 95% within two and 99.7% within three. The calculator counts how many of your values fall within one and two standard deviations so you can compare your data with that pattern. It also lists possible outliers using the interquartile range rule. To see the middle values of a data set, use the mean median mode calculator, and for percentages of totals the percentage calculator is a quick companion.
Limits and Things to Know
The calculator needs at least two numbers for a sample and one for a population. A single extreme value can raise the standard deviation a lot, so check the outlier note. The 68-95-99.7 pattern applies to normal distributions and may not describe skewed data. Enter raw values rather than grouped data, and repeat a value if it occurs more than once. Entries that are not numbers are ignored and listed above the results.
Who Can Use This Calculator?
Students and teachers use it to check statistics homework and to see the method behind an answer. Researchers, analysts and quality engineers use it to describe variation in measurements, and teachers use it to explain grades. Investors and sports analysts use it to compare consistency, since a lower value means more predictable results.
How to Use This Standard Deviation Calculator
Enter your data
Type or paste your numbers separated by commas, spaces or new lines, or tap an example to load one.
Choose sample or population
Pick Sample (n − 1) if your data is part of a larger group, or Population (n) if it covers the whole group.
Set the decimals
Choose how many decimal places to show. Add a value in the z-score box to see how far it is from the mean.
Read the result and working
See the standard deviation, variance, mean, chart, step-by-step working and notes on spread and outliers.