Standard deviation calculator
Sample and population standard deviation, variance, mean, median and mode from a list of numbers.
- ๐บ๐ธ US federal & state rules
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Sample standard deviation (s)
2.138089935
Mean 5, 8 values
- Mean
- 5
- Median
- 4.5
- Sample variance
- 4.571428571
- Mode
- 4 (3 times)
This is an example. Enter your numbers above and press the button to see your own result.
All statistics
| Count (n) | 8 |
|---|---|
| Sum | 40 |
| Mean | 5 |
| Median | 4.5 |
| Mode | 4 (3 times) |
| Min and max | 2 and 9 |
| Range | 7 |
| Population variance (ฯยฒ, รท n) | 4 |
| Sample variance (sยฒ, รท n-1) | 4.571428571 |
| Population SD (ฯ) | 2 |
| Sample SD (s) | 2.138089935 |
| Standard error of the mean | 0.755928946 |
| Q1 | 4 |
| Q3 | 5.5 |
| Interquartile range (IQR) | 1.5 |
| Coefficient of variation | 42.76179871% |
- Quartiles use the inclusive method (same as Excel QUARTILE.INC). Textbooks and calculators like the TI-84 may use a different method and give slightly different Q1 and Q3.
Show the math
| Sorted | 2, 4, 4, 4, 5, 5, 7, 9 |
|---|---|
| Mean = sum รท n | 40 รท 8 = 5 |
| Sum of squared deviations ฮฃ(x - mean)ยฒ | 32 |
| Variance = ฮฃ รท (n - 1) | 32 รท 7 = 4.571428571 |
| Standard deviation = โvariance | 2.138089935 |
What to do next
How it works
The mean is the sum divided by the count. Each value's distance from the mean is squared and the squares are added. Dividing that total by n gives the population variance, and dividing by n - 1 gives the sample variance. The standard deviation is the square root of the variance.
The median is the middle value after sorting, or the average of the two middle values. The mode is the value that appears most often. If every value appears the same number of times, there is no mode.
Things to keep in mind
Standard deviation is in the same units as your data. A coefficient of variation (SD divided by the mean) lets you compare spread between data sets with different units, but it is not meaningful when the data can be negative.
Sources
Frequently asked questions
What is the difference between sample and population standard deviation?
Population SD divides the squared deviations by n. Sample SD divides by n - 1, which corrects for estimating the mean from the same data. For 2, 4, 4, 4, 5, 5, 7, 9 the population SD is 2 and the sample SD is 2.138.
Which one should I use?
Use sample if your data is a subset of a larger group, like 30 survey answers or one class of students. Use population only when you have every value, like all 12 months of a year.
How do you calculate standard deviation by hand?
Find the mean, subtract it from each value, square each result, add them up, divide by n (or n - 1), then take the square root.
Why are my quartiles different from my textbook?
There are several quartile methods. This calculator uses the inclusive method that Excel QUARTILE.INC uses. Some textbooks and the TI-84 split the data in halves instead.