How to use it
- Paste or type your numbers, separated by commas, spaces or line breaks. Use a point for decimals (2.5).
- Everything is calculated as you type: count, sum, mean, median, mode, range, variance, standard deviation, quartiles and outliers.
- The address bar holds the list, so a link brings the same numbers back. Up to 2,000 numbers.
What is calculated
| Measure | How |
|---|---|
| Mean | the sum divided by n |
| Median | the middle value of the sorted list, or the mean of the two middle values |
| Mode | the value or values that occur most often; none if every value is different |
| Range | largest minus smallest |
| Sample variance s² | the sum of squared deviations from the mean, divided by n − 1 |
| Population variance σ² | the same sum, divided by n |
| Standard deviation | the square root of the variance |
| Standard error | s divided by √n |
Use the sample versions when the numbers are a sample from a larger group, the population versions when they are the whole group.
Quartiles and outliers
The first quartile Q1 has a quarter of the values below it, the third quartile Q3 three quarters. The interquartile range is IQR = Q3 − Q1. A value is flagged as an outlier when it lies below Q1 − 1.5·IQR or above Q3 + 1.5·IQR (Tukey’s fences). The whiskers of the box plot end at the most extreme values inside the fences.
Books disagree on how to find the quartiles, so there are two methods. The default interpolates, as Excel’s QUARTILE.INC, R and NumPy do. The other takes the median of the lower half and of the upper half, leaving the median out when n is odd.
Worked example
For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the population standard deviation is exactly 2, the sample variance is 32/7 ≈ 4.571 and the mode is 4.
Common questions
Sample or population standard deviation?
Use the sample version (divide by n − 1) when your numbers are a sample from something larger, which is the usual case. Use the population version (divide by n) when you have every value.
Why are my quartiles different from my textbook?
There is more than one accepted method. Switch the quartile method to match your book.
For tests of whether a result is significant, see the p-value calculator.