How to use it
- Paste or type your numbers into Numbers. Commas, semicolons, spaces and new lines all separate them.
- Under Data is, choose the sample option when the numbers are part of a larger group, or the whole population option when they are everything you have.
- Read the Mean (average), the Median (middle value) and the spread: the Variance and the Standard deviation.
- Check the Outliers and the frequency table, and follow How the mean and the spread are worked out to see each step.
How the mean and the spread are calculated
The mean is the sum of the values divided by their count. The variance is the average of the squared distances from the mean, and the standard deviation is its square root. The calculator first finds the mean and then sums the squared differences from it, the two-pass method, which avoids the rounding errors of the shorter one-pass formula.
A sample divides by n − 1 instead of n. This is Bessel's correction, which makes the sample variance an unbiased estimate of the variance of the whole population. If your numbers are all there is, choose the population option.
mean = Σx ÷ n
sample variance = Σ(x − mean)² ÷ (n − 1)
population variance = Σ(x − mean)² ÷ n
standard deviation = √varianceHow the quartiles are calculated
The quartiles interpolate linearly between the sorted values. This is the method of R's default quantile type 7 and of Excel's QUARTILE.INC. The position of a quartile is h = (n − 1) × p, counting from 0 at the smallest value; if h is not whole, the quartile lies between the two values either side of it, in proportion to the fraction.
The median is the middle value, or the mean of the two middle values when the count is even. Other software uses other rules, such as Tukey's hinges or QUARTILE.EXC, and gives slightly different quartiles for the same numbers.
What this calculator does not cover
- Outliers use the 1.5 × IQR rule. It is a rule of thumb: an outlier is not an error by itself, but it is worth checking.
- The calculator does not test significance, fit a distribution or use weights. For weighted values, use the weighted average tool.
- A text entry stops the calculation, and the message shows the first entry it cannot read as a number.
Worked example: 2, 4, 4, 4, 5, 5, 7, 9
The sum is 40 and the count is 8, so the mean is 5. The squared differences from 5 add up to 32. Divided by 8 that gives a population variance of 4 and a standard deviation of 2; divided by 7 it gives a sample variance of 4.571429 and a sample standard deviation of 2.13809.
The middle values are 4 and 5, so the median is 4.5. The most frequent value is 4. The quartiles are 4 and 5.5, the interquartile range is 1.5, and the fences are 1.75 and 7.75, so the value 9 is an outlier.
| Measure | Value |
|---|---|
| Mean | 5 |
| Median | 4.5 |
| Mode | 4 |
| Variance (population) | 4 |
| Variance (sample) | 4.571429 |
| Standard deviation (population) | 2 |
| Standard deviation (sample) | 2.13809 |
| First and third quartile | 4 and 5.5 |
| Outliers | 9 |
Frequently asked questions
What is the difference between a sample and a population variance?
A population variance divides the squared differences by n, the number of values, because you have every value. A sample variance divides by n − 1, which corrects for the tendency of a sample to understate the spread of the whole group.
Why do my quartiles differ from another calculator's?
Several methods exist for quartiles. This page uses linear interpolation, the method of R's default type 7 and Excel's QUARTILE.INC. Other methods, such as Tukey's hinges, can give different quartiles for the same numbers.
What counts as an outlier here?
A value below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR. This is the rule used for the whiskers of box plots. It flags unusual values; it does not say whether they are mistakes.
Can I paste numbers from a spreadsheet?
Yes. Copy a column and paste it into the box; the values are separated by new lines. Write numbers without thousands separators, so 1000 and not 1,000, because commas separate the values in English.
What does the mode show when no value repeats?
It says that there is no mode, because every value occurs once. When two or more values share the highest count, all of them are listed.
Are the numbers I enter sent to a server?
No. The calculation runs in your browser, and nothing you enter is sent to us.
Sources
- Wikipedia: Standard deviation, with Bessel's correction for the sample
- Wikipedia: Algorithms for calculating variance, the two-pass method
- R documentation: quantile(), the default method is type 7
- Microsoft: QUARTILE.INC function in Excel
- Wikipedia: Interquartile range and the 1.5 × IQR outlier rule
Figures and text last checked on Tuesday, November 10, 2026.