Quick answer: Statistics summarize a list of numbers: the mean is the sum divided by the count, the median is the middle value, the mode is the most common value, and the standard deviation measures how spread out the numbers are. For 2, 4, 4, 4, 5, 5, 7, 9 the mean is 5, the median 4.5, the mode 4 and the population standard deviation 2.
When you have a list of numbers, such as test scores, prices, measurements or survey answers, a few figures describe them: the average, the middle value, the most common value and how spread out they are. This calculator works them all out at once as soon as you paste your data.
You can separate numbers with commas, spaces or new lines, so a column copied from a spreadsheet works. Choose sample or population for the standard deviation, depending on whether your numbers are all of the group or just a sample of it.
What each figure means
- Mean: the sum divided by the count. It is the everyday “average”.
- Median: the middle value when the numbers are sorted. With an even count it is the average of the two middle values. It is less affected by extreme values than the mean.
- Mode: the value that appears most often. There can be more than one, or none.
- Range: the largest value minus the smallest.
- Variance and standard deviation: how far values are from the mean on average. The standard deviation is the square root of the variance.
- Quartiles: Q1 has 25% of values below it and Q3 has 75% below it. The gap between them is the interquartile range.
Population variance = Σ(x − mean)² ÷ n
Sample variance = Σ(x − mean)² ÷ (n − 1)
Standard deviation = √variance
Use the sample standard deviation when your data is a sample taken from a bigger group (most real cases). Use population only when you have every value in the group. Quartiles are calculated by linear interpolation, the same as Excel’s QUARTILE.INC. Other methods can give slightly different quartiles.
A worked example
Example. For the numbers 2, 4, 4, 4, 5, 5, 7, 9: the mean is 5, the median is 4.5, the mode is 4, the range is 7, the population standard deviation is 2 and the sample standard deviation is 2.14. The quartiles are Q1 = 4 and Q3 = 5.5.
A worked data set
| Measure | Value for 2, 4, 4, 4, 5, 5, 7, 9 | How it is found |
|---|---|---|
| Count | 8 | How many numbers |
| Sum | 40 | Add them all |
| Mean | 5 | 40 ÷ 8 |
| Median | 4.5 | Average of the 4th and 5th values (4 and 5) |
| Mode | 4 | Appears three times |
| Range | 7 | 9 − 2 |
| Population standard deviation | 2 | Square root of the average squared distance from the mean |
| Sample standard deviation | 2.14 | Same, but divided by n − 1 |
| Quartiles (Q1, Q3) | 4 and 5.5 | Linear interpolation, as in Excel’s QUARTILE.INC |
Quartile methods differ between programs, so other tools can give slightly different Q1 and Q3.
Common mistakes to avoid
- Using population standard deviation on a sample. When your numbers are a sample from a larger group, divide by n − 1 (sample), not n.
- Reporting the mean of skewed data. House prices and incomes have a few very large values that pull the mean up, so the median is often more meaningful.
- Writing decimals with a comma. Here a comma separates numbers, so write 2.5, not 2,5.
Frequently asked questions
What is the difference between sample and population standard deviation?
Population divides by n and is used when you have the whole group. Sample divides by n − 1, which corrects for the fact that a sample tends to underestimate the spread of the whole group. When unsure, use sample.
When should I use the median instead of the mean?
Use the median when there are extreme values, such as house prices or incomes, because a few very large numbers pull the mean up but barely move the median.
Can a data set have no mode?
Yes. If every value appears only once, there is no mode. It can also have several modes if different values tie for most frequent.
How many numbers can I enter?
Up to 100,000. Separate them with commas, spaces or new lines. Anything that is not a number is flagged so you can fix it.
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Sources and further reading
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