A whole data set, summed up
A list of numbers tells you little until you summarise it. Where is the middle? How spread out are the values? Which value turns up most? This statistics calculator answers all of that in one go: count, sum, mean, median, mode, lowest, highest, range, variance, standard deviation and the standard error of the mean.
It is built for coursework, lab results, survey answers, sales figures or test scores. Paste a column straight from a spreadsheet, and the numbers never leave your browser.
How to use it
- Type or paste your numbers into the box, separated by commas, spaces or new lines.
- Choose whether they are a sample or the whole population.
- Read the standard deviation at the top, and the other statistics underneath.
- Open the working to see the mean, the squared distances and the variance step by step.
Use a point for decimals (2.5, not 2,5), because commas separate the numbers.
The formulas
mean = Σx ÷ n
sample variance s² = Σ(x − mean)² ÷ (n − 1)
population variance σ² = Σ(x − mean)² ÷ N
standard deviation = √variance; standard error = s ÷ √nThe median is the middle value once the list is sorted, or the average of the two middle values when the count is even. The mode is the value that appears most often; a list can have several modes, or none if nothing repeats. The range is the highest value minus the lowest.
Worked example: 2, 4, 4, 4, 5, 5, 7, 9
| Value | Value − mean | Squared |
|---|---|---|
| 2 | −3 | 9 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 4 | −1 | 1 |
| 5 | 0 | 0 |
| 5 | 0 | 0 |
| 7 | 2 | 4 |
| 9 | 4 | 16 |
- Sum 40, count 8, so the mean is 5; the median is (4 + 5) ÷ 2 = 4.5; the mode is 4
- Squared distances add up to 32
- Sample variance: 32 ÷ 7 = 4.5714; sample standard deviation: √4.5714 = 2.1381
- Population variance: 32 ÷ 8 = 4; population standard deviation: 2
Treated as a sample, the standard deviation is 2.1381. Treated as a whole population, it is exactly 2. The gap shrinks as the list gets longer.
Sample or population?
Use population when your list holds every member of the group you care about, such as the marks of every student in one class. Use sample when your list is a selection that stands for something larger, such as 50 customers out of thousands. Dividing by n − 1 instead of n (Bessel’s correction) makes the sample figure a fairer estimate of the larger group, which would otherwise come out too low.
For a quick middle value only, the average calculator is simpler. For quartiles and box plots, see the quartile and outlier calculator.
Frequently asked questions
How do I calculate standard deviation?
Find the mean, take each value’s distance from the mean, square those distances and add them up. Divide by n − 1 for a sample, or by n for a whole population, to get the variance. The standard deviation is the square root of the variance. The calculator shows each of these steps.
What is the difference between variance and standard deviation?
Variance is the average squared distance from the mean, so it is in squared units, such as square centimetres. Standard deviation is its square root, which brings it back to the original units. That makes standard deviation easier to read: it says roughly how far a typical value sits from the mean.
Should I use sample or population standard deviation?
Use sample when your numbers are a selection from a bigger group you want to describe, which is most real data. Use population only when you have every single member of the group. When unsure, choose sample; it is slightly larger and is the safer choice for estimates.
What is the standard error of the mean?
It is the sample standard deviation divided by the square root of the count. It tells you how much the mean itself would likely vary if you took another sample of the same size. More data gives a smaller standard error, which is why bigger samples give steadier averages.
What if my list has no mode?
If no value appears more than once, there is no mode, and the calculator says so. If two or more values tie for the most appearances, all of them are modes. Mode is most useful for repeated values, such as shoe sizes or survey ratings, and less useful for precise measurements.
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