Statistics Calculator
Paste in a data set and get every standard summary measure at once - including the sample-versus-population distinction that most calculators leave you to guess at.
Last updated: August 2026 · Written and verified by Akash Pandey
Summarise a data set
Mean
This runs the same calculation engine as the main scientific calculator, entirely in your browser. Nothing you type is sent anywhere.
Measures of centre
Three different answers to "what is a typical value here?", each useful in different circumstances.
| Measure | Definition | Best when |
|---|---|---|
| Mean | sum ÷ count | Data is roughly symmetric with no extreme outliers |
| Median | the middle value when sorted | Data is skewed or contains outliers |
| Mode | the most frequent value | Data is categorical, or you want the most common case |
The difference matters more than it first appears. In a room of ten people earning £30,000 each, the mean and median salary are both £30,000. Add one person earning £1,000,000 and the mean jumps to about £118,000 while the median stays at £30,000. The median is the honest answer to "what does a typical person here earn?", which is why income statistics are almost always reported as medians.
mean (x̄) = (x₁ + x₂ + … + xₙ) ÷ n
- For the median, sort the values first. With an odd count, take the middle one; with an even count, average the two middle ones.
- A data set can have no mode, one mode, or several.
Measures of spread
Two data sets can share a mean and still look nothing alike. Spread measures how far the values sit from the centre.
variance = Σ(xᵢ − x̄)² ÷ N standard deviation = √variance
- Each deviation is squared so that positive and negative distances do not cancel out.
- Standard deviation takes the square root to return to the original units - a variance in "pounds squared" is not interpretable.
- N is the divisor for a population; a sample divides by n − 1 instead.
Sample or population? The n−1 question
This is the single most confusing part of introductory statistics, and it comes down to one question: is your data the whole group, or a sample drawn from a larger one?
- If you measured every member of the group you care about - the exam scores of all 30 students in a class, when the class is the entire population of interest - divide by N.
- If your data is a sample and you want to estimate the spread of the wider population it came from, divide by n − 1.
The reason is that a sample's own mean sits, by construction, in the middle of that sample - closer to its own points than the true population mean would be. So the squared deviations come out slightly too small, and dividing by n − 1 rather than n corrects the bias. This is called Bessel's correction. With large data sets the two barely differ; with small ones the gap is substantial, which is exactly when it matters. The calculator above reports both, so you never have to guess which one a tool gave you.
A worked example, start to finish
Take the data set 4, 8, 6, 5, 3.
- Mean:
4+8+6+5+3 = 26, and26 ÷ 5 = 5.2. - Median: sorted, the set is
3, 4, 5, 6, 8. Five values, so the middle one is the third:5. - Mode: every value appears once, so there is no mode.
- Range:
8 − 3 = 5. - Deviations from the mean:
−1.2, 2.8, 0.8, −0.2, −2.2. - Squared deviations:
1.44, 7.84, 0.64, 0.04, 4.84, summing to14.8. - Population variance:
14.8 ÷ 5 = 2.96, so σ =√2.96 ≈ 1.7205. - Sample variance:
14.8 ÷ 4 = 3.7, so s =√3.7 ≈ 1.9235.
How far is far? The empirical rule
For data that is roughly bell-shaped, standard deviation has a direct interpretation:
| Within | Contains about | Meaning |
|---|---|---|
| ±1 std dev | 68% of values | Ordinary |
| ±2 std dev | 95% of values | Notable |
| ±3 std dev | 99.7% of values | Rare |
So for a data set with mean 100 and standard deviation 15, roughly two thirds of values fall between 85 and 115, and a value above 145 would be a genuine outlier. This is precisely how IQ scores are constructed.
Frequently Asked Questions
What is the difference between sample and population standard deviation?
The divisor. Population divides the sum of squared deviations by N; sample divides by n − 1. Use the sample form when your data is a subset being used to estimate a larger group, which is the more common case in practice.
Why divide by n − 1 for a sample?
Because the sample mean sits in the middle of the sample by construction, so deviations measured from it are systematically a little too small. Dividing by n − 1 instead of n corrects that bias. It is called Bessel's correction.
When should I use the median instead of the mean?
When the data is skewed or has outliers. One extreme value drags the mean a long way but barely moves the median, which is why house prices and incomes are usually reported as medians.
Can a data set have more than one mode?
Yes. If two or more values tie for the highest frequency, all of them are modes - the set is bimodal or multimodal. If every value appears exactly once, there is no mode at all.
What is the difference between variance and standard deviation?
Standard deviation is the square root of variance. Variance is in squared units, which are hard to interpret; taking the root returns it to the units of the original data.
How do I enter my data?
Type or paste the numbers into the box, separated by commas, spaces or new lines. Any mixture works, so pasting a column from a spreadsheet is fine.
What does a high standard deviation tell me?
That the data points are spread out over a wider range. A low standard deviation means values cluster tightly around the mean. Neither is inherently good or bad - it depends on whether you expect consistency or variation.
Can I use this calculator for percentages or grades?
Yes, as long as you enter the raw numbers. For a set of test scores, enter each score and the calculator will give you the class average, the spread, and whether the results cluster around one value or are scattered.