Mean, Median, Mode and Standard Deviation
Describe a list of numbers, with both the sample and the population standard deviation.
Runs entirely in your browser. Nothing is uploaded, logged or stored.
Mean
5
- Median
- 4.5
- Mode
- 4
- Standard deviation (sample, ÷ n−1)
- 2.13809
- Standard deviation (population, ÷ n)
- 2
- Variance (sample)
- 4.571429
- Variance (population)
- 4
- How many values
- 8
- Sum
- 40
- Smallest
- 2
- Largest
- 9
- Range
- 7
- First quartile (Q1)
- 4
- Third quartile (Q3)
- 6
- Interquartile range
- 2
Paste a column of numbers and get the whole description at once: where the middle is, how spread out they are, and where the quarters fall.
How it works
Both standard deviations are shown, because they answer different questions and most calculators give one without saying which. Divide by n − 1 when the numbers are a sample of something larger and you want to estimate how much that larger thing varies. Divide by n when the numbers are the whole of what you care about. The sample figure is always the larger of the two.
Quartiles are placed by the method of Moore and McCabe, which is what most introductory courses teach: the median splits the list in two, and each quartile is the median of its half, with the overall median belonging to neither half when the count is odd. Other methods exist — statistical software offers as many as nine — and they can disagree by a noticeable amount on a short list.
The mode is every value that occurs as often as the most frequent one. A list where nothing repeats has no mode at all, which is more useful to be told than to be handed the first value.
Examples
| Case | Input | Result |
|---|---|---|
| A short list | 2 4 4 4 5 5 7 9 | mean 5, median 4.5, mode 4, population σ 2, sample s 2.138 |
| Two values tie for most frequent | 1 1 2 2 3 | mode 1 and 2 |
| Nothing repeats | 1 2 3 4 | no mode |
Frequently asked questions
Which standard deviation should I use?
The sample one, divided by n − 1, in almost every practical case: you nearly always have a sample and want to say something about the population behind it. Use the population figure only when your numbers are literally everything — every pupil in the class, every day of the month — and you are describing that set and nothing beyond it.
Why is my quartile different from my calculator's?
Because there is no single definition of a quartile. This page uses the method taught in most introductory courses; spreadsheets and statistical packages often use one that interpolates between values instead, and the two agree on long lists and disagree on short ones. Neither is wrong — they are different conventions, and the fix is to say which you used.
Can I paste a column from a spreadsheet?
Yes. Commas, spaces, semicolons, tabs and newlines all separate values, so a pasted column works as it is.
Why does it refuse a decimal comma?
Because the comma is already a separator here, so "1,5" would be one number or two depending on a guess. Rather than guess, the page asks for a point. Your spreadsheet can be told to export that way.
Good to know
- Up to ten thousand values. Beyond that the page refuses with a sentence rather than timing out, which tells you more.
Sources
- Moore DS, McCabe GP, Craig BA. Introduction to the Practice of Statistics. New York: W. H. Freeman — the quartile convention used here.