Average calculator: mean, median and mode
Paste a list and get all three, plus the range and the spread. Where they disagree the page says which one describes your numbers, because a mean dragged out of position by a single value is a true figure and a misleading answer.
By Alex Seote, Built and maintains Rule Calculator
Formulas checked against the NIST handbook · How we check
Commas, spaces, tabs or new lines. A column pasted straight out of a spreadsheet works.
Mean
14.142857
7 values, summing to 99.
| Mean | 14.142857 |
| Median | 14 |
| Mode | none — no value repeats |
| Range | 7 (11 to 18) |
| Standard deviation, sample | 2.4103 |
| Standard deviation, population | 2.2315 |
| Variance, sample | 5.8095 |
| Variance, population | 4.9796 |
| Standard error of the mean | 0.911 |
Which middle to quote
Use the mean, 14.142857. The mean and the median agree to within a quarter of a standard deviation and nothing falls outside the outlier fences, so the mean is a fair summary of this list.
The working, one value at a time
A standard deviation is the square root of the average squared distance from the mean. The distances are below; squaring them is what stops the ones above and below cancelling out.
| Value | − mean | squared |
|---|---|---|
| 12 | -2.1429 | 4.5918 |
| 15 | 0.8571 | 0.7347 |
| 11 | -3.1429 | 9.8776 |
| 18 | 3.8571 | 14.8776 |
| 14 | -0.1429 | 0.0204 |
| 16 | 1.8571 | 3.449 |
| 13 | -1.1429 | 1.3061 |
Sum of squares: 34.8571
Divided by n − 1 = 6: 5.8095 (variance)
Square root: 2.4103 (standard deviation)
Quartiles and outliers
| Minimum | 11 |
| Q1 | 12 |
| Median (Q2) | 14 |
| Q3 | 16 |
| Maximum | 18 |
| Interquartile range | 4 |
| Outlier fences | 6 and 22 |
| Outliers | none |
Q1 and Q3 above are medians of the halves either side of the median, which is the method US schools and the TI-83 use. Excel's QUARTILE.INC interpolates instead and gives 12.5 and 15.5 for this data. Neither is wrong — they are different conventions, and a calculator that prints one without naming it is why homework and spreadsheet disagree.
Three middles, and when they part company
On a symmetric list the mean, the median and the mode land in roughly the same place and the choice does not matter. On a skewed one they separate, and the gap between them is itself the finding: a mean above the median says the tail runs high, a mean below it says the tail runs low.
The mode is the only one of the three that works on things that are not numbers — the most common answer to a survey question, the most frequent size sold. It is also the only one that can be missing entirely, and a list where nothing repeats has no mode rather than a mode of whatever came first.
Questions people actually ask
- Is the average the mean or the median?
- In ordinary speech it is the mean: add everything up, divide by how many. In statistics "average" is a family that includes the median and the mode, and news reports move between them without saying so. Average household income in the US is a mean and it sits well above the median, because a small number of very large incomes lift the total without moving the middle. When a figure looks too high to describe most people, that is usually the reason.
- When should I use the median instead?
- When the list has a long tail or a value far from the rest: house prices, incomes, response times, anything where a handful of extremes are real rather than mistakes. The median is the value with as many below as above, so an outlier moves it by one position rather than by its size. This page flags the case automatically when a value falls outside the outlier fences.
- Can a list have no mode, or more than one?
- Both. If every value appears exactly once there is no mode at all, and reporting the first number in the list — which some calculators do — is simply wrong. If two values tie for most frequent the list is bimodal and both are modes. The panel above prints every tied value and how many times it occurred.
- How do I average percentages or rates?
- Not by taking the mean of the percentages, in most cases. Averaging 50% of 10 items and 90% of 1,000 items as (50 + 90) / 2 = 70% gives a figure nothing in the data supports; the right answer weights each by its size and comes to about 89.6%. The same trap applies to speeds, where the correct average of 30 mph out and 60 mph back is the harmonic mean, 40 mph, not 45.
- What does the range tell me that the mean does not?
- How far apart the extremes are, which the mean cannot show: 5, 5, 5 and 0, 5, 10 have the same mean and nothing else in common. The range is the crudest measure of spread and the easiest to read; the standard deviation on the companion page is the one that accounts for every value rather than only the two ends.