Average Calculator
Mean, median and mode together — because all three are called the average.
Updated
6 numbers found. Commas, spaces or one per line all work.
Average
17.333333mean
Sum 104 divided by 6.
- Median — the middle value
- 13
- Mode — the most common
- 12
- Count
- 6
- Sum
- 104
- Smallest
- 11
- Largest
- 40
- Range
- 29
The mean and the median are 4.33 apart. That gap is what an outlier looks like: one unusually large or small value drags the mean towards it while leaving the median where it is. For incomes, house prices and response times the median is usually the more honest summary.
Mean, median and mode are all called the average in ordinary speech. The mean uses every value and moves with outliers; the median uses only the middle and does not; the mode is the only one that works on data that is not numeric at all.
In short
What is the difference between mean, median and mode?
The mean adds everything and divides by the count. The median is the middle value once sorted. The mode is the most common. All 3 are called the average, and on skewed data they part company — for 12, 15, 11, 14, 12, 40 they are 17.33, 13 and 12.
When the mean and the median disagree, an outlier is pulling one of them. The median is the one that resists.
How to use the average calculator
Three statistics share one word. The mean is what most people picture: add everything up, divide by how many there are. The median is the value in the middle once the list is sorted. The mode is whichever value appears most often. On tidy, symmetric data they land close together, and on real data they frequently do not.
Which one is right depends on the shape of the data rather than on preference. The mean uses every value, which makes it sensitive to extremes — one enormous number drags it upwards regardless of how ordinary the rest are. The median ignores the size of the values and looks only at their order, so an extreme value moves it barely at all.
17.3
Mean of 12, 15, 11, 14, 12, 40
pulled up by the 40
13
Median of the same six
unmoved by it
12
Mode of the same six
the only value that repeats
Those three figures describe one small list, and they are four points apart. The 40 is doing all the work: it lifts the mean well above five of the six values, while the median sits comfortably among them. Reporting the mean alone would suggest a typical value that nothing in the data is near.
This is why incomes and house prices are reported as medians. A handful of very large values would lift a mean far above what most people experience, so the middle value gives a more honest picture of the typical case. Response times and waiting times are usually reported the same way, and for the same reason.
- Symmetric data, no extremes
- mean — it uses all the information
- Incomes, prices, response times
- median — outliers do not move it
- Categories rather than numbers
- mode — the only one that works
- Reporting to a general audience
- give the mean and the median together
- Data with a hard floor at zero
- expect the mean above the median
The last row is a useful prior. Anything that cannot go below zero but has no ceiling — money, time, counts — tends to be right-skewed, which puts the mean above the median almost as a rule.
Measure how spread out the numbers are
An average says where the middle is and nothing about how tightly the values cluster around it. Standard deviation answers the second half.
Open the standard deviation calculator →Small datasets chosen to separate the three averages, showing how far apart they travel as skew and repetition are introduced.
| Data | Mean | Median / mode |
|---|---|---|
| 1, 2, 3, 4, 5 | 3 | 3 / none |
| 1, 2, 3, 4 | 2.5 | 2.5 / none |
| 2, 2, 3, 4, 9 | 4 | 3 / 2 |
| 12, 15, 11, 14, 12, 40 | 17.33 | 13 / 12 |
| 12, 15, 11, 14, 12 | 12.8 | 12 / 12 |
| 1, 1, 1, 1, 96 | 20 | 1 / 1 |
| 5, 5, 5, 5, 5 | 5 | 5 / 5 |
| 1, 2, 2, 3, 3, 4 | 2.5 | 2.5 / 2 and 3 |
| 10, 20, 30, 40, 1000 | 220 | 30 / none |
| −5, 0, 5 | 0 | 0 / none |
What it means when they disagree
The classic demonstration is a room of ten people whose incomes are ordinary, and then one very wealthy person walks in. The median barely moves, because it only cares about who is in the middle. The mean leaps, because it divides a total that has just changed enormously. Nobody in the room is any richer, and one of the two statistics now says otherwise.
- The list
- 12, 15, 11, 14, 12, 40
- Sum
- 104
- Mean
- 104 ÷ 6 = 17.33
- Sorted
- 11, 12, 12, 14, 15, 40
- Median
- (12 + 14) ÷ 2 = 13
- The gap
- 4.33 — the outlier, showing up
Remove the 40 and the mean drops to 12.8, almost exactly the median of the remaining five. One value in six moved the mean by more than four and the median by half a point, which is the whole difference between the two statistics in one line.
For an even-length list the median is the mean of the two middle values, which is the standard convention in descriptive statistics and is what both NIST and the IUPAC Gold Book specify. It means the median need not be a value that actually appears in the data — 13 does not, in the example above.
The mode, and what to do when nothing repeats
The mode is the most frequent value, and it is the only average that works on data that is not numeric at all — the most common colour, the most common answer, the most common fault code. On numeric data it is often the least useful of the three, because it depends entirely on how the values happen to be rounded.
Do
- Report the median alongside the mean for anything skewed
- Use the mode for categories, where mean and median have no meaning
- Check the count before trusting any of the three
- Say which average you mean rather than writing "average"
Don't
- Average a set of averages without weighting them by their counts
- Report a mean for data with a long tail and call it typical
- Assume the median is one of the values in the list
- Treat several tied modes as a single answer
When no value repeats, this page reports no mode. That follows the convention in NIST's Dictionary of Algorithms and Data Structures, and it is a convention rather than a law — some texts read the definition literally and call every value a mode when all frequencies are equal. Saying "the mode is 1, 2, 3, 4 and 5" restates the list rather than summarising it, which is why the other reading is used here.
The formula, worked line by line
Three definitions, and only the first is arithmetic. The median is a sorting operation and the mode is a counting operation, which is why they behave so differently when the data is awkward.
“The mean asks what everyone would get if the total were shared equally. The median asks who is in the middle. Those are not the same question.”
Neither is more correct than the other in the abstract. The mistake is reporting one, calling it "the average", and letting the reader assume it answers whichever question they had in mind.
mean = sum of the values ÷ count
median = middle value once sorted
median, even count = mean of the two middle values
mode = the value that occurs most often
range = largest − smallest- Class A
- 10 students, mean 60
- Class B
- 30 students, mean 80
- Averaging the averages
- (60 + 80) ÷ 2 = 70
- The real total
- 600 + 2,400 = 3,000
- Over 40 students
- 3,000 ÷ 40 = 75
- Correct combined mean
- 75, not 70
The two classes are not the same size, so their means do not carry equal weight. Averaging averages is only valid when every group has the same count — which, outside textbook examples, it almost never does.
The median can be a value that is not there
With an even number of observations the median is the midpoint between the two central values, so it may sit between them rather than on one. In the list 11, 12, 12, 14, 15, 40 the median is 13, which appears nowhere in the data. That is expected rather than an error, and it is the standard convention for numeric data.
What each average survives
- Adding one very large value
- mean jumps, median barely moves
- Doubling every value
- all three double
- Adding 10 to every value
- all three rise by 10
- Working with categories
- only the mode has meaning
- A single value repeated
- all three are that value
The first row is the only one where they part company, and it is the row that describes most real data. Everything else in this table is a transformation all three handle identically.
That is the practical summary: for well-behaved data the choice does not matter, and for the data most people actually have, it decides what the answer means.
Questions people ask
Sources
Where the constants and formulas on this page come from. Each line names the figure it backs.
Mean, median and mode are the standard measures of location, and the median of an even-sized sample is the average of the two middle values.
NIST/SEMATECH e-Handbook of Statistical Methods — Measures of Location — National Institute of Standards and Technology
If no value in a dataset is repeated, the dataset has no mode.
Dictionary of Algorithms and Data Structures — mode — National Institute of Standards and Technology
The median is less affected by extreme values than the mean, which makes it the more robust summary for skewed distributions.
NIST/SEMATECH e-Handbook of Statistical Methods — Measures of Location — National Institute of Standards and Technology
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