Mean, Median, Mode and Range Calculator

How to use the mean, median, mode and range 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.

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.

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.

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 range, and the two numbers it uses

The range is the largest value minus the smallest, so for 12, 15, 11, 14, 12, 40 it is 40 − 11 = 29. It is the quickest measure of spread there is, and the only one that never touches more than two of your numbers. Everything sitting between the two extremes is discarded before the subtraction happens.

That is also its defect. One unusual value sets an entire end of the range by itself, so the figure describes the two most atypical observations rather than the data. Drop the 40 from that list and the range falls from 29 to 4, a seven-fold change caused by one number in six, while the median moves only from 13 to 12.

Two different things are called the range in mathematics and search results mix them freely. In statistics it is largest minus smallest, which is what this page computes. In algebra the range of a function is the set of values that function can produce, paired with its domain — a different question answered by a different method, with no list of numbers to paste anywhere.

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.

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.

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

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

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