Paste a column of numbers and get the whole picture at once: where the data centres, how far it spreads, which shape it takes, and which values sit outside the crowd.
Result
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Three questions describe almost any set of numbers. Where is the middle? How spread out is it? And is it lopsided? The mean answers the first by balancing the values; the median answers it by counting to the middle. When the two disagree, the data is telling you something — usually that a handful of extreme values are pulling on the mean.
Spread is measured by the standard deviation: the typical distance from the mean. Whether you divide by n or by n − 1 depends on what your numbers are. A complete population divides by n. A sample divides by n − 1, because a sample always hugs its own mean a little too closely and would otherwise understate the true spread.
The interquartile range takes a different route to the same question, describing the middle half of the data and ignoring the tails entirely. That makes it steady where the standard deviation is sensitive, which is why outlier fences are built from it rather than from the mean.
mean = Σx ÷ n
s² = Σ(x − mean)² ÷ (n − 1) s = √s²
For a whole population, divide by n instead of n − 1.
Q1 − 1.5 × IQR and Q3 + 1.5 × IQR
Tukey’s rule. Values outside these bounds are worth a second look.
The series averages 15.5 with a median of 15 — close together, so the distribution is fairly even.
The sample standard deviation is about 3.37, meaning a typical reading sits a little over three units from the middle. Nothing falls outside the fences.
Eight values between 4 and 8, and one of 40. The mean jumps to 9.78 while the median stays at 6.
The gap between them is the warning sign, and the 40 is duly flagged as an outlier — it sits far above the upper fence of 10.
These ten marks are every mark in the class, so the population setting applies and the variance divides by 10, not 9.
The mean is 80.8 and the standard deviation about 8.4 — the choice of divisor shifts it by roughly half a point.
Population if your numbers are everything you care about: every employee in the company, every mark in the class. Sample if they are a subset standing in for something bigger.
The difference only touches the variance and standard deviation, and it shrinks as the series grows. With thirty values it is under 2%; with five it is over 10%.
The median, whenever a few extreme values could distort the picture — incomes and house prices being the classic cases. The mean, when the data is roughly symmetrical and you need it for further calculation.
A large gap between the two is itself a finding, and worth reporting alongside whichever figure you choose.
This calculator uses Tukey’s rule: anything more than one and a half interquartile ranges beyond the first or third quartile.
It is a convention, not a verdict. An outlier can be a typing error, a broken instrument, or the single most interesting observation you have. Investigate before you delete.
Because there are at least nine defensible ways to define them, and different software picks different ones. Spreadsheets and R default to linear interpolation, which is what is used here.
On large series the methods agree closely. On small ones they can differ noticeably, which is worth knowing before comparing two reports.
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