Five Number Summary Explained

The five numbers

A five number summary describes a whole data set with five landmarks, in order: the minimum (smallest value), the first quartile Q1 (the value a quarter of the data sits below), the median (the middle), the third quartile Q3 (three-quarters point) and the maximum. Those five numbers locate the centre, frame the middle half of the data, and give the full span — enough to sketch the distribution without seeing a single raw value. The five number summary calculator produces them, plus the IQR and outlier fences, from one paste.

Finding quartiles by hand

Step 1 — Sort the data. Step 2 — Median: the middle value. Step 3 — Q1: the median of the values below the median. Step 4 — Q3: the median of the values above it.

For 12, 15, 18, 22, 25, 29, 31, 35, 40: the median is 25. The lower half is 12, 15, 18, 22, whose median is (15 + 18) ÷ 2 = 16.5 = Q1. The upper half is 29, 31, 35, 40, whose median is (31 + 35) ÷ 2 = 33 = Q3. Summary: 12 · 16.5 · 25 · 33 · 40. (Quartile conventions differ slightly between textbooks for very small sets; this median-of-halves method is the most common classroom approach, and our calculator shows the halves in its steps so the working can always be followed.)

IQR explained

The interquartile range is the spread measure with armour on: because it spans only the central 50% of values, extremes and outliers cannot stretch it. It pairs naturally with the median ("median 25, IQR 16.5") the way standard deviation pairs with the mean, and it powers the standard outlier rule — fences at Q1 − 1.5 × IQR and Q3 + 1.5 × IQR — covered in what is an outlier?

From summary to box plot

A box plot is the summary drawn: a box from Q1 to Q3, a line inside it at the median, and whiskers extending to the smallest and largest values within the fences (values beyond the fences appear as individual dots — the outliers). Read it at a glance: the box's length is the IQR; the median line off-centre inside the box reveals skew in the middle half; long whiskers or dots reveal long tails. Every box plot you have ever seen in a report is five numbers wearing a picture.

Worked example

Putting it together for 12, 15, 18, 22, 25, 29, 31, 35, 40: min 12, Q1 16.5, median 25, Q3 33, max 40; IQR 16.5; fences at 16.5 − 24.75 = −8.25 and 33 + 24.75 = 57.75; no outliers (nothing outside the fences). The distribution is mildly right-spread in its upper half (Q3 − median = 8 vs median − Q1 = 8.5 — actually near-symmetric in the middle, with the extremes balanced too). That is a complete, honest description of nine numbers in one sentence — the entire point of the summary.

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