Five-Number Summary
Paste a dataset for the min, Q1, median, Q3, and max — plus the IQR, outliers, and a box plot.
The box spans Q1 to Q3, the thick line is the median, whiskers reach the most extreme values inside the fences, and any dots are outliers.
Quartiles use the median-of-halves (Tukey) method, so results may differ slightly from calculators that use the exclusive or inclusive interpolation methods.
The five-number summary is the minimum, first quartile (Q1), median, third quartile (Q3), and maximum of a sorted dataset. For 4, 8, 15, 16, 23, 42 it is min 4, Q1 8, median 15.5, Q3 23, max 42, giving an IQR of 15. A box plot draws these five numbers so you can read the spread at a glance.
What the five numbers describe
The five-number summary compresses a whole dataset into five landmarks that mark how the values are spread out. The minimum and maximum give the full range. The median is the middle value, splitting the data into a lower and upper half. The first quartile (Q1) sits a quarter of the way in and the third quartile (Q3) three-quarters of the way in, so the box between them holds the middle 50% of the data. The width of that box, Q3 − Q1, is the interquartile range (IQR) — a measure of spread that ignores the extremes.
How the quartiles are computed
This tool uses the median-of-halves method, often called the Tukey method. First sort the data and find the median. Then split the sorted values into a lower half and an upper half at the median — when the count is odd, the middle value belongs to neither half and is left out. Q1 is the median of the lower half and Q3 is the median of the upper half. The outlier fences follow the standard rule: anything below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR is flagged as an outlier.
Worked example
Six values: 4, 8, 15, 16, 23, 42.
- 1 Sort the data and read off the ends. Sorted: 4, 8, 15, 16, 23, 42. The minimum is 4 and the maximum is 42.
- 2 Find the median. With 6 values the two middle ones are 15 and 16, so the median is (15 + 16) ÷ 2 = 15.5.
- 3 Take the median of the lower half for Q1. The lower half is 4, 8, 15, whose middle value is 8, so Q1 = 8.
- 4 Take the median of the upper half for Q3. The upper half is 16, 23, 42, whose middle value is 23, so Q3 = 23.
- 5 Compute the IQR and check for outliers. IQR = 23 − 8 = 15. Fences are 8 − 1.5 × 15 = −14.5 and 23 + 1.5 × 15 = 45.5, so no value is an outlier.
What each number means
The five landmarks plus the spread and outlier rule.
| Statistic | What it marks | How it is found |
|---|---|---|
| Minimum | Smallest value | First value after sorting |
| Q1 (25th percentile) | Quarter of the way through the data | Median of the lower half |
| Median (50th percentile) | Middle of the data | Middle value (or mean of the two middle values) |
| Q3 (75th percentile) | Three-quarters through the data | Median of the upper half |
| Maximum | Largest value | Last value after sorting |
| IQR | Spread of the middle 50% | Q3 − Q1 |
| Outlier | Unusually extreme value | Below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR |
Reading the box plot
The box is the middle half of the data. Its left edge is Q1, its right edge is Q3, and the line inside it is the median. A wide box means the central values are spread out; a narrow box means they are tightly packed. If the median sits off-centre inside the box, the data is skewed toward the longer side.
The whiskers and dots show the tails. The whiskers reach the most extreme values that still fall inside the fences, and any point plotted beyond them is a flagged outlier. Because the quartiles resist extreme values, the box and median stay stable even when a few points are far out.
Quartile methods differ. This calculator uses the Tukey median-of-halves method; other tools use the exclusive or inclusive interpolation methods, so Q1 and Q3 (and therefore the fences) can differ slightly for the same data. The minimum, maximum, and median never change between methods.