Statistics From Zero · Lesson 19 of 22
Percentiles, Quartiles and Box Plots: Reading a Distribution at a Glance
In one sentence: Percentiles tell you where a value sits in the line-up, quartiles cut the line-up into four, and a box plot draws those cuts so you can see the middle, the spread and the oddballs in one picture.
The idea
The average (Lesson 2) and the standard deviation (Lesson 3) summarise data in two numbers. But sometimes you want to know where one particular value stands, or what the whole spread looks like, including the odd ones. That's what percentiles, quartiles and box plots are for.
They all start the same way: line the values up from smallest to largest.
Percentiles: where you stand in the line-up
The 90th percentile is the value below which 90% of the data falls. If your exam score is at the 90th percentile, you scored higher than about 90% of the people who took it. It says nothing about whether the score was good in absolute terms, only how it compares with everyone else.
The median is simply the 50th percentile: half below, half above.
The pub quiz league
Uncle Gary's quiz team, The Usual Suspects, scored 21 in the league final. Eleven teams took part. The scores, lined up:
12, 15, 17, 18, 20, 21, 23, 24, 26, 29, 45
Gary says 21 makes them "a solid top-half team". The team of retired teachers that scored 45 has not spoken to anyone all evening.
Quartiles by hand
Quartiles cut the line-up into four equal parts:
- The median (Q2): the middle value of the 11. That's the 6th: 21. Gary is exactly in the middle, not the top half. He disputes this.
- The lower quartile (Q1): the median of the lower half (12, 15, 17, 18, 20): 17.
- The upper quartile (Q3): the median of the upper half (23, 24, 26, 29, 45): 26.
The middle half of the teams scored between 17 and 26. The width of that middle half, 26 - 17 = 9, is the interquartile range (IQR). It's a measure of spread that, unlike the range or the standard deviation, doesn't care about one extreme value.
A warning for exams: calculators and spreadsheets use slightly different rules for quartiles, especially with small data. A spreadsheet might give 17.5 and 25 here instead of 17 and 26. Use the method your course uses, and don't panic if software disagrees by a little.
Spotting the oddball: the 1.5 × IQR rule
A common rule marks a value as an outlier if it's more than 1.5 IQRs beyond the quartiles:
- Upper fence: 26 + 1.5 × 9 = 39.5.
- Lower fence: 17 - 1.5 × 9 = 3.5.
The teachers' 45 is above 39.5, so it's an outlier. Notice what it does to the average: the mean score is 22.7, pulled up by that one team, while the median stays at 21. Another reminder from Lesson 2 that the median is harder to fool.
"Outlier" doesn't mean "mistake" or "throw it away". The teachers really did score 45. It means "look at this one separately and ask why".
The box plot
A box plot draws all of this in one picture. A box stretches from Q1 to Q3 (17 to 26), with a line inside it at the median (21). Whiskers reach out to the furthest values that aren't outliers (12 at the bottom, 29 at the top). Outliers get their own dot (45).
In one glance you can see where the middle is, how wide the typical spread is, whether the data leans to one side (here the upper whisker and the outlier stretch further up than down) and which values are odd. Put several box plots side by side and you can compare whole groups at once, which is much more honest than comparing two averages.
Try it yourself
Using the same eleven quiz scores:
1. Aunt Linda's team scored 26. Roughly what percentile is that? 2. Cousin Kevin's team scored 12. Is 12 an outlier by the 1.5 × IQR rule? 3. If the teachers had scored 30 instead of 45, what would change: the median, the IQR, or the mean?
Show the answer
1. 8 of the 11 teams scored less than 26, so Linda's team is at roughly the 73rd percentile (8 ÷ 11). It's also exactly the upper quartile: top quarter, just.
2. No. The lower fence is 3.5, and 12 is above it. Low, but within the normal spread. Kevin will be relieved, or possibly disappointed.
3. Only the mean would change (it would drop from about 22.7 to 21.4). The median stays 21 and the quartiles stay 17 and 26, because they only care about positions in the line-up, not how extreme the top value is. And 30 would no longer be an outlier.
The words your lecturer uses
| Lecturer says | It means |
|---|---|
| Percentile (p-th percentile) | The value below which p% of the data falls |
| Quartiles (Q1, Q2, Q3) | The 25th, 50th and 75th percentiles; Q2 is the median |
| Interquartile range (IQR) | Q3 minus Q1: the spread of the middle half of the data |
| Five-number summary | Minimum, Q1, median, Q3 and maximum |
| Box plot (box-and-whisker plot) | A chart of the five-number summary, with outliers shown separately |
| Outlier (1.5 × IQR rule) | A value more than 1.5 IQRs below Q1 or above Q3 |
| Robust statistic | One that isn't thrown off by a few extreme values, like the median and the IQR |
Three things to remember
- A percentile tells you where a value stands compared with the others, not whether it's good.
- The median and the IQR ignore extreme values; the mean and the standard deviation don't.
- An outlier is a value to investigate, not automatically a value to delete.
Part 2 so far
Lessons 11 to 19 take you beyond the basics: probabilities that depend on what you know, lines through data, the value of a bet, comparing averages, the two ways a test can fail, testing counts, why averages make bell curves, how totals can mislead, and reading a whole distribution at a glance. Together with Lessons 1 to 10, that's most of what a first statistics course covers, explained without the panic.
Part of Statistics From Zero, a free beginner course by WrongStat. Not a substitute for your lecturer, but hopefully a good translator.