Averages are useful, but they hide a lot. Two classes can have the same average test score while one has everyone clustered around the middle and the other has a mix of top scores and failing grades. To describe how data is spread out, statisticians often turn to quartiles, percentiles and the interquartile range (IQR). These tools are simple to calculate, easy to explain and, unlike the range or the mean, they are not thrown off by a single extreme value. This guide walks you through each concept with a worked example.
Percentiles: where a value stands
A percentile tells you the percentage of values in a dataset that fall at or below a given point. If your score is at the 80th percentile, roughly 80% of scores are at or below yours. Percentiles are about position, not about the score itself: being at the 80th percentile does not mean you got 80% of the questions right.
You meet percentiles often in everyday life: children’s growth charts, standardized test reports and website performance metrics (for example, “95th percentile load time”) all use them.
Quartiles: cutting the data into four parts
Quartiles are simply three special percentiles that split sorted data into four groups of roughly equal size:
| Quartile | Equivalent percentile | Meaning |
|---|---|---|
| Q1 (first or lower quartile) | 25th | About a quarter of the values are at or below it |
| Q2 (second quartile) | 50th | The median: half the values are below, half above |
| Q3 (third or upper quartile) | 75th | About three quarters of the values are at or below it |
Together with the minimum and maximum, the three quartiles form the well-known five-number summary, which is exactly what a box plot draws.
Calculating quartiles step by step
Let’s use a small dataset: the number of minutes nine students spent on homework one evening.
Data (already sorted): 2, 4, 4, 5, 7, 8, 9, 11, 15
- Find the median (Q2). With nine values, the middle one is the 5th: 7.
- Split the data into a lower and an upper half. One common textbook method leaves the median out: the lower half is 2, 4, 4, 5 and the upper half is 8, 9, 11, 15.
- Find Q1, the median of the lower half: (4 + 4) ÷ 2 = 4.
- Find Q3, the median of the upper half: (9 + 11) ÷ 2 = 10.
So the five-number summary is: minimum 2, Q1 = 4, median 7, Q3 = 10, maximum 15.
A note on methods: there are several accepted ways to compute quartiles, and they can give slightly different answers, especially with small datasets. Spreadsheet programs reflect this; Excel, for example, has both QUARTILE.INC and QUARTILE.EXC. Neither is “wrong”. What matters is using one method consistently and mentioning which one you used when precision matters.
The interquartile range (IQR)
The interquartile range is the distance between the upper and lower quartiles:
IQR = Q3 − Q1
In our example, IQR = 10 − 4 = 6 minutes. This tells us that the middle half of the students spent within a 6-minute band of each other.
Why not just use the range (maximum minus minimum)? Because the range depends entirely on the two most extreme values. If one student had spent 90 minutes, the range would jump from 13 to 88, even though nothing changed for everyone else. The IQR ignores the top and bottom quarters, so a single unusual value barely moves it. In statistical language, the IQR is a robust measure of spread.
Using the IQR to spot outliers
A widely used rule of thumb, popularized by the statistician John Tukey, flags a value as a potential outlier if it lies more than 1.5 × IQR beyond the quartiles:
- Lower fence: Q1 − 1.5 × IQR
- Upper fence: Q3 + 1.5 × IQR
For our data: 1.5 × 6 = 9, so the lower fence is 4 − 9 = −5 and the upper fence is 10 + 9 = 19. Every value lies between −5 and 19, so there are no outliers by this rule. If a tenth student had reported 40 minutes, that value would sit well beyond the upper fence and deserve a closer look.
Remember that “outlier” does not automatically mean “error”. It means “check this”. It might be a typo, a measurement problem, or a perfectly real case that tells an interesting story.
Reading a box plot
A box plot (or box-and-whisker plot) turns these numbers into a picture:
- The box stretches from Q1 to Q3, so its length is the IQR.
- A line inside the box marks the median.
- The whiskers extend toward the smallest and largest values that are still within the fences.
- Points plotted beyond the whiskers are potential outliers.
Box plots shine when comparing groups side by side. A tall box means a lot of variation in the middle of the data; a median line close to one edge of the box suggests the data is skewed.
IQR or standard deviation?
| Interquartile range | Standard deviation | |
|---|---|---|
| Based on | Positions (quartiles) | Distances from the mean |
| Sensitive to outliers? | Very little | Yes, quite sensitive |
| Usually paired with | The median | The mean |
| Best for | Skewed data, data with extreme values | Roughly symmetric, bell-shaped data |
A good habit: when data is skewed (incomes, house prices, response times), report the median and IQR. When it is roughly symmetric, the mean and standard deviation work well.
Conclusion
Percentiles tell you where a value stands, quartiles divide data into four parts, and the interquartile range measures how spread out the middle half is, without being distorted by extremes. With the 1.5 × IQR rule and a box plot, you can summarize and compare datasets quickly and honestly. If you’d like to build a solid foundation in descriptive statistics, probability and data analysis, take a look at the free statistics courses available on Cursa.
