Foundations · Spread
Quartile Calculator
The first and third quartiles, the median, the interquartile range and the full five-number summary — drawn as a box plot, with outliers flagged. And because the textbooks define quartiles three slightly different ways that genuinely disagree, this shows all three, so you know exactly which Q1 and Q3 you are quoting.
“Tukey” (the median-excluded method) is the usual textbook default; “Interpolation” is what Excel’s PERCENTILE and most software use. They can give different Q1 and Q3 — see the comparison below.
Result
In plain English
Quartiles cut your sorted data into four equal-sized quarters. The three cut points — Q1, the median (Q2) and Q3 — together with the smallest and largest values make the five-number summary, the most honest quick picture of a distribution: where the middle sits, how wide the bulk is, and how far the tails reach. It is what a box plot draws.
- Q1, Q2, Q3
- The values below which 25%, 50% and 75% of the data fall. Q2 is just the median.
- interquartile range (IQR)
- Q3 − Q1, the spread of the middle half. Unlike the range or standard deviation, a couple of outliers cannot inflate it, so it is the robust measure of spread.
- five-number summary
- Minimum, Q1, median, Q3, maximum — five numbers that sketch the whole distribution, and the skeleton of a box plot.
- outliers (1.5 × IQR)
- Points more than 1.5 IQRs beyond Q1 or Q3 — the usual flag for “unusually far out”. It is a convention for a second look, not a verdict that the point is wrong.
- why three methods
- For odd-sized data, textbooks disagree on whether the median joins each half (inclusive) or not (Tukey), and software interpolates between values. The median and IQR shift a little depending which you use.
Frequently asked
How do you find the first and third quartiles?
Sort the data and find the median (Q2). The first quartile Q1 is the median of the lower half, and Q3 is the median of the upper half. The one wrinkle is what to do with the middle value when the count is odd: the Tukey (exclusive) method leaves it out of both halves, the inclusive method puts it in both, and software usually interpolates between the two nearest values. This calculator shows all three so the choice is explicit.
Why do Excel and my textbook give different quartiles?
Because they use different methods. Excel’s QUARTILE and PERCENTILE functions (and QUARTILE.INC) use linear interpolation; most intro-stats textbooks use the Tukey median-excluded method. On the same data the two can return different Q1 and Q3 — neither is “wrong”, they are just different conventions. Quote which one you used, especially for small datasets where the gap is largest.
What is the 1.5 × IQR rule for outliers?
A point is flagged as a (mild) outlier if it lies below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR; beyond 3 × IQR it is an “extreme” outlier. These fences are John Tukey’s convention for the whiskers of a box plot. A flagged point is worth a closer look — a typo, a special case, a genuine extreme — but the rule is a prompt to investigate, not proof that the value is an error.
What is the difference between quartiles and percentiles?
Quartiles are just specific percentiles. Q1 is the 25th percentile, Q2 (the median) the 50th, and Q3 the 75th. Percentiles slice the data into a hundred parts; quartiles into four. So every quartile is a percentile, but the quartiles are the handful that define the box plot and the interquartile range — and the same choice-of-method subtlety affects both.