- Why does my calculator give a different IQR than my textbook?
- Because there is no single definition of a quartile. There are four common ones and they disagree on the same data. For 1, 2, 3, 4, 5, 6, 7, 8, 9 the IQR is 5 under Tukey's hinges, 4 if you include the median in both halves, 4 under Excel's QUARTILE.INC and 5 under QUARTILE.EXC. None of those is a mistake; they split the data differently. Almost every calculator picks one silently, which is why the mismatch feels like a bug. This page shows all four at once and names where each one comes from, so you can read the row that matches whatever you are being checked against.
- Which quartile method should I use?
- Use whichever your course or your source uses. If you are in AP Statistics or working on a TI-83 or TI-84, that is Tukey's hinges — the headline number here. If you are checking against a spreadsheet, Python or R, it is almost certainly QUARTILE.INC, also called R type 7, which is what NumPy's percentile function returns by default. Some textbooks, Mendenhall and Sincich among them, keep the median in both halves. If nobody told you, use Tukey's hinges for anything involving outliers, because the 1.5 × IQR rule was written for it.
- How does the 1.5 × IQR rule find outliers?
- Take the IQR, multiply it by 1.5, subtract that from Q1 and add it to Q3. Those two numbers are the fences. Anything strictly below the lower fence or strictly above the upper fence is an outlier. For 1, 2, 2, 3, 3, 4, 5, 60 the quartiles are 2 and 4.5, so the IQR is 2.5, and the fences land at −1.75 and 8.25. The 60 is outside, so it gets flagged. The rule is John Tukey's, and the 1.5 is a convention rather than a law — it is simply the multiplier that flags roughly one value in 150 for data that follows a normal distribution.
- What is the difference between a mild and an extreme outlier?
- The multiplier. A point beyond 1.5 × IQR from the nearer quartile is a mild outlier; a point beyond 3 × IQR is an extreme one. In the example above the 3 × IQR fences sit at −5.5 and 12, and 60 clears the upper one easily, so it is extreme rather than mild. The distinction matters when you are deciding what to do about it. A mild outlier is usually a real observation at the edge of the spread. An extreme one is worth checking for a typo, a unit mix-up or a sensor that dropped out before you decide to keep it.
- Why does Tukey's method have no answer for a single number?
- Because it works by splitting your data into a lower half and an upper half and taking the median of each. One number gives you one half with nothing in it, so there is no Q1 to find. This is worth saying out loud because the naive way to write that code does not return zero — it crashes. The other methods still have an answer at n = 1: including-the-median and QUARTILE.INC both return the number itself for Q1 and Q3, which puts the IQR at 0.
- Why does QUARTILE.EXC refuse to answer for two numbers?
- Its rank formula points outside your data. The exclusive method looks for the value at position 0.25 × (n + 1). With two values that is position 0.75, which is before the first number you have, so there is nothing there to read. Excel returns #NUM! for exactly this reason. Three values is the smallest set it can handle. Some libraries quietly clamp the rank to the first value instead of admitting there is no answer — NumPy's weibull method does — which is how two tools can disagree on data this small.
- What does the interquartile range actually tell me?
- It is the width of the middle half of your data. Sort the numbers, cut off the bottom quarter and the top quarter, and the IQR is what is left over. That makes it a measure of spread like the range or the standard deviation, but a far more stubborn one: one absurd value cannot move it, because the point sits outside the middle half and changing it does not move either quartile. A range or a standard deviation, by contrast, moves a long way on a single bad reading. That resistance is why the IQR is the basis of the outlier rule rather than a victim of it.
- Why does this page report a different Q1 than the Percentile Calculator?
- Because the two pages lead with different conventions, and both do it deliberately. The Percentile Calculator reports Q1 and Q3 using the inclusive method, since that is the one a spreadsheet uses. This page leads with Tukey's hinges, since that is what the outlier rule needs. On an even-length list they usually agree; on an odd-length one they can differ. The QUARTILE.INC row here is the same calculation the Percentile Calculator runs, so if you want the two pages to match, read that row.
- My numbers use a decimal comma. Why did 1,5 become two values?
- Commas separate values here, so 1,5 reads as 1 and 5 rather than as one-and-a-half. That is a real limitation if your keyboard writes decimals with a comma. The fix is to use dots for decimals and keep commas, spaces or new lines between values: 1.5 2.3 2.8. The summary strip shows n, so a count that is double what you expected is the tell that this happened.
- Is my data sent anywhere?
- No. Everything is worked out in your browser, so nothing you paste leaves your machine and closing the tab clears it. There is no sign-up, no row limit behind a plan and no export you have to pay to unlock. Paste a hundred thousand numbers if you have them — the page sorts them and answers in well under a fifth of a second.
