Interquartile Range Calculator - Quartiles and Outliers

Interquartile range, quartiles and outliers from one pasted list

Eleven delivery times in minutes, 22 25 27 28 30 31 33 35 38 41 95, give Q1 = 27.5, Q3 = 36.5 and an interquartile range of 9 minutes. Anything above 50 minutes sits outside the upper fence, so the 95-minute delivery is flagged as an outlier, and as an extreme one. The result also draws the box plot, lists the fences and shows what the same list gives under the four quartile conventions that Excel, a TI-84, R and SPSS use.

One list, four numbers worth remembering

IQR of the delivery times, in minutes

-70.7%

drop in standard deviation when the 95 is set aside

-19.4%

drop in IQR for the same change

1.349 σ

IQR of a normal distribution

Those middle two tiles are the reason the IQR exists. One late delivery drags the standard deviation from 5.89 to 20.09. The IQR moves from 7.25 to 9, because a quartile only cares how many values lie on each side of it, not how far away they are.

The five-number summary behind the box plot of those deliveries:

Minimum	Q1	Median	Q3	Maximum	Fences
22	27.5	31	36.5	95	14 to 50

Three boxes and a method

Data - paste the numbers, separated by spaces, commas or new lines. A spreadsheet column works. Decimals take a dot, and units have to be left out.
Quartile method - linear interpolation to match Excel QUARTILE.INC, Google Sheets and R; QUARTILE.EXC to match Minitab or SPSS; halves without the median to match a TI-84 or an AP Statistics answer key; Tukey's hinges for a classic box plot.
Decimal places - automatic follows your data; the choice only changes how numbers are printed.
Label (optional) - a name for the data set in the result heading.
Read the result - the IQR, the three quartiles, the box plot, the fences, any outliers, and the four-method comparison.

Why Excel, a TI-84 and R disagree about Q1

There is no single definition of a quartile. Hyndman and Fan catalogued nine in statistical software, and four of them cover almost every classroom and spreadsheet. Interpolation methods place Q1 a quarter of the way along the sorted list and blend the two nearest values; the halves methods split the list at the median and take the median of each half, differing only in whether the middle value of an odd-sized list joins the halves. With large data sets the answers converge. With eight values they do not:

Method, for 3 5 7 8 9 11 13 15	Q1	Q3	IQR
Linear interpolation (QUARTILE.INC, Sheets, R)	6.5	11.5	5
QUARTILE.EXC (Minitab, SPSS)	5.5	12.5	7
Halves without the median (TI-84)	6	12	6
Tukey's hinges (R fivenum)	6	12	6

Three different IQRs, 5, 6 and 7, from one list of eight numbers, and every one of them is correct under its own rule. When a homework answer disagrees with a spreadsheet, the method is almost always the explanation, which is why the result prints all four side by side.

Facts about the IQR that textbooks bury

For normally distributed data the IQR is 1.349 standard deviations, so IQR / 1.349 is a quick, outlier-proof estimate of σ.

The 1.5 × IQR fences sit 2.698 standard deviations from the mean of a normal distribution, so about 0.7% of perfectly normal data gets flagged anyway. In a list of a thousand values, expect around seven "outliers" that are nothing of the sort.

The whiskers of a box plot do not reach the minimum and maximum when there are outliers. They stop at the most extreme values still inside the fences, and the outliers are drawn as separate dots.

Who needs an interquartile range

Students checking a five-number summary against an answer key, where the TI-84 method is usually the one expected. Analysts summarizing skewed data such as response times, house prices or incomes, where a mean and standard deviation would mislead. Quality and lab staff screening a batch of measurements for values to re-check before anything is averaged.

IQR questions, briefly

How do you find outliers with the IQR?

Compute Q1 - 1.5 × IQR and Q3 + 1.5 × IQR. Values below the first or above the second are outliers; beyond 3 × IQR they are called extreme. For the delivery times the fences are 14 and 50 minutes.

Is the IQR better than the standard deviation?

For skewed data or data with outliers, yes. For symmetric, clean data the standard deviation uses every value and is the more precise measure, and most formulas (confidence intervals, t-tests) are built on it.

Which method does my textbook use?

If it splits the data at the median and leaves the median out of both halves, it matches the TI-84 option. If it says "use QUARTILE" or shows a spreadsheet, it is linear interpolation.

Can the IQR be zero?

Yes, when the middle half of the sorted values is one repeated number, as in survey answers where most people chose 3. For 2 3 3 3 3 3 3 9 40 the IQR is 0, both fences sit at 3, and 2, 9 and 40 are all flagged, so read the outlier list with care.

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