Quartile Calculator

What quartiles measure

Quartiles cut an ordered list into four groups of equal size. Q1 sits at the 25 percent mark, Q2 is the median at 50 percent, and Q3 sits at 75 percent. For the scores 62, 70, 74, 79, 85, 88 and 91 the four groups hold one or two scores each, and the cut points land at 72, 79 and 86.5. Those three numbers describe the shape of the data without a single average.

That is the appeal. An average salary, home price or test score tells you the middle of a distribution and nothing about the spread around it. Compare two school districts with a median of 78: in one, half the scores fall between 74 and 82, while in the other half fall between 58 and 95. Same median, completely different classrooms. The quartile spread exposes the difference, and a box plot draws it in one line.

Quartiles also survive extremes. Add a single score of 300 to a class of thirty, and the mean jumps by several points while the median barely moves and the quartile cut points shift by less than one rank. For pay tables, house prices and test results, that resistance to one wild value is the reason quartiles appear in published statistics while means get footnoted.

How this calculator finds the quartiles

The method is linear interpolation between the two scores that bracket each cut point, the same rule spreadsheets use when you call QUARTILE.INC or PERCENTILE.INC:

position = 1 + (n - 1) x p

Here n is the count of values and p is 0.25, 0.5 or 0.75. A whole number position lands exactly on a score. A fractional position sits partway between two scores, and the calculator takes that fraction of the gap.

Sorting happens first and is easy to forget by hand. If your list is unsorted the ranks above point at the wrong scores entirely, which is the most common source of a wrong quartile on a spreadsheet someone built from scratch.

A worked example with 7 scores

Seven test scores, already sorted: 62, 70, 74, 79, 85, 88, 91. With n = 7 each cut point sits at a half position, so every quartile is the midpoint of two scores.

Read the Q1 row in full: the position is 2.5, which sits halfway between the second score (70) and the third (74), so it sits halfway between 70 and 74, which is 72. The Q3 row works the same way between 85 and 88: 85 and 88, so Q3 is 86.5.

The interquartile range is the distance between them: 86.5 - 72 = 14.5 points. Half of the class sits in that 14.5 point band, a tighter picture than the full 29 point spread from 62 to 91 suggests.

Even counts and fractional cut points

With an even count no cut point lands on a single score, so both the median and the quartiles come from interpolation. Take ten daily deliveries: 3, 5, 7, 8, 9, 12, 14, 15, 16, 20.

The median sits between the fifth and sixth values, which is the familiar average of the two middle scores. Q1 sits from the third to the fourth value, and Q3 three quarters of the way from the seventh to the eighth. Interpolation means a quartile can be a number that never appears in the data, 7.25 deliveries in this case, and that is expected rather than an error.

The interquartile range here is 14.75 - 7.25 = 7.5 deliveries. The mean of the same ten days is 10.9, slightly above the median because the two largest days pull it up.

Fences that flag outliers

The interquartile range doubles as an outlier test. Multiply it by 1.5 and place a fence that far below Q1 and that far above Q3. Anything outside the fences is worth a second look.

The value 30 sits far above the upper fence and gets flagged, while 2 sits above the lower fence of -0.875 and stays in. A count of 2 is low but not unusual against this spread, which is exactly what the 1.5 rule is designed to separate: the extreme value that deserves a check from the merely small one.

The rule is a convention, not a law. Tukey's 1.5 multiplier flags moderate outliers, and 3.0 marks points that are extreme. Neither number knows your subject matter, so a flagged delivery day or a flagged lab result means start asking questions, not delete the row.

Quartiles, percentiles and competing methods

Quartiles are percentiles with fixed names: Q1 is the 25th percentile, Q2 the 50th and Q3 the 75th. Any percentile uses the same position rule with p set to the percentile over 100, so the 90th percentile of the delivery list above sits at position 9.1, one tenth of the way past the ninth value.

Different software answers the same question with slightly different numbers because the cut point can be defined several ways:

The gap between 70 and 72 on the same seven scores is not a bug in either method. Each defines a quartile as a different compromise between equal group sizes and simple arithmetic. When you compare your number with a published table, match the method before you match the digits; the NIST handbook section on percentiles lays out the definitions side by side.

Where quartiles show up in public data

Wage tables are the most common place an ordinary reader meets quartiles. The Bureau of Labor Statistics occupational employment statistics publish the 10th, 25th, 50th, 75th and 90th percentile wage for each occupation, so an electrician's pay table reads as a ladder of percentiles rather than one average. The 25th percentile is Q1 and the 75th is Q3, and the distance between them shows how wide the pay band runs.

School reporting uses the same cut points. Test score summaries often give the median and the interquartile range because a single average hides a class split between strong and struggling readers, and quartiles show both halves at once.

One caution applies to every table of this kind. Quartiles describe the sample they were computed from, so a Q3 wage for a metro area covers full time workers in that area only, and a Q3 test score covers the students who sat that test. Read the footnote that says who is counted before you compare two tables that look alike.