The Quartile Calculator splits a sorted data set into four parts and returns the first quartile (Q1), the median (Q2), the third quartile (Q3), the interquartile range and the five-number summary. Paste values separated by commas, spaces or line breaks; choose a quartile method; and read every cut with the sorted list beside it.
At least four accepted methods exist, and they disagree on small sets. Naming the method in use is the difference between a trustworthy tool and a silent mismatch with a textbook.
Calculate quartiles from a data set
Quartiles cut ordered data into four segments of roughly equal count. Q1 marks the 25th percentile region, Q2 is the median, and Q3 marks the 75th. Paste the list, confirm the count, and pick a method before comparing to homework answers.
Confirm the count under the paste field before comparing with a homework key. A missing value shifts every position in the ordered list and can move Q1 by a full observation on short sets. Sorted order appears beside the results so positions are visible without retyping the list.
Sorted order is required. The calculator sorts ascending automatically and shows that order so you can verify positions by eye on short lists.
Find the five number summary
The five-number summary is minimum, Q1, median, Q3 and maximum in that order. It is the skeleton of a box plot and a compact description of both centre and spread without squared units.
Unlike variance, every figure stays in the original measurement units, which makes it easy to read beside the raw list and to transfer onto graph paper by hand.
For 6, 7, 15, 36, 39, 40, 41 under the exclusive method: min 6, Q1 7, median 36, Q3 40, max 41. Those five numbers alone show a long lower stretch (6 up to 36) and a tight upper cluster (36 to 41).
Calculate the interquartile range
Interquartile range equals Q3 minus Q1 and measures the width of the middle 50 percent of the ordered data. Because it ignores the tails, a single extreme high or low value cannot inflate it the way it inflates the full range. IQR is also the scale used for the common 1.5 times IQR outlier fences described later on this page.
On the worked set, IQR = 40 − 7 = 33. That wide middle band reflects the jump from the teens into the thirties. IQR is also the scale used for the common 1.5 × IQR outlier fences.
Choose a quartile method
Four methods ship in the selector because classrooms and software packages disagree on how to form the halves around the median. Exclusive excludes the median from both halves; inclusive includes it when n is odd; linear interpolation places Q1 and Q3 at fractional positions; nearest rank returns an observed value with no interpolation. Match the method your course names.
| Method | Behaviour | Familiar from |
|---|---|---|
| Exclusive (default) | Median excluded from both halves | Excel QUARTILE.EXC, many US textbooks |
| Inclusive | Median included in both halves when n is odd | Excel QUARTILE.INC, many TI calculators |
| Linear interpolation | Position (n−1)/4 + 1 with interpolation | R type 7, NumPy default |
| Nearest rank | No interpolation; returns an observed value | Rank-based courses |
None is universally "correct." The correct method is the one your course, software or standard operating procedure names. The calculator defaults to exclusive and always prints the method next to Q1 and Q3.
Understand why methods disagree on small data sets
Take {1, 2, 3, 4, 5}. Exclusive puts Q1 at 1.5; inclusive puts Q1 at 2. Both are taught in real classrooms. On large n the gap shrinks; on homework-sized n it is often the entire mark difference between matching the key and missing it.
Software defaults differ too. Excel QUARTILE.EXC tracks exclusive; QUARTILE.INC tracks inclusive; R's default quantile type uses interpolation. Students who check a hand answer in two programs without reading the method name often conclude one of them is broken.
Odd counts are where inclusive and exclusive diverge most, because the median either is or is not reused when the lower and upper halves are formed. Linear interpolation can land between observed values even when exclusive returns a hinge that matches a data point. If your answer disagrees with a solution key by a small amount on a short list, switch methods before assuming arithmetic error.
Find quartiles for 6, 7, 15, 36, 39, 40, 41
This seven-value set is already sorted and is the fixture for the exclusive method default. Median position is (7 + 1) / 2 = 4, so Q2 is 36. The lower half excluding the median is 6, 7, 15 and the upper half is 39, 40, 41. Every cut below can be checked without relying on the interactive tool.
Median (Q2). Position (7+1)/2 = 4 → 36.
Lower half (median excluded): 6, 7, 15. Middle value → Q1 = 7.
Upper half (median excluded): 39, 40, 41. Middle value → Q3 = 40.
IQR. 40 − 7 = 33.
Five-number summary. 6, 7, 36, 40, 41.
Inclusive would fold 36 into both halves and can shift Q1 and Q3. Try both in the selector if a textbook solution disagrees.
Identify outliers using the IQR
Outlier fences extend 1.5 times the IQR below Q1 and above Q3. Points beyond those fences are flagged as mild outliers; points beyond 3 times IQR are often labelled extreme. The rule is a screen for investigation, not an automatic delete key.
Typing errors and genuine rare events both land outside fences and need different responses. lower fence = Q1 − 1.5 × IQR upper fence = Q3 + 1.5 × IQR ```
For the worked set: 1.5 × 33 = 49.5, so lower = 7 − 49.5 = −42.5 and upper = 40 + 49.5 = 89.5. Every value sits inside; none is flagged.
A point beyond the fence is a mild outlier; beyond Q1 − 3×IQR or Q3 + 3×IQR is often called extreme. Investigate before removing. The Descriptive Statistics Calculator applies the same fences inside a fuller summary.
Read the box plot
The box runs from Q1 to Q3. A line inside marks the median. Whiskers reach the most extreme points still inside the fences; outer points plot individually. A median line near one end of the box signals skew inside the middle 50%. A long whisker signals a stretched tail.
Compare whisker lengths for a quick skew read. A long lower whisker with a high median line inside the box often matches a left lean; the reverse matches a right lean. Outlier dots beyond the whiskers should match the fence table as a cross-check.
On the worked set the box spans 7 to 40 with the median at 36, so most of the box sits below the median line's neighbours on the high side: the upper half of the middle 50% is compressed into 36-40 while the lower half stretches 7-36.
Quartiles in reports and coursework
Report Q1, median, Q3 and IQR together when the distribution is skewed or contains outliers that would distort a mean-and-standard-deviation story. Many lab rubrics ask for the five-number summary explicitly because it supports a box plot without further calculation. State the quartile method in one short clause whenever n is small.
For arbitrary percentiles such as the 10th or 95th, switch to the Percentile Calculator. For a full numeric portrait that includes skewness and a histogram, use the Descriptive Statistics Calculator, which calls the same quartile engine and labels the method the same way.
Frequently asked questions
What is a quartile?
A quartile is one of the three cut points (Q1, Q2, Q3) that divide ordered data into four parts. Q1 is the lower quarter cut, Q2 is the median, and Q3 is the upper quarter cut.
What is the five-number summary?
Minimum, Q1, median, Q3 and maximum. Those five values feed a box plot and summarise centre and spread without squared units.
How is IQR calculated?
IQR = Q3 − Q1. It is the range of the middle 50% of the data and the scale for the 1.5 × IQR outlier rule.
Why does my textbook get a different Q1?
Different quartile definitions. Exclusive, inclusive, interpolation and nearest-rank methods disagree on small sets. Switch the method selector to match the book before reworking the arithmetic.
Which method matches Excel?
QUARTILE.EXC matches exclusive. QUARTILE.INC matches inclusive. Older QUARTILE behaved like inclusive. Check which function the assignment used.
Can Q1 equal a data value?
Yes, often. Nearest-rank always returns an observed value. Exclusive and inclusive often do on odd-length halves. Interpolation may land between two values.
How do quartiles relate to percentiles?
Q1 corresponds to the 25th percentile, Q2 to the 50th, and Q3 to the 75th, though percentile interpolation rules can still differ slightly from a given quartile method. See the Percentile Calculator for rank formulas.
Are outliers removed automatically?
No. They are flagged. Removal is a separate decision and should be documented when it happens.
Summary
The Quartile Calculator returns Q1, Q2, Q3, IQR, the five-number summary and outlier fences from any pasted list, with the quartile method named in the output. Exclusive is the default; inclusive, linear interpolation and nearest rank are one click away because small data sets disagree across definitions.
IQR = Q3 − Q1 measures the middle 50% and sets the 1.5 × IQR outlier fences. Match the method to your course or software, then read the box plot for skew and tail length.