Quartile in Math: Definition, How to Find & Examples

Definitionnoun

What is a quartile in math?

A quartile is one of three values that split an ordered data set into four parts with about the same number of values. The first quartile, Q₁, is the median of the lower half; Q₂ is the median of all the data; Q₃ is the median of the upper half. The interquartile range is Q₃ − Q₁.

In Spanish: cuartil
Also called: Q1, Q2, Q3, lower quartile, upper quartile, first quartile, third quartile
Grade level: 6-8

Grade 6Grade 7StatisticsCCSS 6.SP.B.4CCSS 6.SP.B.5.cSATACTGEDTEASAP

Quartile in a picture

The median splits the data in half; the quartiles split each half again. The box of a box plot runs from Q₁ to Q₃ and holds the middle 50% of the data.

Formal definition and notation

To find the quartiles, order the data and find the median, Q₂. Split the data into a lower half and an upper half; when the number of values is odd, leave the median out of both halves. Q₁ is the median of the lower half and Q₃ is the median of the upper half. About 25% of the data lie below Q₁, 50% below Q₂ and 75% below Q₃, so Q₁ and Q₃ are the 25th and 75th percentiles. Some calculators and spreadsheets use slightly different rules, so their quartiles can differ a little for small data sets.

Symbol	Read it as	Meaning
Q₁	Q one, the first or lower quartile	the middle of the lower half; about a quarter of the data are below it
Q₂	Q two	the median
Q₃	Q three, the third or upper quartile	the middle of the upper half; about three quarters of the data are below it
IQR = Q₃ − Q₁	interquartile range	the spread of the middle 50% of the data

Key facts

Always put the data in order first, from least to greatest.
Q₂ is the median, so the quartiles extend the idea of the median to the halves of the data.
The five-number summary is minimum, Q₁, median, Q₃, maximum; a box plot draws it, with the box from Q₁ to Q₃.
A quartile does not have to be a data value: for 4, 6, 9, 11, Q₁ = 6 + 92 = 7.5.
Quartiles and the IQR resist outliers: changing the maximum from 21 to 210 leaves Q₁, Q₂ and Q₃ the same.
Each of the four parts holds about 25% of the values, but the parts can have very different widths. A wide part means the data there are spread out.

Using quartiles: IQR and outlier fences

IQR = Q₃ − Q₁; outliers are below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR

Q₁, Q₃: the first and third quartiles
1.5 × IQR: how far beyond the box a value must be to count as an outlier

For the diagram’s data, IQR = 15 − 5 = 10, so the fences are 5 − 15 = −10 and 15 + 15 = 30. No value is outside them, so there are no outliers.

Try it: move the quartiles

Interactive

This box plot shows quiz scores. Drag or edit values and watch Q1, the median and Q3 move. Then press Add an outlier: the quartiles and IQR barely change.

This interactive needs JavaScript. The worked examples below show the same ideas step by step.

Quartile examples, step by step

Example 1: An odd number of values

Warm-up

Find the quartiles of 3, 7, 8, 5, 12, 14, 21.

Order: 3, 5, 7, 8, 12, 14, 21. There are 7 values.
The median is the 4th value: Q₂ = 8.
Lower half (without the median): 3, 5, 7, so Q₁ = 5.
Upper half: 12, 14, 21, so Q₃ = 14.

Answer: Q₁ = 5, Q₂ = 8 and Q₃ = 14.

Example 2: An even number of values

Core

Ten players scored 12, 15, 18, 20, 22, 25, 27, 30, 34 and 40 points. Find the quartiles and the interquartile range.

The data are in order, and n = 10. The median is the average of the 5th and 6th values: Q₂ = 22 + 252 = 23.5.
Lower half: 12, 15, 18, 20, 22, so Q₁ = 18.
Upper half: 25, 27, 30, 34, 40, so Q₃ = 30.
IQR = 30 − 18 = 12.

Answer: Q₁ = 18, Q₂ = 23.5, Q₃ = 30, and the IQR is 12.

Example 3: Is it an outlier?

Test-style

A data set has Q₁ = 42 and Q₃ = 58. Is the value 85 an outlier by the 1.5 × IQR rule?

IQR = 58 − 42 = 16.
1.5 × 16 = 24.
Upper fence: 58 + 24 = 82.
85 is greater than 82, so it lies beyond the fence.

Answer: Yes. 85 is above the upper fence of 82, so it is an outlier.

Common mistakes

✗ MistakeFinding quartiles from the data as listed: for 9, 2, 7, 4, 5, taking 2 as Q1.

✓ InsteadOrder first: 2, 4, 5, 7, 9. Then Q1 = 3 (the median of 2 and 4).

Quartiles describe positions in the ordered data, so the original order means nothing.

✗ MistakeThinking Q₁ is one quarter of the largest value: for a maximum of 40, Q₁ = 10.

✓ InsteadQ₁ is the middle of the lower half of the ordered data, such as 18 for the scores 12 to 40.

Quartiles split the count of values into four parts. They are not fractions of the biggest number.

✗ MistakeComputing the IQR as Q₃ + Q₁, or as maximum minus minimum.

✓ InsteadIQR = Q₃ − Q₁. For Q₁ = 18 and Q₃ = 30, it is 12.

The IQR measures the spread of the middle half only. Maximum minus minimum is the range.

Not to be confused with

Quartilevs.Quarter

A quarter is the fraction 14 of something. A quartile is a cut point in ordered data. The quartiles cut the data into four quarters, but each quartile is a single value, such as Q₁ = 18.

Quartilevs.Interquartile range

The interquartile range is one number made from two quartiles: Q₃ − Q₁. It measures spread, while each quartile marks a position.

Quartilevs.Median

The median is the middle of all the data and is the same as Q₂. Q₁ and Q₃ are the middles of the lower and upper halves.

Quick check: 3 questions

Question 1 of 3

What is Q₁ for 4, 6, 9, 11, 15, 18, 20, 25?

Show answer and explanation

Answer: A. The lower half is 4, 6, 9, 11, and its median is 6 + 92 = 7.5. 6 and 9 are only one of the two middle values, and 13 is the median of the whole set.

Question 2 of 3

What percent of a data set lies between the first and third quartiles?

Show answer and explanation

Answer: B. About 25% of the data lie below Q1 and 75% lie below Q3, so about 75% − 25% = 50% lie between them. 25% is one section, 75% is everything below Q3, and 100% is the whole data set.

Question 3 of 3

A box plot shows minimum 10, Q₁ = 22, median 30, Q₃ = 41 and maximum 60. What is the interquartile range?

Show answer and explanation

Answer: C. IQR = Q₃ − Q₁ = 41 − 22 = 19. 50 is the range, 11 is 41 − 30, and 30 is the median.

Quartile: frequently asked questions

How do you find the quartiles of a data set?

Put the data in order and find the median; that is Q₂. Split the list into a lower half and an upper half, leaving the median out if the count is odd. The median of the lower half is Q₁, and the median of the upper half is Q₃. For 1, 3, 4, 6, 8, 9, 12: Q₁ = 3, Q₂ = 6, Q₃ = 9.

Is the second quartile the same as the median?

Yes. Q₂ is the median: half the data lie at or below it and half at or above it. The name fits the pattern of the other two quartiles, which split each half again. So the three quartiles together cut the data into four parts, with the median in the middle.

Why does my calculator give a different quartile?

There is more than one accepted rule for quartiles. The rule taught in most US classes leaves the median out of both halves, while some calculators, spreadsheets and statistics programs include it or interpolate between values. For large data sets the answers are almost the same; for small ones they can differ, so follow the rule your class or test uses.

Practice and tools

Mean, median, mode and range lessonPractice on Effortless Math Box and Whisker Plot MakerDraw a box plot from Q1, the median and Q3 Descriptive Statistics CalculatorGet the quartiles and IQR of your data Percentile CalculatorQ1 and Q3 are the 25th and 75th percentiles

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.SP.B.4, CCSS 6.SP.B.5.c

Cite this entry

Effortless Math Team. “Quartile.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/quartile/

Flashcard

TermQuartileQ₁, Q₂, Q₃

DefinitionThree values that cut ordered data into four parts. For 2, 4, 5, 7, 8, 10, 12, 13, 15, 18, 21: Q₁ = 5, Q₂ = 10, Q₃ = 15.

Study the full deck

Browse all terms in the Math Dictionary · More statistics terms · Random term

DataData are facts or values collected to answer a question, such as survey answers, measurements or counts. Numerical data are numbers you measure or count, like heights; categorical data are labels or groups, like favorite colors. Data are organized in tables and shown in graphs so patterns are easy to see.Statistics · Grade 2, Grade 3, Grade 6MedianThe median is the middle value when data are listed in order. If there is an even number of values, the median is the average of the two middle values. Half the data lie at or below the median and half at or above it, and outliers barely move it.Statistics · Grade 5, Grade 6, Grade 7NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2PercentileA percentile tells what percent of the values in a data set are at or below a given value. If your test score is at the 80th percentile, about 80% of the scores are at or below yours. The 25th, 50th and 75th percentiles are the first quartile, the median and the third quartile.Statistics · Grade 7, Algebra 1, StatisticsBox PlotA box plot displays a data set through five numbers: minimum, first quartile, median, third quartile, and maximum. The box holds the middle half of the data, a line inside it marks the median, and whiskers reach out to the smallest and largest values. It makes spread easy to compare between sets.Statistics · Grade 6, Grade 7, Grade 8OutlierAn outlier is a data value that is far away from most of the other values in a set. In 4, 5, 6, 7, 50, the value 50 is an outlier. A common test flags any value more than 1.5 times the interquartile range below the first quartile or above the third. Outliers pull the mean strongly.Statistics · Grade 6, Grade 7, Grade 8Interquartile RangeThe interquartile range (IQR) is the spread of the middle half of a data set: the third quartile minus the first quartile, IQR = Q₃ − Q₁. Because it ignores the lowest and highest quarters of the data, outliers barely change it. In a box plot it is the length of the box.Statistics · Grade 6, Grade 7, Algebra 1AverageAn average is one number that stands for the typical size of the values in a data set. In everyday math, average means the mean: add all the values, then divide by how many values there are. In statistics, the median and the mode are also kinds of average.Statistics · Grade 5, Grade 6, Grade 7RangeThe range of a data set is the difference between the greatest and least values: maximum minus minimum. For 2, 5, 9, the range is 9 − 2 = 7. Range is a simple measure of spread: it tells how far apart the extreme values are, though one outlier can make it misleading about the rest of the data.Statistics · Grade 5, Grade 6, Grade 7Line PlotA line plot shows data values along a number line, often with marks above each value. Each X or dot stands for one data value, and repeated values stack in a column, so the tallest stack is the mode. Line plots make it easy to see clusters, gaps, outliers and the spread of a small data set.Statistics · Grade 3, Grade 4, Grade 5Standard DeviationStandard deviation is a number that measures how spread out data values are from their mean: roughly, the typical distance of a value from the mean. A small standard deviation means the values cluster near the mean; a large one means they are spread out. It is the square root of the variance.Statistics · Algebra 1, Statistics

Quartile

What is a quartile in math?

Quartile in a picture

Formal definition and notation

Key facts

Using quartiles: IQR and outlier fences

Try it: move the quartiles

Quartile examples, step by step

Example 1: An odd number of values

Example 2: An even number of values

Example 3: Is it an outlier?

Common mistakes

Not to be confused with

Quick check: 3 questions

Quartile: frequently asked questions

How do you find the quartiles of a data set?

Is the second quartile the same as the median?

Why does my calculator give a different quartile?

Related terms

Practice and tools

Sources

Flashcard

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