Interquartile Range (IQR): Definition, Formula & Examples

Definitionnoun phrase

What is the interquartile range in math?

The 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.

In Spanish: rango intercuartílico
Also called: IQR, midspread, middle 50%, H-spread
Grade level: 6-8

Grade 6Grade 7Algebra 1StatisticsCCSS 6.SP.A.3CCSS 6.SP.B.5.cCCSS HSS-ID.A.2SATACTGEDAP

Interquartile Range in a picture

The IQR is the width of the box: the middle half of the data. The far value 34 stretches the range to 32 but leaves the IQR at 10.

Formal definition and notation

Order the data and find the median. The first quartile Q₁ is the median of the lower half and the third quartile Q₃ is the median of the upper half; with an odd count, leave the median itself out of both halves. The interquartile range is IQR = Q₃ − Q₁, the distance between the first and third quartiles. It is a measure of variability that is resistant to extreme values.

Symbol	Read it as	Meaning
IQR	I Q R, or interquartile range	Q₃ − Q₁
Q₁	Q one	first quartile, the 25th percentile
Q₃	Q three	third quartile, the 75th percentile

Key facts

About half of the data lie between Q₁ and Q₃, inside a span of one IQR.
The IQR is never negative, and it is 0 when the middle half of the values are all equal.
The range uses the two most extreme values; the IQR skips them, so it is the better spread for skewed data or data with outliers.
Pair the IQR with the median, and the standard deviation with the mean.
The 1.5 × IQR rule flags possible outliers; box plots draw them as separate dots.
Different calculators find quartiles in slightly different ways, so a small data set can give slightly different IQRs.

Interquartile range formula

IQR = Q₃ − Q₁

Q₁: median of the lower half of the ordered data
Q₃: median of the upper half of the ordered data

Outlier test: a value is a possible outlier if it is below Q₁ − 1.5 × IQR or above Q₃ + 1.5 × IQR.

Try it: the middle half

Interactive

The box shows Q₁ to Q₃, and the readout gives the IQR. Drag the largest dot far to the right: the range jumps, but the IQR stays put.

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

Interquartile Range examples, step by step

Example 1: IQR with an even count

Warm-up

Find the IQR of 3, 5, 6, 8, 10, 12, 15, 20.

There are 8 values, already in order. Lower half: 3, 5, 6, 8. Upper half: 10, 12, 15, 20.
Q₁ = 5 + 62 = 5.5 and Q₃ = 12 + 152 = 13.5.
IQR = 13.5 − 5.5 = 8.

Answer: The IQR is 8.

Example 2: IQR with an odd count

Core

Find the IQR of the ages 21, 14, 9, 30, 17, 12, 25.

Order: 9, 12, 14, 17, 21, 25, 30. The median is 17.
Leave 17 out. Lower half: 9, 12, 14, so Q₁ = 12. Upper half: 21, 25, 30, so Q₃ = 25.
IQR = 25 − 12 = 13.

Answer: The IQR is 13 years.

Example 3: Is it an outlier?

Test-style

For a set of commute times, Q₁ = 40 minutes and Q₃ = 52 minutes. Is a commute of 75 minutes an outlier by the 1.5 × IQR rule?

IQR = 52 − 40 = 12.
1.5 × 12 = 18.
Upper fence: 52 + 18 = 70.
75 is greater than 70.

Answer: Yes. 75 minutes is above the upper fence of 70, so it is an outlier.

Common mistakes

✗ MistakeFinding quartiles in unordered data: for 21, 14, 9, 30, 17, 12, 25, taking Q₁ = 14 from the first three values.

✓ InsteadOrder first: 9, 12, 14, 17, 21, 25, 30, so Q₁ = 12.

Quartiles are medians of halves of the ordered data.

✗ MistakeSubtracting the wrong numbers: IQR = max − min = 30 − 9 = 21.

✓ InsteadThat is the range. IQR = Q₃ − Q₁ = 25 − 12 = 13.

The IQR measures only the middle half; the range uses the extremes.

✗ MistakeGiving the quartiles as the answer: “IQR = 12 to 25.”

✓ InsteadThe IQR is one number, the difference: 25 − 12 = 13.

Like the range, the IQR is a distance, not an interval.

Not to be confused with

Interquartile Rangevs.Range

The range is maximum minus minimum and uses every extreme value. For 2, 4, 5, 7, 8, 10, 12, 13, 15, 18, 34 the range is 32 but the IQR is only 10.

Interquartile Rangevs.Quartile

A quartile is a single cut point, like Q₁ = 5. The IQR is the distance between two of them, Q₃ − Q₁.

Quick check: 3 questions

Question 1 of 3

Find the IQR of 1, 3, 4, 6, 9, 11, 14, 18.

Show answer and explanation

Answer: D. Lower half 1, 3, 4, 6 gives Q₁ = 3.5; upper half 9, 11, 14, 18 gives Q₃ = 12.5. IQR = 12.5 − 3.5 = 9. 17 is the range, and 12.5 and 3.5 are the quartiles themselves, not the distance between them.

Question 2 of 3

A box plot has Q₁ = 20 and Q₃ = 32. What is the IQR?

Show answer and explanation

Answer: A. IQR = 32 − 20 = 12. 26 is the midpoint of the box, 52 is the sum, and “20 to 32” names the quartiles instead of the distance between them.

Question 3 of 3

Q₁ = 10 and Q₃ = 18. Which value is an outlier by the 1.5 × IQR rule?

Show answer and explanation

Answer: C. IQR = 8 and 1.5 × 8 = 12, so the fences are 10 − 12 = −2 and 18 + 12 = 30. Only 31 is outside. 0 and 28 are inside the fences, and 25 is well inside.

Interquartile Range: frequently asked questions

Why use the IQR instead of the range?

The range depends only on the two most extreme values, so one unusual value can make it huge. The IQR measures the middle 50% of the data and ignores the top and bottom quarters, so outliers barely move it. That makes it the better measure of spread for skewed data, such as incomes or home prices.

Do you include the median when finding the quartiles?

In the method taught with Common Core and most US textbooks, no: with an odd number of values, leave the median out and find the medians of the two halves. With an even count, the data split evenly, so nothing is left out. Some calculators and spreadsheets use other rules and can give slightly different quartiles.

Practice and tools

Mean, median, mode and range lessonPractice on Effortless Math Box and Whisker Plot MakerDraw the box and read the IQR Descriptive Statistics CalculatorQuartiles, IQR and more

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.SP.A.3, CCSS 6.SP.B.5.c, CCSS HSS-ID.A.2

Cite this entry

Effortless Math Team. “Interquartile Range.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/interquartile-range/

Flashcard

TermInterquartile RangeQ₃ − Q₁

DefinitionThe spread of the middle half of the data. For 2, 4, 5, 7, 8, 10, 12, 13, 15, 18, 34: Q₁ = 5, Q₃ = 15, IQR = 10.

Study the full deck

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

OutlierAn 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 8Box 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 8DataData 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 7QuartileA 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.Statistics · Grade 6, Grade 7, StatisticsPercentileA 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, StatisticsRangeThe 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 7Standard 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, StatisticsNumberA 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 2DifferenceA difference is the result of subtracting one number from another. In 12 − 7 = 5, the difference is 5. It tells how much bigger one number is than another, or how far apart they are on a number line. “The difference of a and b” means a − b, in that order.Operations · Grade 1, Grade 2, Grade 3Odd NumberAn odd number is a whole number that cannot be split into two equal groups: dividing it by 2 always leaves a remainder of 1. Odd numbers end in 1, 3, 5, 7 or 9, such as 7, 23 and 409. Every odd number can be written as 2k + 1, one more than a multiple of 2.Number System · Grade 2, Grade 3, Grade 4Mean Absolute DeviationThe mean absolute deviation (MAD) is the average distance between each data value and the mean. Find the mean, find how far each value is from it, then average those distances. A small MAD means the data cluster close to the mean; a large MAD means they are spread out.Statistics · Grade 6, Grade 7

Interquartile Range

What is the interquartile range in math?

Interquartile Range in a picture

Formal definition and notation

Key facts

Interquartile range formula

Try it: the middle half

Interquartile Range examples, step by step

Example 1: IQR with an even count

Example 2: IQR with an odd count

Example 3: Is it an outlier?

Common mistakes

Not to be confused with

Quick check: 3 questions

Interquartile Range: frequently asked questions

Why use the IQR instead of the range?

Do you include the median when finding the quartiles?

Related terms

Practice and tools

Sources

Flashcard

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