Outlier in Math: Definition, IQR Rule & How to Find One

Definitionnoun

What is an outlier in math?

An 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.

In Spanish: valor atípico
Also called: unusual value, extreme value, anomaly
How to say it: OUT-ly-er
Grade level: 6-8

Grade 6Grade 7Grade 8Algebra 1CCSS 6.SP.B.5.cCCSS 8.SP.A.1GEDSATACT

Outlier in a picture

The fences sit 1.5 IQRs outside the box. Every value inside them is ordinary; 22 is past the upper fence (13), so it is an outlier and is drawn as a separate dot.

Formal definition and notation

Informally, an outlier is an observation that differs markedly from the rest of the data. The standard 1.5 × IQR rule calls a value x a potential outlier if x < Q₁ − 1.5 × IQR or x > Q₃ + 1.5 × IQR, where Q₁ and Q₃ are the first and third quartiles and IQR = Q₃ − Q₁. In a scatter plot, an outlier is a point far from the overall pattern. An outlier may be a recording error, a rare but real value, or a sign of a different group, so it calls for investigation.

Symbol	Read it as	Meaning
Q₁, Q₃	Q one, Q three	the first and third quartiles: the medians of the lower and upper halves of the ordered data
IQR	interquartile range	Q₃ − Q₁, the spread of the middle half of the data
1.5 × IQR	one and a half I Q R	how far past a quartile a value must be to count as an outlier

Key facts

Outliers move the mean and the range a lot but barely move the median: for 4, 5, 6, 7, 50 the mean is 14.4 while the median is 6.
On a box plot, outliers are drawn as separate dots beyond the whiskers, and the whiskers stop at the last value inside the fences.
Outliers can be high or low: in 2, 29, 30, 31, 32, 33 the outlier is 2.
A data set can have no outliers, one, or several.
In a scatter plot, a point far from the trend of the others is an outlier, even if its x- and y-values are each ordinary on their own.

The 1.5 × IQR rule (outlier fences)

lower fence = Q₁ − 1.5 × IQR, upper fence = Q₃ + 1.5 × IQR

Q₁: first quartile (median of the lower half)
Q₃: third quartile (median of the upper half)
IQR: Q₃ − Q₁

Any value below the lower fence or above the upper fence is an outlier. Values exactly on a fence are not. Some books find quartiles slightly differently, which can move the fences a little.

Try it: add an outlier

Interactive

Press Add an outlier, type your own extreme value, or drag a dot far to the right. Values outside the fences turn orange; compare how far the mean moves with the median.

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

Outlier examples, step by step

Example 1: A low outlier

Warm-up

Which value is an outlier: 31, 29, 33, 30, 2, 32?

Order the data: 2, 29, 30, 31, 32, 33.
Five of the values are between 29 and 33. The value 2 is far below them.
Check with fences: Q₁ = 29, Q₃ = 32, IQR = 3, and the lower fence is 29 − 4.5 = 24.5.
2 is below 24.5.

Answer: The outlier is 2, a low value far below the rest.

Example 2: Use the 1.5 × IQR rule

Core

Use the 1.5 × IQR rule to test for outliers in 10, 12, 13, 13, 14, 15, 16, 26.

The data are in order and n = 8. Lower half: 10, 12, 13, 13, so Q₁ = 12.5. Upper half: 14, 15, 16, 26, so Q₃ = 15.5.
IQR = 15.5 − 12.5 = 3, and 1.5 × 3 = 4.5.
Lower fence: 12.5 − 4.5 = 8. Upper fence: 15.5 + 4.5 = 20.
Nothing is below 8. Only 26 is above 20.

Answer: The fences are 8 and 20, so 26 is an outlier and no other value is.

Example 3: How an outlier moves the mean and median

Test-style

Five test scores are 82, 85, 88, 90 and 35. The 35 is an outlier. If it is left out, which changes more, the mean or the median?

With 35: the mean is 3805 = 76; ordered, 35, 82, 85, 88, 90 has median 85.
Without 35: the mean is 3454 = 86.25; 82, 85, 88, 90 has median 85 + 882 = 86.5.
The mean changes by 86.25 − 76 = 10.25. The median changes by 86.5 − 85 = 1.5.

Answer: The mean changes more (by 10.25 points, against 1.5 for the median).

Common mistakes

✗ MistakeDeleting the 22 from 4, 5, 6, 7, 7, 8, 9, 22 just because it looks strange.

✓ InsteadCheck the source first. If 22 was a typing error for 2, fix it. If it is real, keep it and report it, or show results with and without it.

Do not remove an outlier unless there is a valid reason, such as a recording error.

✗ MistakeIn 3, 4, 5, 6, 7, the outlier is 7 because it is the largest value.

✓ InsteadQ₁ = 3.5, Q₃ = 6.5, IQR = 3, so the fences are −1 and 11. No value is outside them: there is no outlier.

Every data set has a largest and a smallest value. An outlier must be far from the rest, not just at the end.

✗ MistakeA student who is 6 feet 8 inches tall in a class height survey must be a mistake.

✓ InsteadUnusual is not the same as wrong. Confirm the measurement; a real but rare value belongs in the data.

Outliers may be errors, but they may also be the most interesting part of the data.

Not to be confused with

Outliervs.Range

The range is greatest minus least, so it always includes any outlier: for 4, 5, 6, 7, 50 the range is 46. An outlier is one value; the range is a measure of spread.

Outliervs.Maximum

Every data set has a maximum, its largest value. The maximum is an outlier only when it is far from the rest, as 22 is in 4, 5, 6, 7, 7, 8, 9, 22.

Quick check: 3 questions

Question 1 of 3

Which value is most likely an outlier in 52, 55, 49, 51, 90, 53?

Show answer and explanation

Answer: C. Ordered: 49, 51, 52, 53, 55, 90. Q₁ = 51, Q₃ = 55, IQR = 4, so the upper fence is 55 + 6 = 61, and only 90 is beyond it. 49 and 55 are just the smallest and second-largest values.

Question 2 of 3

A data set has Q₁ = 20 and Q₃ = 30. Which value is an outlier?

Show answer and explanation

Answer: B. IQR = 10, so the fences are 20 − 15 = 5 and 30 + 15 = 45. Only 46 is outside. 44 and 6 are close to the fences but still inside, and 35 is an ordinary value.

Question 3 of 3

For 3, 4, 4, 5, 5, 6, 30, which measure of center changes most if the 30 is removed?

Show answer and explanation

Answer: A. The mean drops from about 8.14 to 4.5. The median only moves from 5 to 4.5, and the modes (4 and 5) stay the same. Outliers pull the mean far more than the median or the mode.

Outlier: frequently asked questions

How do you find an outlier?

First look: a value far from the cluster on a dot plot or box plot is a candidate. To decide, use the 1.5 × IQR rule. Find Q₁ and Q₃, subtract to get the IQR, multiply it by 1.5, then subtract that from Q₁ and add it to Q₃. Any value outside those two fences is an outlier.

Should you remove outliers?

Only with a reason. If an outlier comes from a typing or measuring error, correct it or remove it and say so. If it is a real value, keep it, because it is part of what happened. You can report results both with and without it, or use the median and IQR, which outliers barely affect, to describe the data.

Can a data set have more than one outlier?

Yes. Any value beyond either fence counts, so a set can have several high outliers, several low ones, or both. It can also have none. In 1, 20, 21, 22, 23, 24, 60 both 1 and 60 are outliers, while in 3, 4, 5, 6, 7 there are none.

Practice and tools

Effortless Math practicePractice on Effortless Math Box and Whisker Plot MakerDraw a box plot and see outliers as separate dots Descriptive Statistics CalculatorQuartiles, IQR, mean and median for your data

Sources

Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 6.SP.B.5.c, CCSS 8.SP.A.1

Cite this entry

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

Flashcard

TermOutlierfar from the rest

DefinitionA value far away from the other data values. In 4, 5, 6, 7, 50 the outlier is 50; it pulls the mean up to 14.4.

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 6QuartileA 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, StatisticsScatter PlotA scatter plot displays paired numerical data as points on a coordinate plane, one point for each pair. It helps show whether two quantities have a pattern or relationship: points rising from left to right show a positive association, points falling show a negative association, and a shapeless cloud shows no association.Statistics · Grade 8, Algebra 1MedianThe 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 7MeanThe mean is the sum of the values divided by the number of values. It is the arithmetic average: add every value and divide by how many there are. For 3, 7, 5 and 9, the mean is 24 ÷ 4 = 6. The mean is the balance point of the data, so one outlier can pull it far.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 7Box 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 8InchAn inch is a US customary unit of length equal to one twelfth of a foot, about the width of an adult thumb. Its abbreviation is in. Rulers split each inch into halves, quarters, eighths and sixteenths. One inch is exactly 2.54 centimeters, and 12 inches make 1 foot.Measurement · Grade 2, Grade 3, Grade 4HeightHeight is how tall something is: the distance from its bottom to its top, measured straight up and down. In geometry, the height of a shape is the perpendicular distance from a base to the opposite vertex or side, so it always meets the base at a right angle. Area and volume formulas use it.Measurement · Grade 2, Grade 5, Grade 6Dot PlotA dot plot is a graph that shows each data value as a dot above its place on a number line. Dots for repeated values stack up in a column, so the tallest stack shows the mode, and you can see the center, spread, gaps and outliers of the data at a glance.Statistics · Grade 3, Grade 4, Grade 5HistogramA histogram uses bars to show how numerical data are grouped into intervals. The data are split into equal-width intervals, called bins, and each bar’s height is the count of values in that interval. The bars touch, because the intervals are continuous, which is what distinguishes a histogram from a bar graph.Statistics · Grade 6, Grade 7, Grade 8Standard 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, StatisticsInterquartile 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 1

Outlier

What is an outlier in math?

Outlier in a picture

Formal definition and notation

Key facts

The 1.5 × IQR rule (outlier fences)

Try it: add an outlier

Outlier examples, step by step

Example 1: A low outlier

Example 2: Use the 1.5 × IQR rule

Example 3: How an outlier moves the mean and median

Common mistakes

Not to be confused with

Quick check: 3 questions

Outlier: frequently asked questions

How do you find an outlier?

Should you remove outliers?

Can a data set have more than one outlier?

Related terms

Practice and tools

Sources

Flashcard

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