Box Plot in Math: Definition & How to Read It

Definitionnoun phrase

What is a box plot in math?

A 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.

In Spanish: diagrama de caja
Also called: box-and-whisker plot, box and whisker diagram, box-whisker plot
Grade level: 6-8

Grade 6Grade 7Grade 8Algebra 1CCSS 6.SP.B.4CCSS 6.SP.B.5.cSATACTAP

Box Plot in a picture

The five numbers split the ordered data into four groups of about the same size. The box is the middle 50%; its length, the IQR, measures how spread out the typical values are.

Formal definition and notation

A box plot is drawn over a number line from the five-number summary of a data set. The box spans from the first quartile Q₁ to the third quartile Q₃, so its length is the interquartile range IQR = Q₃ − Q₁; a line inside the box marks the median Q₂; and whiskers extend to the minimum and maximum. In a modified box plot, values more than 1.5 × IQR beyond a quartile are plotted as separate points (outliers).

Symbol	Read it as	Meaning
Q₁	Q one, the first quartile	the median of the lower half of the data; about 25% of values lie below it
Q₃	Q three, the third quartile	the median of the upper half; about 75% of values lie below it
IQR	interquartile range	Q₃ − Q₁, the length of the box

Key facts

Each of the four sections (whisker, half box, half box, whisker) holds about 25% of the data, even when the sections have different lengths.
A longer section means the values there are more spread out, not that it holds more values.
The range is max minus min; the IQR is the length of the box. The IQR is not affected by extreme values.
If the median line sits left of center and the right whisker is long, the data are skewed right.
A box plot hides the individual values, so two very different data sets can have the same box plot.

Five-number summary and outlier fences

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

min, max: the smallest and largest values: the ends of the whiskers
Q₁, Q₃: the ends of the box
median: the line inside the box
1.5 × IQR: the usual rule for flagging outliers

Order the data first. Find the median, then find the median of each half (leave the middle value out of both halves when the count is odd).

Try it: build a box plot

Interactive

The data start as a box plot. Edit the numbers or press Add an outlier and watch the box, whiskers and median move; switch to Dot plot to see every value and drag dots.

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

Box Plot examples, step by step

Example 1: Read range and IQR

Warm-up

A box plot of quiz scores has minimum 12, Q1 = 20, median 26, Q3 = 31 and maximum 40. Find the range and the interquartile range.

Range = maximum − minimum: 40 − 12 = 28.
IQR = Q3 − Q1: 31 − 20 = 11.
The middle half of the scores spans 11 points; all the scores span 28.

Answer: The range is 28 and the IQR is 11.

Example 2: Five-number summary with an even count

Core

Find the five-number summary of 7, 21, 4, 15, 9, 25, 12, 18, 10, 16.

Order the data: 4, 7, 9, 10, 12, 15, 16, 18, 21, 25. Min = 4, max = 25.
There are 10 values, so the median is the mean of the 5th and 6th: 12 + 152 = 13.5.
Lower half: 4, 7, 9, 10, 12, so Q1 = 9. Upper half: 15, 16, 18, 21, 25, so Q3 = 18.
Write them in order: min, Q1, median, Q3, max.

Answer: The five-number summary is 4, 9, 13.5, 18, 25.

Example 3: Is it an outlier?

Test-style

A data set has Q1 = 40 and Q3 = 52. Its largest value is 75. Using the 1.5 × IQR rule, is 75 an outlier?

IQR = 52 − 40 = 12.
1.5 × 12 = 18.
Upper fence = Q₃ + 18 = 52 + 18 = 70.
75 is greater than 70, so it is an outlier and would be drawn as a separate point.

Answer: Yes. The upper fence is 70, and 75 is above it, so 75 is an outlier.

Common mistakes

✗ MistakeReading the box plot as if every value were drawn: ‘there are 5 data values.’

✓ InsteadA box plot shows 5 summary numbers. The data set behind it might have 11 values or 1,000.

The box plot does not show every data value; it summarizes the distribution.

✗ Mistake‘The right whisker is longer, so more students scored there.’

✓ InsteadEach whisker holds about 25% of the values. A longer whisker means those values are more spread out.

Section lengths show spread, not counts. Every quarter has about the same number of values.

✗ MistakeTaking the middle of the box as the median.

✓ InsteadRead the line inside the box. In the picture above, the box runs from 6 to 15 but the median is 10, not 10.5.

The median line is often off-center. Its position shows whether the middle half is skewed.

✗ MistakeFinding quartiles from unsorted data.

✓ InsteadPut the values in order before finding the median and quartiles.

Quartiles are positions in the ordered list, just like the median.

Not to be confused with

Box Plotvs.Histogram

A histogram shows how many values fall in each interval, so you can see the shape in detail. A box plot shows only five summary numbers.

Box Plotvs.Line plot (dot plot)

A line plot shows every value as a dot or X. A box plot replaces the dots with a box and whiskers.

Box Plotvs.Bar graph

A bar graph compares counts in categories. A box plot summarizes one set of numerical data along a number line.

Quick check: 3 questions

Question 1 of 3

What does the line inside the box of a box plot show?

Show answer and explanation

Answer: B. The line inside the box is the median, the middle value of the ordered data. The mean is not shown on a box plot, and the first quartile is the left edge of the box.

Question 2 of 3

About what percent of the data lies inside the box?

Show answer and explanation

Answer: D. The box runs from Q1 (25% of the data below) to Q3 (75% below), so it holds the middle 50%. 25% is one whisker or one half of the box, and 75% is everything below Q3.

Question 3 of 3

Class A’s box runs from 70 to 90 and Class B’s from 78 to 84. Both medians are 80. Which class’s middle half is more spread out?

Show answer and explanation

Answer: A. IQR is the box length: 90 − 70 = 20 for A and 84 − 78 = 6 for B. Equal medians say the centers match, not the spreads, and the box plot alone is enough to compare IQRs.

Box Plot: frequently asked questions

How do you read a box plot?

Read left to right. The end of the left whisker is the minimum and the left edge of the box is Q1. The line inside the box is the median, the right edge is Q3, and the end of the right whisker is the maximum. About a quarter of the data lies in each section, and the box holds the middle half.

How do you make a box plot?

Put the data in order and find the five-number summary: minimum, Q1, median, Q3, maximum. Draw a number line that covers the data. Draw a box from Q1 to Q3, a line at the median, and whiskers from the box to the minimum and maximum. If you check for outliers, plot them as dots and stop the whiskers at the last non-outlier.

What does a box plot tell you about skew?

If the median sits near the left of the box and the right whisker is long, the upper values are spread out: the data are skewed right, as with incomes. A median near the right with a long left whisker means skewed left. A centered median and equal whiskers suggest roughly symmetric data.

Practice and tools

Effortless Math practicePractice on Effortless Math Box and Whisker Plot MakerDraw a box plot from your own data Descriptive Statistics CalculatorQuartiles, IQR and the five-number summary 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. “Box Plot.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/box-plot/

Flashcard

TermBox Plotfive numbers, one box

DefinitionA graph of min, Q1, median, Q3 and max. The box holds the middle half of the data; the line inside is the median.

Study the full deck

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

MedianThe 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 7Number LineA number line is a straight line with numbers placed at equal distances along it, so every point stands for a number. Zero sits in the middle, positive numbers go to the right and negative numbers to the left. Numbers increase from left to right, so a number line shows order, distance and adding as moving.Number System · Grade 2, Grade 3, Grade 6DataData 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, StatisticsLengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3Interquartile 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 1OutlierAn 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 8NumberA 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 2HistogramA 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 8Line 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 5Bar GraphA bar graph uses bars to compare amounts in different categories. The taller or longer the bar, the larger the value it represents. Each bar stands for one category, such as a favorite sport, and its length is read against a number scale. Bars can be vertical or horizontal, with gaps between them.Statistics · Grade 3, Grade 4, Grade 5RangeThe 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 7

Box Plot

What is a box plot in math?

Box Plot in a picture

Formal definition and notation

Key facts

Five-number summary and outlier fences

Try it: build a box plot

Box Plot examples, step by step

Example 1: Read range and IQR

Example 2: Five-number summary with an even count

Example 3: Is it an outlier?

Common mistakes

Not to be confused with

Quick check: 3 questions

Box Plot: frequently asked questions

How do you read a box plot?

How do you make a box plot?

What does a box plot tell you about skew?

Related terms

Practice and tools

Sources

Flashcard

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