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Pre-Algebra • Statistics & Probability

Interpreting Box Plots — I can interpret a box plot by identifying quartiles and IQR and relating them to context.

Learn to read box plots so you can quickly compare data and spot patterns.

Explore the lesson 10 min read
Middle School Math emblem with sigma notation, geometric shapes, and a ruler.
Inside this lesson 10 sections 5 questions 8 figures

SECTION 1

Where Did Box Plots Come From?

Have you ever tried to describe a big pile of numbers to a friend? Maybe you collected test scores for your whole class. You could list every single number, but that takes forever. People have been looking for faster, smarter ways to summarize data for hundreds of years.

The idea of splitting data into quarters goes way back. But it was a mathematician named John Tukey who invented the box-and-whisker plot in the 1970s. He wanted a simple picture that could show the shape and spread of data at a glance.

1700s • Early Statistics

Mathematicians begin using averages and ranges to describe sets of numbers.

1800s • Quartiles Appear

Statisticians start dividing data into four equal parts called quartiles to better understand spread.

1977 • Tukey's Box Plot

John Tukey publishes his book Exploratory Data Analysis and introduces the box-and-whisker plot.

Today • Box Plots Everywhere

Box plots are used in science, sports, business, and schools to quickly compare groups of data.

So the big question Tukey answered was: How can we draw one small picture that shows the center, the spread, and the overall shape of a data set? That picture is the box plot.

SECTION 2

Key Vocabulary and Core Ideas

Before you can read a box plot, you need to know a few key words. These ideas all work together. Think of them as the building blocks of the box plot.

1

Median (Q2)

The median is the middle value when you put all the numbers in order. It splits the data in half.
2

Lower Quartile (Q1)

The lower quartile (Q1) is the median of the lower half of the data. About 25% of values fall below Q1.
3

Upper Quartile (Q3)

The upper quartile (Q3) is the median of the upper half. About 75% of values fall below Q3.
4

IQR (Interquartile Range)

The interquartile range is Q3 − Q1. It tells you how spread out the middle 50% of the data is.
5

Minimum & Maximum

The smallest and largest values in the data set. The whiskers stretch out to these values.
✦

KEY TAKEAWAY

Think of a box plot like a sandwich. The two slices of bread are Q1 and Q3. The filling in the middle is the median line. The "whiskers" are like toothpicks sticking out to the smallest and largest values. The thicker the sandwich (wider the box), the more spread out the middle half of your data is.

SECTION 3

Anatomy of a Box Plot

The diagram below shows every part of a box plot. Study it carefully. Once you can name each piece, reading any box plot becomes easy.

Anatomy of a Box Plot102030405060Minimum13Q125Median (Q2)37Q350Maximum58IQR = Q3 − Q1 = 25whiskerwhiskerbox
The box stretches from Q1 to Q3. The median line sits inside the box. The whiskers reach out to the minimum and maximum. The IQR is the width of the box.

Notice that the box holds the middle 50% of all the data. If you have 20 data values, about 10 of them live inside that box. The left whisker covers the lowest 25%, and the right whisker covers the highest 25%.

💡

Remember

Each section of a box plot — the left whisker, the left half of the box, the right half of the box, and the right whisker — holds roughly 25% of the data, even though the sections may look different widths.

SECTION 4

The Math Behind Box Plots

Building a box plot starts with putting your data in order from smallest to largest. Then you find three key numbers: Q1, Q2 (the median), and Q3. Let's see the formulas.

MEDIAN (Q2)

Q2=middlevalueoftheordereddatasetQ2 = middle value of the ordered data set

If there is an even number of values, Q2 is the average of the two middle values.

LOWER QUARTILE (Q1)

Q1=medianofthelowerhalf(valuesbelowQ2)Q1 = median of the lower half (values below Q2)

The lower half is every value to the left of Q2 in the ordered list.

UPPER QUARTILE (Q3)

Q3=medianoftheupperhalf(valuesaboveQ2)Q3 = median of the upper half (values above Q2)

The upper half is every value to the right of Q2 in the ordered list.

INTERQUARTILE RANGE (IQR)

IQR=Q3−Q1IQR = Q3 − Q1

A larger IQR means the middle 50% is more spread out. A smaller IQR means the data is more bunched together.

You also need the minimum (smallest value) and the maximum (largest value). Together, these five numbers — minimum, Q1, median, Q3, maximum — are called the five-number summary.

SECTION 5

What Box Plots Tell Us About Data

Box plots really shine when you compare two or more groups side by side. The diagram below shows the test scores for two classes. Notice how you can instantly see which class scored higher overall and which class had more spread.

Comparing Test Scores: Class A vs. Class B50556065707580Class A5258657277Class B6065697478Class A IQR = 72 − 58 = 14Class B IQR = 74 − 65 = 9Class B has a smaller IQR → scores are more consistent.
Class A's box is wider (IQR = 14), meaning scores are more spread out. Class B's box is narrower (IQR = 9), meaning students scored more consistently. Class B's median (69) is also higher than Class A's median (65).

Here are some important things to look for when you compare box plots:

  • ✓ Center: Which group has a higher median? That group tends to have higher values overall.
  • ✓ Spread: Which group has a larger IQR? A larger IQR means more variety in the middle 50%.
  • ✓ Range: Which group has a larger overall range (max − min)? This tells you the total spread.
  • ✓ Overlap: Do the boxes overlap? If not, the groups are very different.

SECTION 6

Worked Example: Basketball Free Throws

A coach recorded the number of free throws each player made during 11 practice sessions: 3, 5, 7, 8, 10, 12, 14, 15, 18, 20, 22. Let's find the five-number summary and the IQR, then describe what they tell us.

Finding the Five-Number Summary & IQR

  1. 1

    Step 1 — Order the Data

    The data is already in order from least to greatest: 3, 5, 7, 8, 10, 12, 14, 15, 18, 20, 22.
    Ordered data: 3, 5, 7, 8, 10, 12, 14, 15, 18, 20, 22
  2. 2

    Step 2 — Find the Median (Q2)

    There are 11 values. The middle value is the 6th number.
    Q2 (median) = 12
  3. 3

    Step 3 — Find Q1

    The lower half is the five numbers below the median: 3, 5, 7, 8, 10. The middle of these five is the 3rd number.
    Q1 = 7
  4. 4

    Step 4 — Find Q3

    The upper half is the five numbers above the median: 14, 15, 18, 20, 22. The middle of these five is the 3rd number.
    Q3 = 18
  5. 5

    Step 5 — Find the IQR

    Subtract Q1 from Q3: IQR = 18 − 7 = 11.
    IQR = 11
  6. 6

    Step 6 — Write the Five-Number Summary

    Minimum = 3, Q1 = 7, Median = 12, Q3 = 18, Maximum = 22.
    Five-Number Summary: 3, 7, 12, 18, 22
  7. 7

    Step 7 — Interpret in Context

    The typical (median) number of free throws made is 12. The middle 50% of sessions had between 7 and 18 free throws. An IQR of 11 shows quite a bit of variation from session to session.
    The player's performance varied a lot — the middle half of sessions ranged from 7 to 18 free throws.

SECTION 7

Strengths and Limitations of Box Plots

Box plots are powerful, but they don't show everything. Here's a quick comparison of what box plots are great at and what they leave out.

StrengthsLimitations
Show center (median) and spread (IQR) at a glance.Don't show individual data values.
Easy to compare two or more groups side by side.Don't tell you how many data points there are.
Help spot outliers (values far from the rest).Can't show the exact shape of the distribution (like a histogram can).
Work well for large data sets.Two very different data sets can sometimes produce similar-looking box plots.

Strengths vs. Limitations of Box Plots

✦

KEY TAKEAWAY

A box plot is like a movie trailer — it gives you the highlights and the feel of the story, but it doesn't show you every scene. If you need to see every data point, you might want a dot plot or a histogram instead. But for a quick, clean comparison, box plots are hard to beat.

SECTION 8

From Box Plots to More Advanced Statistics

The ideas behind box plots connect to bigger ideas you'll see in high school and beyond. The table below shows how each box plot concept grows into something more advanced.

Box Plot ConceptAdvanced VersionWhat Changes?
MedianMean (average)The mean uses every value, not just the middle. Median is better when there are outliers.
IQRStandard deviationStandard deviation measures spread using every data point, not just the quartiles.
Five-number summaryFull distribution / histogramHistograms show the shape of the data in much more detail.
Comparing two box plotsHypothesis testingStatisticians use formal tests to decide if two groups are truly different.

How box plot ideas grow into advanced statistics

You don't need to learn all of that right now. The important thing is that the skills you're building — finding medians, calculating IQR, and comparing data visually — are the same skills professional scientists and analysts use every day. You're building a strong foundation!

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A box plot has a very wide box and short whiskers. What does this tell you about the data?
Show answer
A wide box means the IQR is large, so the middle 50% of the data is spread over a big range. Short whiskers mean the minimum and maximum aren't far from Q1 and Q3. In other words, most of the variation in this data set happens in the middle half, and the extreme values aren't too far from the quartiles.

Question 2: BASIC CALCULATION

Here is an ordered data set of daily high temperatures (°F): 58, 62, 65, 68, 70, 73, 75, 79, 84. Find Q1, Q2 (median), Q3, and the IQR.
Show answer
There are 9 values, so the median (Q2) is the 5th value: 70. The lower half is 58, 62, 65, 68. The median of these four values is (62 + 65) ÷ 2 = 63.5, so Q1 = 63.5. The upper half is 73, 75, 79, 84. The median of these is (75 + 79) ÷ 2 = 77, so Q3 = 77. The IQR = Q3 − Q1 = 77 − 63.5 = 13.5°F.

Question 3: INTERMEDIATE

A box plot of students' reading times (in minutes) shows: minimum = 10, Q1 = 18, median = 25, Q3 = 34, maximum = 50. About what percent of students read for more than 34 minutes? Calculate the IQR and explain what it means in context.
Show answer
About 25% of the students read for more than 34 minutes, because Q3 = 34 means 75% of values fall at or below 34. The IQR = 34 − 18 = 16 minutes. This means the middle half of students' reading times spanned a 16-minute window, from 18 to 34 minutes. This is a moderate amount of spread — students' habits varied, but not wildly.

Question 4: APPLIED

Two soccer teams tracked the number of goals scored per game over a season. Team X has a five-number summary of 0, 1, 2, 4, 6. Team Y has a five-number summary of 1, 2, 3, 3, 5. Which team is more consistent? Which team has a higher typical performance? Explain using the median and IQR.
Show answer
Team Y is more consistent because its IQR = 3 − 2 = 1 goal, while Team X's IQR = 4 − 1 = 3 goals. A smaller IQR means less variation in the middle 50% of games. Team Y also has a higher median (3 goals vs. 2 goals), meaning its typical performance is better. Even though Team X had a higher maximum (6 vs. 5), Team Y was more reliable game after game. A coach would say Team Y's scoring was more predictable.

Question 5: CRITICAL THINKING

Two data sets both have a median of 50 and an IQR of 10. Does this mean the data sets look the same? Could the box plots still look different? Give an example to explain your thinking.
Show answer
No, the data sets do not have to look the same. Even though the median and IQR match, the minimum and maximum could be very different, which would change the length of the whiskers. For example, Data Set A could have a five-number summary of 20, 45, 50, 55, 80 (long whiskers), while Data Set B could have 44, 45, 50, 55, 57 (very short whiskers). Both have a median of 50 and an IQR of 55 − 45 = 10, but their box plots would look quite different. This shows that the IQR only describes the middle 50%, not the full picture.

SUMMARY

Lesson Summary

Summary

A box plot is a visual display built from the five-number summary: the minimum, Q1 (lower quartile), Q2 (median), Q3 (upper quartile), and the maximum. The box stretches from Q1 to Q3 and shows where the middle 50% of data lives. The line inside the box marks the median. The whiskers reach to the smallest and largest values.

The interquartile range (IQR) equals Q3 − Q1 and measures how spread out the middle half of the data is. A small IQR means the data is bunched together; a large IQR means more spread. When you compare box plots side by side, focus on the median (center), the IQR (spread), and the overall range to draw conclusions about the data in context.

Varsity Tutors • Pre-Algebra • Interpreting Box Plots