How to make a box and whisker plot

How to Make a Box and Whisker Plot

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A box-and-whisker plot displays spread by using five numbers: minimum, first quartile, median, third quartile, and maximum. The box runs from \(Q_1\) to \(Q_3\), the line inside the box is the median, and the whiskers extend to the least and greatest non-outlier values.

Core Formulas

  • Five-number summary: minimum, \(Q_1\), median, \(Q_3\), maximum.
  • Interquartile range: \(IQR=Q_3-Q_1\).
  • Outlier fences: lower fence \(=Q_1-1.5(IQR)\), upper fence \(=Q_3+1.5(IQR)\).

Worked Example

For \(4, 7, 9, 10, 13\), the median is \(9\), \(Q_1=5.5\), and \(Q_3=11.5\). Thus the five-number summary is \(4, 5.5, 9, 11.5, 13\).

Reference Diagram

box and whisker plot

Box and Whisker Plots

Think of this lesson as more than a rule to memorize. Box and Whisker Plots is about organizing data and choosing the right summary measure. A strong student does not rush to the first formula on the page; they pause, identify the structure of the problem, and then choose the tool that matches that structure. That pause is what prevents most mistakes.

Statistics is about describing data clearly. First organize the values, then choose the measure or graph that best answers the question being asked.

Here is the teacher way to approach the topic. First, read the problem slowly and underline the information that is actually given. Next, name the target: are you finding a value, simplifying an expression, comparing two quantities, solving for a variable, or interpreting a graph? Once the target is clear, the calculation becomes much less mysterious because every step has a job.

  • Read the scale and labels first.
  • Identify the key values the graph shows.
  • Connect the graph to the formula or data table.
  • Answer using the units and context.

A helpful habit is to say what each number represents before using it. For example, if a number is a denominator, a radius, a slope, a common difference, or a coefficient, it should not be treated like an ordinary loose number. Its role tells you where it belongs in the formula. This is especially important on ACT-style questions because many wrong answer choices come from using the right numbers in the wrong places.

Another good habit is to keep the work organized vertically. Write one clean line for substitution, one line for simplifying, and one line for the final answer. If the problem has units, keep the units attached. If the problem has signs, exponents, or parentheses, copy them carefully from one line to the next. Most errors in this topic are not caused by a hard idea; they are caused by dropping a negative sign, combining unlike terms, using the wrong denominator, or skipping a check.

When you finish, ask a quick reasonableness question. Should the answer be positive or negative? Should it be larger or smaller than the original number? Does it fit the graph, table, shape, or equation? Can you plug it back into the original problem? This final check turns the lesson from a procedure into understanding.

On a test, the goal is not to write the longest solution. The goal is to write enough clear work that you can see the structure, avoid traps, and recover quickly if one line goes wrong. Practice the examples below with that mindset: identify the type of problem, choose the matching rule, show the substitution, simplify carefully, and check the answer before moving on.

Exercises

Solve each ACT-style practice problem. The questions increase in difficulty.

1) For the data set \( 4, 7, 9, 10, 13 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

2) For the data set \( 3, 5, 8, 12, 14, 16 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

3) For the data set \( 11, 13, 15, 17, 19, 21, 23 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

4) For the data set \( 2, 6, 7, 9, 10, 15, 18, 20 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

5) For the data set \( 22, 24, 25, 29, 31, 34, 38, 41 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

6) For the data set \( 5, 5, 8, 10, 12, 12, 15, 18, 20 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

7) For the data set \( 30, 32, 35, 36, 38, 40, 44, 48, 50 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

8) For the data set \( 7, 9, 12, 12, 14, 17, 19, 21, 25, 28 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

9) For the data set \( 40, 42, 45, 47, 50, 52, 55, 58, 60, 65 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

10) For the data set \( 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

11) For the data set \( 3, 4, 4, 5, 7, 9, 11, 13, 15, 40 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

12) For the data set \( 18, 19, 21, 22, 24, 25, 27, 29, 30, 55 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

13) For the data set \( 60, 62, 64, 65, 66, 68, 70, 72, 74, 90 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

14) For the data set \( 1, 4, 6, 8, 10, 12, 14, 16, 18, 35, 38 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

15) For the data set \( 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 100 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

16) For the data set \( 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

17) For the data set \( 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 60 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

18) For the data set \( 45, 46, 47, 49, 50, 52, 53, 55, 56, 58, 80, 82 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

19) For the data set \( 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

20) For the data set \( 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 80 \), find the five-number summary and interquartile range. Identify any outliers using the \(1.5IQR\) rule.

 
1) Order the data: \( 4, 7, 9, 10, 13 \).
The median is \(9\). The lower-half median is \(Q_1=\frac{11}{2}\), and the upper-half median is \(Q_3=\frac{23}{2}\).
Five-number summary: \(4, \frac{11}{2}, 9, \frac{23}{2}, 13\).
\(IQR=\frac{23}{2}-\frac{11}{2}=6\). The fences are \(\frac{-7}{2}\) and \(\frac{41}{2}\), so the outliers are none.
2) Order the data: \( 3, 5, 8, 12, 14, 16 \).
The median is \(10\). The lower-half median is \(Q_1=5\), and the upper-half median is \(Q_3=14\).
Five-number summary: \(3, 5, 10, 14, 16\).
\(IQR=14-5=9\). The fences are \(\frac{-17}{2}\) and \(\frac{55}{2}\), so the outliers are none.
3) Order the data: \( 11, 13, 15, 17, 19, 21, 23 \).
The median is \(17\). The lower-half median is \(Q_1=13\), and the upper-half median is \(Q_3=21\).
Five-number summary: \(11, 13, 17, 21, 23\).
\(IQR=21-13=8\). The fences are \(1\) and \(33\), so the outliers are none.
4) Order the data: \( 2, 6, 7, 9, 10, 15, 18, 20 \).
The median is \(\frac{19}{2}\). The lower-half median is \(Q_1=\frac{13}{2}\), and the upper-half median is \(Q_3=\frac{33}{2}\).
Five-number summary: \(2, \frac{13}{2}, \frac{19}{2}, \frac{33}{2}, 20\).
\(IQR=\frac{33}{2}-\frac{13}{2}=10\). The fences are \(\frac{-17}{2}\) and \(\frac{63}{2}\), so the outliers are none.
5) Order the data: \( 22, 24, 25, 29, 31, 34, 38, 41 \).
The median is \(30\). The lower-half median is \(Q_1=\frac{49}{2}\), and the upper-half median is \(Q_3=36\).
Five-number summary: \(22, \frac{49}{2}, 30, 36, 41\).
\(IQR=36-\frac{49}{2}=\frac{23}{2}\). The fences are \(\frac{29}{4}\) and \(\frac{213}{4}\), so the outliers are none.
6) Order the data: \( 5, 5, 8, 10, 12, 12, 15, 18, 20 \).
The median is \(12\). The lower-half median is \(Q_1=\frac{13}{2}\), and the upper-half median is \(Q_3=\frac{33}{2}\).
Five-number summary: \(5, \frac{13}{2}, 12, \frac{33}{2}, 20\).
\(IQR=\frac{33}{2}-\frac{13}{2}=10\). The fences are \(\frac{-17}{2}\) and \(\frac{63}{2}\), so the outliers are none.
7) Order the data: \( 30, 32, 35, 36, 38, 40, 44, 48, 50 \).
The median is \(38\). The lower-half median is \(Q_1=\frac{67}{2}\), and the upper-half median is \(Q_3=46\).
Five-number summary: \(30, \frac{67}{2}, 38, 46, 50\).
\(IQR=46-\frac{67}{2}=\frac{25}{2}\). The fences are \(\frac{59}{4}\) and \(\frac{259}{4}\), so the outliers are none.
8) Order the data: \( 7, 9, 12, 12, 14, 17, 19, 21, 25, 28 \).
The median is \(\frac{31}{2}\). The lower-half median is \(Q_1=12\), and the upper-half median is \(Q_3=21\).
Five-number summary: \(7, 12, \frac{31}{2}, 21, 28\).
\(IQR=21-12=9\). The fences are \(\frac{-3}{2}\) and \(\frac{69}{2}\), so the outliers are none.
9) Order the data: \( 40, 42, 45, 47, 50, 52, 55, 58, 60, 65 \).
The median is \(51\). The lower-half median is \(Q_1=45\), and the upper-half median is \(Q_3=58\).
Five-number summary: \(40, 45, 51, 58, 65\).
\(IQR=58-45=13\). The fences are \(\frac{51}{2}\) and \(\frac{155}{2}\), so the outliers are none.
10) Order the data: \( 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42 \).
The median is \(27\). The lower-half median is \(Q_1=18\), and the upper-half median is \(Q_3=36\).
Five-number summary: \(12, 18, 27, 36, 42\).
\(IQR=36-18=18\). The fences are \(-9\) and \(63\), so the outliers are none.
11) Order the data: \( 3, 4, 4, 5, 7, 9, 11, 13, 15, 40 \).
The median is \(8\). The lower-half median is \(Q_1=4\), and the upper-half median is \(Q_3=13\).
Five-number summary: \(3, 4, 8, 13, 40\).
\(IQR=13-4=9\). The fences are \(\frac{-19}{2}\) and \(\frac{53}{2}\), so the outliers are 40.
12) Order the data: \( 18, 19, 21, 22, 24, 25, 27, 29, 30, 55 \).
The median is \(\frac{49}{2}\). The lower-half median is \(Q_1=21\), and the upper-half median is \(Q_3=29\).
Five-number summary: \(18, 21, \frac{49}{2}, 29, 55\).
\(IQR=29-21=8\). The fences are \(9\) and \(41\), so the outliers are 55.
13) Order the data: \( 60, 62, 64, 65, 66, 68, 70, 72, 74, 90 \).
The median is \(67\). The lower-half median is \(Q_1=64\), and the upper-half median is \(Q_3=72\).
Five-number summary: \(60, 64, 67, 72, 90\).
\(IQR=72-64=8\). The fences are \(52\) and \(84\), so the outliers are 90.
14) Order the data: \( 1, 4, 6, 8, 10, 12, 14, 16, 18, 35, 38 \).
The median is \(12\). The lower-half median is \(Q_1=6\), and the upper-half median is \(Q_3=18\).
Five-number summary: \(1, 6, 12, 18, 38\).
\(IQR=18-6=12\). The fences are \(-12\) and \(36\), so the outliers are 38.
15) Order the data: \( 25, 28, 31, 34, 37, 40, 43, 46, 49, 52, 55, 100 \).
The median is \(\frac{83}{2}\). The lower-half median is \(Q_1=\frac{65}{2}\), and the upper-half median is \(Q_3=\frac{101}{2}\).
Five-number summary: \(25, \frac{65}{2}, \frac{83}{2}, \frac{101}{2}, 100\).
\(IQR=\frac{101}{2}-\frac{65}{2}=18\). The fences are \(\frac{11}{2}\) and \(\frac{155}{2}\), so the outliers are 100.
16) Order the data: \( 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28 \).
The median is \(17\). The lower-half median is \(Q_1=11\), and the upper-half median is \(Q_3=23\).
Five-number summary: \(6, 11, 17, 23, 28\).
\(IQR=23-11=12\). The fences are \(-7\) and \(41\), so the outliers are none.
17) Order the data: \( 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 60 \).
The median is \(25\). The lower-half median is \(Q_1=19\), and the upper-half median is \(Q_3=31\).
Five-number summary: \(14, 19, 25, 31, 60\).
\(IQR=31-19=12\). The fences are \(1\) and \(49\), so the outliers are 60.
18) Order the data: \( 45, 46, 47, 49, 50, 52, 53, 55, 56, 58, 80, 82 \).
The median is \(\frac{105}{2}\). The lower-half median is \(Q_1=48\), and the upper-half median is \(Q_3=57\).
Five-number summary: \(45, 48, \frac{105}{2}, 57, 82\).
\(IQR=57-48=9\). The fences are \(\frac{69}{2}\) and \(\frac{141}{2}\), so the outliers are 80, 82.
19) Order the data: \( 100, 105, 110, 115, 120, 125, 130, 135, 140, 145, 150 \).
The median is \(125\). The lower-half median is \(Q_1=110\), and the upper-half median is \(Q_3=140\).
Five-number summary: \(100, 110, 125, 140, 150\).
\(IQR=140-110=30\). The fences are \(65\) and \(185\), so the outliers are none.
20) Order the data: \( 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31, 80 \).
The median is \(21\). The lower-half median is \(Q_1=14\), and the upper-half median is \(Q_3=28\).
Five-number summary: \(9, 14, 21, 28, 80\).
\(IQR=28-14=14\). The fences are \(-7\) and \(49\), so the outliers are 80.

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