Lesson 54: Displaying Data in a Box-and-Whisker Plot

New Concept

A box-and-whisker plot displays data that are divided into four groups.

What’s next

Next, you’ll learn to find the key values—median, quartiles, and outliers—that form the structure of these powerful visual summaries.

Property

A box-and-whisker plot displays data that are divided into four groups. A line inside the box shows the median, the ends of the box show the quartiles, and the ends of the whiskers show the minimum and maximum values.

Explanation

Think of it as a treasure map for your data! The 'box' holds the middle 50% of your treasure, with a line marking the exact middle (the median). The 'whiskers' are lines stretching out to the smallest and largest values, showing the full range of your data from end to end, making it easy to see data distribution.

Examples

• For test scores {65, 70, 72, 80, 85, 90, 98}, the box shows the middle scores from 70 to 90, with a median at 80.
• In a survey of pet ages {1, 2, 2, 5, 7, 8, 12}, the plot visually separates the youngest, middle, and oldest pets.
• For hockey goals {1, 1, 3, 5, 7, 8, 9}, the box spans from $Q_1=1$ to $Q_3=8$, showing where half the goal counts lie.

Let's turn a list of video game scores into a clear visual summary. This is a great example of the first key idea, displaying data in a box-and-whisker plot.

Example Problem
Make a box-and-whisker plot for these game scores: $150, 220, 180, 300, 250, 400, 450, 500, 260, 210, 190, 520, 160, 230, 280$. Half the scores fall between which two values?

Step-by-Step

1. First, order the data from least to greatest.

2. Find the minimum, maximum, median, and quartiles.
   • Minimum: $150$
   • Maximum: $520$
   • Median (the 8th value): $250$
   • First Quartile ($Q_1$, median of the lower half): $190$
   • Third Quartile ($Q_3$, median of the upper half): $400$
3. Draw a number line. Then, draw a box from $Q_1$ ($190$) to $Q_3$ ($400$), add a vertical line inside the box at the median ($250$), and draw whiskers from the box to the minimum ($150$) and maximum ($520$).
4. The box represents the middle half of the data. Therefore, half of the game scores are between $190$ and $400$.

Property

An outlier is a data value that is much greater or much less than the other data values in the set.

Explanation

Imagine a group of penguins, and suddenly there's a flamingo! That flamingo is the outlier. It is a data point so different from the rest that it stands out. We identify outliers to see values that are unusual or might even be errors, so our main data story is not skewed by a wild, stray number.

Examples

• In the data set of ages {14, 15, 14, 13, 52, 15}, the age 52 is an outlier compared to the teenagers.
• For daily temperatures {72, 75, 73, 71, 35, 74}, the temperature 35 is an outlier for a warm week.
• If player points are {12, 15, 10, 18, 51}, the 51 points are an outlier for that game's statistics.

Sometimes one data point is far from the others; let's see how to find it. This example uses the second key idea, identifying outliers.

Example Problem
The daily commute times (in minutes) for 10 employees are: $15, 18, 20, 21, 21, 25, 28, 30, 31, 55$. Display the data and identify any outliers.

Step-by-Step

1. Find the median, $Q_1$, $Q_3$, and the interquartile range (IQR). The data is already in order.
   • The two middle numbers are $21$ and $25$. The median is their mean: $\frac{21+25}{2} = 23$.
   • The first quartile, $Q_1$, is the median of the lower half $(15, 18, 20, 21, 21)$, which is $20$.
   • The third quartile, $Q_3$, is the median of the upper half $(25, 28, 30, 31, 55)$, which is $30$.
   • The interquartile range is $IQR = Q_3 - Q_1 = 30 - 20 = 10$.
2. Use the outlier formulas to find the boundaries. A value $x$ is an outlier if $x < Q_1 - 1.5(IQR)$ or $x > Q_3 + 1.5(IQR)$.
   • Lower boundary: $20 - 1.5(10) = 5$.
   • Upper boundary: $30 + 1.5(10) = 45$.
3. Check for outliers. The value $55$ is an outlier because it is greater than the upper boundary of $45$. There are no outliers on the low end.
4. When drawing the plot, the upper whisker extends only to the highest value that is not an outlier, which is $31$. The outlier at $55$ is marked separately with an asterisk (*).