Lesson 3: Display Data in Box Plots

Property

A box plot (or box-and-whisker plot) is a visual representation of the five-number summary, and tells us much more about the spread of the data.
A box is drawn from the 25th percentile to the 75th percentile, and 'whiskers' are drawn from the lowest data value to the 25th percentile and from the 75th percentile to the highest data value.

Examples

• Given a five-number summary of {Min: 4, Q1: 7, Med: 10, Q3: 12, Max: 18}, a box plot would show a box extending from 7 to 12 with a line at 10, and whiskers reaching to 4 and 18.
• A data set of quiz scores is {5, 6, 6, 7, 8, 9, 10}. The five-number summary is {Min:5, Q1:6, Med:7, Q3:9, Max:10}. Its box plot has a box from 6 to 9 and whiskers from 5 to 10.
• If one box plot has a much longer box than another, it means the middle half of its data is more spread out. A longer right whisker means the top 25% of values are more spread out than the bottom 25%.

Explanation

A box plot turns the five-number summary into a picture. The 'box' contains the middle 50% of your data. The 'whiskers' stretch out to the highest and lowest values, showing the full range and how spread out everything is.

Property

To find the five-number summary (minimum, Q1, median, Q3, maximum), first order the data.

• Odd number of values: The median is the middle value. Q1 is the median of the lower half (excluding the median), and Q3 is the median of the upper half (excluding the median).
• Even number of values: The median is the average of the two middle values. Q1 is the median of the lower half, and Q3 is the median of the upper half.

Examples

• Odd set: $\{1, 5, 2, 7, 6\}$

Order the data: $\{1, 2, 5, 6, 7\}$.
Minimum: $1$, Maximum: $7$
Median: $5$
Lower half: $\{1, 2\}$, so $Q1 = \frac{1+2}{2} = 1.5$
Upper half: $\{6, 7\}$, so $Q3 = \frac{6+7}{2} = 6.5$
Summary: $(1, 1.5, 5, 6.5, 7)$

• Even set: $\{9, 3, 1, 8, 5, 10\}$

Order the data: $\{1, 3, 5, 8, 9, 10\}$.
Minimum: $1$, Maximum: $10$
Median: $\frac{5+8}{2} = 6.5$
Lower half: $\{1, 3, 5\}$, so $Q1 = 3$
Upper half: $\{8, 9, 10\}$, so $Q3 = 9$
Summary: $(1, 3, 6.5, 9, 10)$

Explanation

The method for finding the median and quartiles depends on whether your data set has an odd or even number of values. For an odd-sized set, the median is a single value from the set, which is then excluded when you find the medians of the lower and upper halves for Q1 and Q3. For an even-sized set, the median is calculated as the average of the two middle values and is not part of the original data, so all values are included when finding Q1 and Q3 from the lower and upper halves. This distinction ensures the data is divided into four equal parts.