Question

Topic: Examining data distributions in a

box-and-whisker plot.

Make a box-and-whisker plot for the following test scores.
60, 64, 68, 68, 72, 76, 76, 80, 80, 80, 84, 84, 84, 84, 88, 88, 88, 92, 92, 96, 96, 96, 96, 96, 96, 96, 100, 100

How much of the data is represented by the box?

How much is represented by each whisker?

What does the graph tell you about the student success on the test?

Answer 1

• 50% of the data is represented by the box.

• The left whisker represents the bottom 25% of the data and the right whisker represents the top 25% of the data.

• The median score was 86, and 50% of the students scored between 78 and 96. Therefore, it was a high scoring test and the students succeeded on the test.

Explanation:

A box and whisker plot (also known as a "box plot"), is a graph displaying the distribution of a set of data based on a five number summary.

Five-number summary

• Minimum value = The value at the end of the left whisker.
• Lower quartile (Q₁) = The left side of the box.
• Median (Q₂) = The vertical line inside the box.
• Upper quartile (Q₃) = The right side of the box
• Maximum = The value at the end of the right whisker.

To calculate the values of the five-number summery, first order the given data values from smallest to largest (this has already been done for us).

Median

The median is the middle value when all data values are placed in order of size.

There are 28 data values in the data set, so this is an even data set.

The middle two values are 84 and 88.

`\textsf{60, 64, 68, 68, 72, 76, 76, 80, 80, 80, 84, 84, 84, \boxed{\sf 84, 88,} 88, 88,}\\ \textsf{92, 92, 96, 96, 96, 96, 96, 96, 96, 100, 100.}`

As there are an even number of data values, the median is the mean of the middle two values:

`\implies \sf Median\;(Q_2) = (84+88)/(2)=86`

Lower Quartile

The lower quartile (Q₁) is the median of the data points to the left of the median. Again, as there is an even number of data points to the left of the median, the lower quartile is the mean of the the middle two values:

`\textsf{60, 64, 68, 68, 72, 76, \boxed{\sf 76, 80,} 80, 80, 84, 84, 84, 84, $|$ 88, 88, 88,}\\ \textsf{92, 92, 96, 96, 96, 96, 96, 96, 96, 100, 100.}`

`\implies \sf Lower\;Quartile\;(Q_1) = (76+80)/(2)=78`

Upper Quartile

The upper quartile (Q₃) is the median of the data points to the right of the median. Again, as there is an even number of data points to the right of the median, the upper quartile is the mean of the the middle two values:

`\textsf$ 88, 88, 88,\\ \textsf{92, 92, 96, \boxed{\sf 96, 96,} 96, 96, 96, 96, 100, 100.}`

`\implies \sf Upper\;Quartile\;(Q_3) = (96+96)/(2)=96`

Outliers

An outlier is any value that lies more than one and a half times the length of the box from either end of the box. The length of the box is called the interquartile range (IQR). IQR = Q₃ - Q₁.

`\begin{aligned}\sf Q_1- (1.5)(IQR) &= 78-(1.5)(96-78)\\&=78-(1.5)(18)\\&=78-27\\&=51\end{aligned}`

`\begin{aligned}\sf Q_3+ (1.5)(IQR) &= 96+(1.5)(96-78)\\&=96+(1.5)(18)\\&=96+27\\&=123\end{aligned}`

As 51 is less than the minimum value of 60, and 123 is more than the maximum value of 100, there are no outliers in the data set.

Minimum and Maximum

The minimum data value is 60.

The maximum data value is 100.

Therefore, the five-number summary is:

• Minimum value = 60
• Lower quartile (Q₁) = 78
• Median (Q₂) = 86
• Upper quartile (Q₃) = 96
• Maximum = 100

Construct the box and whisker plot:

• Draw a box from the lower quartile (78) to the upper quartile (86).
• Add the median (86) as a vertical line through the box.
• The whiskers are horizontal lines from each quartile to the minimum (60) and maximum values (100).

The data represented by the box is the interquartile range (IQR) where 50% of the data is found.

The left whisker represents the bottom 25% of the data and the right whisker represents the top 25% of the data.

The median score was 86, and 50% of the students scored between 78 and 96. Therefore, it was a high scoring test and the students succeeded on the test.