Question

Which modified box plot represents the data set?

39, 39, 3, 40, 46, 48, 60, 62, 62



Responses

Box and whisker plot on a number line from 0 to 65 in increments of 5. The whisker ranges from 39 to 62. The box ranges from 39 to 60 with the vertical bar inside the box at 48. One dot marks above three.
Image with alt text: Box and whisker plot on a number line from 0 to 65 in increments of 5. The whisker ranges from 39 to 62. The box ranges from 39 to 60 with the vertical bar inside the box at 48. One dot marks above three.

Box and whisker plot on a number line from 0 to 65 in increments of 5. The whisker ranges from 3 to 62. The box ranges from 30 to 60.5 with the vertical bar inside the box at 43.
Image with alt text: Box and whisker plot on a number line from 0 to 65 in increments of 5. The whisker ranges from 3 to 62. The box ranges from 30 to 60.5 with the vertical bar inside the box at 43.

Box and whisker plot on a number line from 0 to 65 in increments of 5. The whisker ranges from 3 to 62. The box ranges from 10.75 to 57 with the vertical bar inside the box at 39.5.

Image with alt text: Box and whisker plot on a number line from 0 to 65 in increments of 5. The whisker ranges from 3 to 62. The box ranges from 10.75 to 57 with the vertical bar inside the box at 39.5.,

Box and whisker plot on a number line from 0 to 65 in increments of 5. The whisker ranges from 39 to 62. The box ranges from 39 to 61 with the vertical bar inside the box at 46. One dot marks above three.

Image with alt text: Box and whisker plot on a number line from 0 to 65 in increments of 5. The whisker ranges from 39 to 62. The box ranges from 39 to 61 with the vertical bar inside the box at 46. One dot marks above three.,

Answer 1

D) Box and whisker plot on a number line from 0 to 65 in increments of 5. The whisker ranges from 39 to 62. The box ranges from 39 to 61 with the vertical bar inside the box at 46. One dot marks above three.

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:

• 3, 39, 39, 40, 46, 48, 60, 62, 62

Median

The median is the middle value when all data values are placed in order of size. Therefore, the median is 46.

Lower Quartile

The lower quartile (Q₁) is the median of the data points to the left of the median. 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:

`\sf Lower\;quartile\;(Q_1)=(39+39)/(2)=39`

Upper Quartile

The upper quartile (Q₃) is the median of the data points to the right of the median. 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:

`\sf Lower\;quartile\;(Q_2)=(60+62)/(2)=61`

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)&=39-1.5(61-39)\\&=39-1.5(22)\\&=39-33\\&=6\end{aligned}`

`\begin{aligned}\sf Q_3+(1.5* IQR)&=61+1.5(61-39)\\&=61+1.5(22)\\&=61+33\\&=94\end{aligned}`

Therefore, the lower fence is 6 and the upper fence is 94. Any data value that lies outside the fences is an outlier.

As the lower fence is more than the minimum value of 3, this means that 3 is an outlier. Therefore, the next data value up from 3 is the minimum value.

Minimum and Maximum

The minimum data value is 39.

The maximum data value is 62.

Therefore, the five-number summary is:

• Minimum value = 39
• Lower quartile (Q₁) = 39
• Median (Q₂) = 46
• Upper quartile (Q₃) = 61
• Maximum = 62

In summary:

• The whiskers range from 39 to 62.
• The box ranges from 39 to 61 with the vertical bar inside the box at 46.
• One dot marks above 3.