Question

1. the test scores from ms chen’s math class are shown below: 72, 73, 66, 71, 82, 85, 95, 65, 86, 89, 91, 92. construct a box-and-whisker plot to display these data

Explanation:

Step1: Order the data

First, we need to order the data from least to greatest. The data is: 66, 71, 72, 72, 73, 82, 85, 85, 86, 89, 91, 95.

Step2: Find the minimum and maximum

The minimum value (smallest number) is 66, and the maximum value (largest number) is 95.

Step3: Find the median (Q2)

Since there are 12 data points (even number), the median is the average of the 6th and 7th values. The 6th value is 82 and the 7th value is 85. So the median $Q2=\frac{82 + 85}{2}=\frac{167}{2} = 83.5$.

Step4: Find the lower quartile (Q1)

The lower half of the data is: 66, 71, 72, 72, 73. There are 5 data points (odd number), so the median of the lower half (Q1) is the 3rd value, which is 72.

Step5: Find the upper quartile (Q3)

The upper half of the data is: 85, 85, 86, 89, 91, 95. Wait, no, after finding the median (Q2 = 83.5), the lower half is the first 6 data points? Wait, no, when n is even, the lower half is the first $\frac{n}{2}$ data points and the upper half is the last $\frac{n}{2}$ data points. Wait, n = 12, so lower half: first 6 data points: 66, 71, 72, 72, 73, 82. Upper half: last 6 data points: 85, 85, 86, 89, 91, 95. Now, for the lower half (6 data points, even), Q1 is the median of the lower half. The median of 66, 71, 72, 72, 73, 82 is the average of the 3rd and 4th values. The 3rd value is 72, the 4th value is 72. So $Q1=\frac{72 + 72}{2}=72$. For the upper half (6 data points, even), Q3 is the median of the upper half. The 3rd value is 86, the 4th value is 89. So $Q3=\frac{86+89}{2}=\frac{175}{2} = 87.5$.

Step6: Construct the box - and - whisker plot

• The minimum value (66) is the left - most whisker.
• The lower quartile (Q1 = 72) is the left - hand side of the box.
• The median (Q2 = 83.5) is the line inside the box.
• The upper quartile (Q3 = 87.5) is the right - hand side of the box.
• The maximum value (95) is the right - most whisker.

To draw the plot on the given number line (with marks at 65, 70, 75, 80, 85, 90, 95, 100):

• Plot a dot at 66 (minimum).
• Draw a box from 72 (Q1) to 87.5 (Q3). Inside the box, draw a vertical line at 83.5 (median).
• Plot a dot at 95 (maximum). Then connect the minimum to the box with a line (whisker) and the maximum to the box with a line (whisker).

Answer:

The box - and - whisker plot has minimum = 66, Q1 = 72, median = 83.5, Q3 = 87.5, maximum = 95. The plot is constructed with a whisker from 66 to 72, a box from 72 to 87.5 with a line at 83.5, and a whisker from 87.5 to 95 on the given number line.