Question

the data shows a sample of states percentage of public high school graduation rates for a recent year.
75 79 75 90 86 84 65 71 94 91 85 86 74 90
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part 1 of 2
draw the boxplot for the given data.
high school graduation rates
60 65 70 75 80 85 90 95 100
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Explanation:

Step1: Order the data

First, we order the data set: \( 65, 71, 74, 75, 75, 79, 84, 85, 86, 86, 90, 90, 91, 94 \)

Step2: Find the median (Q2)

The number of data points \( n = 14 \), which is even. The median is the average of the \( \frac{n}{2} \)-th and \( (\frac{n}{2}+1) \)-th values.
\( \frac{14}{2}=7 \)-th value is \( 84 \), \( \frac{14}{2}+1 = 8 \)-th value is \( 85 \). So median \( Q2=\frac{84 + 85}{2}=84.5 \)

Step3: Find Q1 (median of lower half)

Lower half data (first 7 values): \( 65, 71, 74, 75, 75, 79, 84 \)
\( n = 7 \) (odd), median is the \( \frac{7 + 1}{2}=4 \)-th value. So \( Q1 = 75 \)

Step4: Find Q3 (median of upper half)

Upper half data (last 7 values): \( 85, 86, 86, 90, 90, 91, 94 \)
\( n = 7 \) (odd), median is the \( \frac{7+1}{2}=4 \)-th value. So \( Q3 = 90 \)

Step5: Find Min and Max

Minimum value \( \text{Min}=65 \), Maximum value \( \text{Max}=94 \)

Step6: Draw the boxplot

• The box spans from \( Q1 = 75 \) to \( Q3 = 90 \), with a line at \( Q2 = 84.5 \) inside the box.
• The whiskers extend from \( \text{Min}=65 \) to \( Q1 \) and from \( Q3 \) to \( \text{Max}=94 \)

Answer:

To draw the boxplot:

• The minimum value (left whisker end) is at 65.
• The first quartile (Q1, left box edge) is at 75.
• The median (Q2, line inside box) is at 84.5.
• The third quartile (Q3, right box edge) is at 90.
• The maximum value (right whisker end) is at 94.

On the given graph:

• The left whisker should start at 65 (on the x - axis at 65) and end at 75 (Q1).
• The box should start at 75 (Q1), have a line at 84.5 (median), and end at 90 (Q3).
• The right whisker should start at 90 (Q3) and end at 94 (maximum).