Question

here are the scores of 16 students on an algebra test. 58, 64, 65, 69, 69, 70, 71, 73, 74, 75, 76, 80, 80, 82, 86, 89 notice that the scores are ordered from least to greatest. make a box - and - whisker plot for the data.

Explanation:

Step1: Find minimum and maximum

The minimum is $58$ and maximum is $89$ as they are the smallest and largest values in the ordered data set.

Step2: Calculate the median

For $n = 16$ data points, median is average of 8th and 9th values: $Q_2=\frac{73 + 74}{2}=73.5$.

Step3: Calculate first quartile

For the lower 8 - point sub - set, $Q_1$ is average of 4th and 5th values: $Q_1=\frac{69+69}{2}=69$.

Step4: Calculate third quartile

For the upper 8 - point sub - set, $Q_3$ is average of 4th and 5th values: $Q_3=\frac{80 + 80}{2}=80$.

Step5: Construct the box - and - whisker plot

Draw box from $Q_1$ to $Q_3$ with line at $Q_2$ inside, and whiskers to minimum and maximum.

Answer:

To make a box - and - whisker plot, we need to find the five - number summary: minimum, first quartile ($Q_1$), median ($Q_2$), third quartile ($Q_3$), and maximum.

1. Minimum and Maximum:

• The minimum value of the data set is $58$.
• The maximum value of the data set is $89$.

1. Median ($Q_2$):

• Since we have $n = 16$ data points, the median is the average of the $\frac{n}{2}=8$th and $(\frac{n}{2}+1) = 9$th ordered data values.
• The 8th value is $73$ and the 9th value is $74$. So, $Q_2=\frac{73 + 74}{2}=73.5$.

1. First Quartile ($Q_1$):

• The lower half of the data consists of the first 8 data points: $58,64,65,69,69,70,71,73$.
• Since $n = 8$ (in the lower half), the median of the lower half (first quartile) is the average of the $\frac{n}{2}=4$th and $(\frac{n}{2}+1)=5$th ordered data values in the lower - half.
• The 4th value is $69$ and the 5th value is $69$. So, $Q_1=\frac{69+69}{2}=69$.

1. Third Quartile ($Q_3$):

• The upper half of the data consists of the last 8 data points: $74,75,76,80,80,82,86,89$.
• Since $n = 8$ (in the upper half), the median of the upper half (third quartile) is the average of the $\frac{n}{2}=4$th and $(\frac{n}{2}+1)=5$th ordered data values in the upper - half.
• The 4th value is $80$ and the 5th value is $80$. So, $Q_3=\frac{80 + 80}{2}=80$.

On the box - and - whisker plot:

• The left - most point (whisker) is at the minimum value $58$.
• The left - hand side of the box is at $Q_1 = 69$.
• The line inside the box is at the median $Q_2=73.5$.
• The right - hand side of the box is at $Q_3 = 80$.
• The right - most point (whisker) is at the maximum value $89$.

We draw a number line that includes the range from $55$ to $90$. Then we draw a box from $69$ to $80$ with a vertical line at $73.5$ inside the box. We draw whiskers from the box to the minimum ($58$) and maximum ($89$) values.