Question

unit 1: one - variable data
l7 practice - boxplots and outliers

1. the following final exam scores are from a sample of students who are taking stats 101 at university of chicago.

81 70 93 85 75 68 93 80 90 100
a) find and label the 5 - number summary.
b) draw a boxplot.

Explanation:

Step1: Arrange data in ascending order

$68,70,75,80,81,85,90,93,93,100$

Step2: Find the minimum value

The minimum value is $68$.

Step3: Find the first - quartile ($Q_1$)

The position of $Q_1$ is $\frac{n + 1}{4}=\frac{10+ 1}{4}=2.75$. Interpolating between the 2nd and 3rd ordered values: $Q_1=70+(75 - 70)\times0.75 = 73.75$.

Step4: Find the median ($Q_2$)

The position of the median for $n = 10$ (even) is $\frac{n}{2}=5$ and $\frac{n}{2}+1 = 6$. The median $Q_2=\frac{81 + 85}{2}=83$.

Step5: Find the third - quartile ($Q_3$)

The position of $Q_3$ is $\frac{3(n + 1)}{4}=\frac{3\times(10 + 1)}{4}=8.25$. Interpolating between the 8th and 9th ordered values: $Q_3=93+(93 - 93)\times0.25 = 93$.

Step6: Find the maximum value

The maximum value is $100$.
The 5 - number summary is: Minimum = $68$, $Q_1=73.75$, Median = $83$, $Q_3=93$, Maximum = $100$.

To draw a boxplot:

1. Draw a number line that includes the range from $68$ to $100$.
2. Draw a box from $Q_1 = 73.75$ to $Q_3=93$.
3. Draw a vertical line inside the box at the median $83$.
4. Draw whiskers from the box to the minimum ($68$) and maximum ($100$) values.

Answer:

a) 5 - number summary: Minimum = $68$, $Q_1 = 73.75$, Median = $83$, $Q_3=93$, Maximum = $100$.
b) Boxplot is drawn as described above with a box from $73.75$ to $93$, a vertical line at $83$ inside the box and whiskers to $68$ and $100$.