Question

5. The box-and-whisker diagram shows the average times taken for a class of students to walk to school

45
15
20
2
a. Find the range.
30
35
40
45
60
b. Find the interquartile range.
c. Find the percentage of students who took between 15 and 37 minutes to walk to school.
d. It was found that another student took 60 minutes to walk to school. Determine whether this time would be counted as an outlier. Show how

Answer 1

a) Range = 30

b) IQR = 15

c) 75%

d) Yes

Explanation:

A box plot is a graphical representation of the distribution of a dataset. The whiskers of a box plot extend from the box to the minimum and maximum values within a dataset. The box represents the central 50% of the data distribution, where the first quartile (Q1) is the lower boundary of the box, and the third quartile (Q3) is the upper boundary of the box. The line in the middle of the box represents the median (Q2).

Part (a)

The range of a box plot represents the span between the minimum and maximum values within a dataset.

To find the range of a box plot, subtract the minimum value from the maximum value represented by the whiskers of the plot.

`\begin{aligned}\textsf{Range}&=\sf Max - Min\\&=\sf 45-15\\&=\sf 30\end{aligned}`

Part (b)

The interquartile range (IQR) of a box plot is the range between the first quartile (Q1) and the third quartile (Q3).

To calculate the interquartile range (IQR) of a dataset, subtract the first quartile (Q1) from the third quartile (Q3).

`\begin{aligned}\textsf{IQR}&=\textsf{Q3}-\textsf{Q1}\\&=\sf37-22\\&=15\end{aligned}`

Part (c)

The number of students who took 15 minutes to walk to school corresponds to the minimum data value, while those who took 37 minutes represent the Q3 data value. Given that the range from the minimum to Q1 accounts for the first 25% of the data, and Q1 to Q3 (the box) represents the central 50%, it follows that the percentage of students who took between 15 and 37 minutes to walk to school comprises 75% of the dataset.

Part (d)

An outlier is an observation in a dataset that significantly deviates from the overall pattern of the data.

To calculate potential outliers in a box plot, we identify values beyond 1.5 times the interquartile range (IQR) from either the first quartile (Q1) or third quartile (Q3). Therefore, to determine whether 60 minutes is an outlier, we need to calculate the upper bound:

`\begin{aligned}\textsf{Upper Bound} &= \sf Q3 + 1.5 * IQR\\&=\sf 37+1.5* 15\\&=\sf 37+22.5\\&=\sf 59.5\end{aligned}`

Since 60 is greater than 59.5, it would be counted as a potential outlier in this dataset.