Question

The following table shows the number of hours some high school students in two towns spend riding the bus each week:

Town A 5 3 9 4 20 9 5 7 6
Town B 8 8 3 5 5 6 3 9 9


Part A: Create a five-number summary and calculate the interquartile range for the two sets of data. (6 points)

Part B: Are the box plots symmetric? Justify your answer. (4 points)

Answer 1

Part A:

The five number summary is consisting of : Median or 2nd quartile(Q2), 1st quartile(Q1), 3rd quartile(Q3), Lowest and Highest values.

For creating the five number summary for each set, first we need to rearrange the data set in ascending order:

Town A : 3 4 5 5 6 7 9 9 20

Median is the middle value in the set. Here, the middle value is 6. So the median or second quartile(Q2) = 6. The lowest value is 3 and highest value is 20.

The median splits the remaining data into two sets. The first set is 3, 4, 5, 5. The median of this set is
`(4+5)/(2)= (9)/(2) =4.5`

So, first quartile (Q1) = 4.5

The other set is 7, 9, 9, 20. The median of this set is:
`(9+9)/(2)=(18)/(2)=9`

So, third quartile (Q3) = 9

The interquartile range(IQR)= Q3 - Q1 = 9 - 4.5 = 4.5

Town B : 3 3 5 5 6 8 8 9 9

The middle value in the set is 6. So, the median or second quartile(Q2) =6. The lowest value is 3 and the highest value is 9.

The first set in the left of median is: 3 3 5 5 and median of this set=
`(3+5)/(2)=(8)/(2)=4`

So, first quartile (Q1) = 4

The second set in the right of median is : 8 8 9 9 and the median of this set =
`(8+9)/(2)=(17)/(2)=8.5`

So, third quartile (Q3) = 8.5

The interquartile range(IQR)= Q3 - Q1 =8.5 - 4 = 4.5

Part B:

A box plot is called symmetric when the median is in the middle of the box, that means Q2 should be equally distanced from Q1 and Q3.

For town A, distance between Q1 and Q2 = 6 - 4.5 = 1.5 and distance between Q2 and Q3 = 9 - 6 = 3. As the distances are different, so the box plot is not symmetric.

For town B, distance between Q1 and Q2 = 6 - 4 = 2 and distance between Q2 and Q3 = 8.5 - 6 = 2.5. As the distances are different, so this box plot is also not symmetric.