Question

The box-and-whisker plot below shows the numbers of text messages received in one day by students in the seventh and eighth grades at Lincoln Middle School. Two box and whisker plots are shown above a number line. The number line is scaled from 2 to 38 by twos. The first plot above the number line is labeled seventh grade. On the seventh grade plot: The left point is at 6. The left side of the box is at 14. The vertical segment inside the box is at 18 . The right side of the box is at 22. The right point of the plot is at 26. The eighth grade plot is above the seventh grade plot. On the eighth grade plot: The left point is at 22. The left side of the box is at 26. the vertical segment inside the box is at 30. The right side of the box is at 34. The right point of the plot is at 38. a) Describe the overlap of the two sets of data. b) Find the IQR of each set. c) Find the difference between the medians of the sets. d) What number multiplied by the IQR equals the difference between the medians of the two sets?

Answer 1

Answer with explanation:

From the box and whisker plot of seventh grade we have:

The minimum value =6

First quartile or lower quartile i.e. $Q_1$ = 14

Median or second quartile i.e. $Q_2$ = 18

Third quartile or upper quartile i.e. $Q_3$ =22

and maximum value = 26

From the box and whisker plot of eighth grade we have:

The minimum value =22

First quartile or lower quartile i.e. $Q_1$ = 26

Median or second quartile i.e. $Q_2$ = 30

Third quartile or upper quartile i.e. $Q_3$ = 34

and maximum value = 38

a)

The overlap of the two sets of data is as follows.

• The upper quartile or third quartile of seventh grade is same as the minimum value of the data of eighth grade.
• And the maximum value of seventh grade is same as the lower quartile of eighth grade.

b)

IQR is calculated as the difference of the Upper quartile and the lower quartile

i.e. $Q_3-Q_1$

so, IQR of seventh grade is:

22-14=8

IQR of seventh grade=8

IQR of eighth grade is:

34-26=8

Hence, IQR of eighth grade=8

c)

The difference of the median of the two data sets is:

30-18=12

Hence, the difference of median is: 12

d)

As the IQR of both the sets is same i.e. 8.

Hence, the number that must be multiplied by IQR so that it is equal to the difference between the medians of the two sets is:

$8\times n=12\\\\n=\dfrac{12}{8}\\\\n=\dfrac{3}{2}\\\\n=1.5$

Hence, the number is : 1.5

Answer 2

You can use the fact that the leftmost point denotes minimum, from left to right the lines denote first , second and third quartile. The last point which is rightmost, denotes the maximum point.

The answers are:

• a) The overlap is found by seeing the difference between leftmost and rightmost point.
• b) The IQR for seventh grade plot is 8
• The IQR for eighth grade plot is 8
• c) The difference between median of the two sets is 12
• d) The needed number is 1.5
  

How does a boxplot shows the  data points?

A box plot has 5 data description.

The leftmost whisker shows the minimum value in the data.

The rightmost whisker shows the maximum value in the data.

The leftmost line in the box shows the first quartile.

The middle line shows the median, also called second quartile.

The last line of the box shows the third quartile.

How to find the interquartile range?

IQR(inter quartile range)  is the dfference between third and first quartile.

What are the descriptions for the seventh and eighth grade plot?

For seventh grade:

Using the given description, we get:

Minimum value = 6,First quartile = 14, second quartile = 18, third quartile = 22, maximum value = 26

IQR = 22 - 14 = 8

For eighth grade:

Using the given description, we get:

Maximum value = 22,first quartile = 26, second quartile or median = 30, third quartile = 34, fourth quartile = 38

IQR = 34 -26 = 8

Thus, we have:

a) The overlap is found by seeing the difference between leftmost and rightmost point.

Overlap for 7th grade = 26 - 6 = 20

b) The IQR for seventh grade plot  is 8 and for eighth grad plot it is 8 (same for both)

Overlap for 8th grade = 38 - 22 = 16

c) The difference between both's median is 30 - 18 = 12

d) Since IQR is 8 and the difference between the medians is 12, let the number be x, then we have:

$12 = 8x\\\\x = \dfrac{12}{8} = \dfrac{3}{2} = 1.5$

Thus, the needed number is 1.5

Thus,

The answers are:

• a) The overlap is found by seeing the difference between leftmost and rightmost point.
• b) The IQR for seventh grade plot is 8
• The IQR for eighth grade plot is 8
• c) The difference between median of the two sets is 12
• d) The needed number is 1.5

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