# The data for each grade have the same interquartile range (IQR). Which of the following best compares the two best score distributions? With reference

Data distributions
The data for each grade have the same interquartile range (IQR). Which of the following best compares the two best score distributions?
With reference to line plots the data for Sixth grade geography test score is
7 8 8 9 9 9 9 9 10 10 10 11 11 11 12 12 12 14 14 15
The data of seventh grade geography test score is
7 10 10 11 11 11 11 12 12 13 13 13 13 13 14 14 14 15 16 17

2021-02-14

The median is the middle value of the data distribution. Here there are 20 data points for sixth and seventh grade test scores each. So the median will be the average of $$\displaystyle{10}^{{{t}{h}}}$$ and $$\displaystyle{11}^{{{t}{h}}}$$ data point when arranged in ascending order.
The median of sixth grade geography test score is calculated as shown below
$$\displaystyle{10}^{{{t}{h}}}$$ data point $$=10$$
$$\displaystyle{11}^{{{t}{h}}}$$ data point $$=10$$
Median = $$\displaystyle\frac{{{10}+{10}}}{{2}}={10}$$
The median of seventh grade geography test score is calculated as shown below
$$\displaystyle{10}^{{{t}{h}}}$$ data point $$=13$$
$$\displaystyle{11}^{{{t}{h}}}$$ data point $$=13$$
Median $$= \frac{13+13}{2}=13$$
The IQR is difference between Quartile 3 and Quartile 1.
For sixth grade geography test scores
7 8 8 9 9 9 9 9 10 10 10 11 11 11 12 12 12 14 14 15
The Quartile 1 divide the data distribution in such way that $$25\%$$ of data lie less than it and $$75\%$$ lie more than it. In a way we can say it is median of first half of data distribution.
Quartile 1 is average of 5th and 6th data points when arranged in ascending order
$$\displaystyle{Q}_{{1}}=\frac{{{9}+{9}}}{{2}}={9}$$
The Quartile 3 divide the data distribution in such way that $$75\%$$ of data lie less than it and $$25\%$$ lie more than it. In a way we can say it is median of second half of data distribution.
Quartile 3 is average of $$\displaystyle{15}^{{{t}{h}}}$$ and $$\displaystyle{16}^{{{t}{h}}}$$ data points when arranged in ascending order
$$\displaystyle{Q}_{{1}}=\frac{{{12}+{12}}}{{2}}={12}$$
Inter Quartile Range $$=Q3−Q1=12−9=3$$
For seventh grade geography test scores
7 10 10 11 11 11 11 12 12 13 13 13 13 13 14 14 14 15 16 17
The Quartile 1 divide the data distribution in such way that 25% of data lie less than it and 75% lie more than it. In a way we can say it is median of first half of data distribution.
Quartile 1 is average of $$\displaystyle{5}^{{{t}{h}}}$$ and $$\displaystyle{6}^{{{t}{h}}}$$ data points when arranged in ascending order
$$\displaystyle{Q}_{{1}}=\frac{{{11}+{11}}}{{2}}={11}$$
The Quartile 3 divide the data distribution in such way that 75% of data lie less than it and 25% lie more than it. In a way we can say it is median of second half of data distribution.
Quartile 3 is average of $$\displaystyle{15}^{{{t}{h}}}$$ and $$\displaystyle{16}^{{{t}{h}}}$$ data points when arranged in ascending order
$$\displaystyle{Q}_{{1}}=\frac{{{14}+{14}}}{{2}}={14}$$
Inter Quartile Range $$=Q3−Q1=14−11=3$$
Thus the median of Seventh grade is 13 and median of sixth grade is 10, while the Inter quartile range for both of them is 3. So option 3 is correct.
The median score of the seventh grade class is 3 points greater than the median score of the sixth grade class. The difference is the same as the IQR.