Answer:
The interquartile range for Group A students is 1 more than the interquartile range for Group B students. (Please read explanation it took me a long time and I think it may be helpful)
Step-by-step explanation:
The interquartile range is the difference between the third quartile and the first quartile. To find these values, we have to list the values out (typically from lowest to highest).
For group A, the values can be ordered as:
1 , 1 , 1 , 2 , 2 , 2 , 3 , 3 , 3
For group B, the values can be ordered as:
1 , 1 , 2 , 2 , 2 , 2 , 2 , 3 , 3
To find the quartiles, you first need to find the "second quartile"--the median of the data set. The median of a data set is the middle number if you list the values from lowest to highest. (If there are two numbers in the middle, you find the mean/average between the two by dividing the sum of the values by 2, which gets one number for the median).
In the data set: 1 , 2 , 3 , 4 , 5 , 6 , 7 ,
Your median would be 4.
(It is three away from both ends of the data set.)
Once you find the median/Q2[Q2 = 2nd Quartile], you can split off the data into two different groups.
You could consider the data set to be 1 , 2 , 3 , | | 5 , 6 , 7
Now, the Q1 is the median (middle number) of the first split of the data, and the Q3 is the median of the second half of the data.
So, the Q1 would be 2, and the Q3 would be 6.
To find the interquartile range, you find the difference between these two values: 6 - 2 = 4 ; IQR = 4
Going back to your problem,
For group A, the values can be ordered as:
1 , 1 , 1 , 2 , 2 , 2 , 3 , 3 , 3
The middle number here is 2. If we split the data into two halves, we end up with:
1 , 1 , 1 , 2 | | 2 , 3 , 3 , 3
Now, the median of the first half is 1, and the median of the second half is 3.
So, your median is 2 (Q2 = 2)
your first quartile is 1 (Q1 = 1)
and your third quartile is 3 (Q3 = 3)
Finding the IQR of this data set means finding the difference/range between 1 and 3, which we know is 2 (3 - 1 = 2)
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For group B, the values can be ordered as:
1 , 1 , 2 , 2 , 2 , 2 , 2 , 3 , 3
The middle number here is also 2. If we split this data set into two halves, we end up with:
1 , 1 , 2 , 2 | | 2 , 2 , 3 , 3
Now the median of the first half is 1.5 (the mean/average between 1 and 2), and the median of the second half is 2.5 (the mean/average between 2 and 3).
So, your median is 2 (Q2 = 2)
your first quartile is 1.5 (Q1 = 1.5)
and your third quartile is 2.5 (Q3 = 2.5)
The IQR can be found by finding the range between the first quartile and the third quartile. For this data set, we find the IQR by finding the difference/range between 1.5 and 2.5, which we know is 1 (2.5 - 1.5 = 1).
So, the interquartile range for Group A is 2, and the interquartile range for Group B is 1. This means that the interquartile range for Group A is 1 more than the interquartile range for Group B.
[the first quartile (Q1) is the 25th percentile,
the second quartile (Q2) is the 50th percentile,
and the third quartile (Q3) is the 75th percentile].
[IQR = Interquartile Range]
(You can look it up for a more thorough explanation, but simply put, the interquartile range tells you how spread out the middle values are. Finding the Q1 and Q3 can essentially be used to find outliers, as you can assume the data outside of them are not the main set of data. Although this is not the technical way to find outliers, it can help you determine what data is actually important. If your data has a large range / large interquartile range, your middle data is spread out--and your values have a larger difference between them. The reasoning that only considering the range of the interquartile is valuable is that it isn't heavily impacted by the extreme outliers (like, for example, if a student spent 15 hours reading per day) like the average or overall range could be. )
