Question 1201889: Two data sets have the same range, can you assume the interquartile ranges are about the same? Please include an example to justify your answer.
Also, I'm sorry I don't know what the topic would be "(
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Consider this seven element list
{1,2,3,4,5,6,7}
The median is 4 because it's the middle most value.
This can be seen fairly quickly, or we could cross off the first and last terms (1 and 7) to get {2,3,4,5,6}
Then repeat again to get {3,4,5} and at this point it's more clear 4 is at the very middle.
What's another way to find the median?
We have n = 7 items.
The midpoint is at slot 4 because (n+1)/2 = (7+1)/2 = 8/2 = 4
Or you could say n/2 = 7/2 = 3.5 which rounds to 4.
Either formula only works when n is odd.
We'll use the median to split the data into two halves
Split the data into a lower set L and upper set U
L = lower set
L = stuff smaller than the median
L = {1,2,3}
U = upper set
U = stuff larger than the median
U = {5,6,7}
The median is not part of either subset.
The median of set L is 2, which is the first quartile Q1.
The median of set U is 6, which is the third quartile Q3.
Q1 = 2
Q3 = 6
IQR = interquartile range
IQR = Q3 - Q1
IQR = 6-2
IQR = 4
And,
range = max - min
range = 7 - 1
range = 6
Summary: The data set {1,2,3,4,5,6,7} has range = 6 and IQR = 4.
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We can add some constant to each item of that set to shift things over.
I'll add 10 to each value
{1,2,3,4,5,6,7} turns into {11,12,13,14,15,16,17}
The range is still 6 because max-min = 17-11 = 6
The IQR is still 4 because Q3 - Q1 = 16-12 = 4
This demonstrates that shifting the values the same amount won't affect the range, and won't change the IQR either.
But what if we replaced 12 and 16 with say 12.5 and 15.5 respectively?
What if we had {11,12.5,13,14,15,15.5,17}
Again the range is still 6 because max-min = 17-11 = 6
But the IQR is now different
IQR = Q3 - Q1 = 15.5-12.5 = 3
The IQR is smaller since I moved Q1 and Q3 closer together.
Earlier we had IQR = 4.
Here is a box plot, aka box-and-whisker plot, of {11,12,13,14,15,16,17} in blue
Compared to a box plot of {11,12.5,13,14,15,15.5,17} in red.
The distance from left whisker (11) to right whisker (17) is the same for both box plots.
That distance is the range 6.
However, the width of the boxes themselves aren't the same.
The red box at the bottom is slightly more skinny compared to the blue box up top.
Q1 = left edge of the box
Q3 = right edge of the box
IQR = Q3 - Q1 = distance from Q1 to Q3 = width of the box (ignore whiskers)
Conclusion: Two data sets having the same range does NOT guarantee the IQR's will be the same.
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