Question 1194420: Find the following: Q1, Q2, Q3, inter quartile range (IQR) and Q.D. of the following set of data using ungrouped:
19, 2, 16, 0, 14, 9, 6, 1, 12, 13, 10, 19, 7, 5, 8
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Answer by math_tutor2020(3816)   (Show Source):
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Original List
19, 2, 16, 0, 14, 9, 6, 1, 12, 13, 10, 19, 7, 5, 8
Sort the values from smallest to largest
0, 1, 2, 5, 6, 7, 8, 9, 10, 12, 13, 14, 16, 19, 19
There are n = 15 items in that list
n/2 = 15/2 = 7.5 which rounds to 8
The middle most number is at slot 8
It has 7 items below it, and 7 items above it.
7+1+7 = 15
The value at slot 8 is 9
The median of this list is 9, which is the value of Q2.
Another way to find the median is to cross off the first and last items of the sorted list. Repeat this process until you steadily march inwards to narrow things down to the middle-most item. You should arrive at the "9".
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After we determine the median, break the sorted list into two groups
L = {0, 1, 2, 5, 6, 7, 8}
U = {10, 12, 13, 14, 16, 19, 19}
Everything in L is below the median
Everything in U is above the median
We can think of them as
L = lower set
U = upper set
The median of set L is 5
The median of set U is 14
Therefore,
Q1 = 5
Q3 = 14
they represent the first and third quartiles respectively.
The difference of which gives the interquartile range (IQR)
IQR = Q3 - Q1
IQR = 14 - 5
IQR = 9
Coincidentally this value matches up with Q2, but this won't always happen.
Dividing the IQR in half will give the quartile deviation (QD)
QD = IQR/2
QD = 9/2
QD = 4.5
Or you could use the steps below
QD = (Q3 - Q1)/2
QD = (14 - 5)/2
QD = 4.5
These set of steps are effectively identical to the previous paragraph. The IQR portion is replaced by Q3 - Q1.
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Answers:
Q1 = 5
Q2 = 9
Q3 = 14
IQR = 9
QD = 4.5
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