Question 1203218: Please show steps of how to solve the question. Thanks!
The following Stem & Leaf plot shows the length distribution (in centimetres) of adult cats.
2 | 3
3 | 1223456677799
4 | 0257778
5 | 368
6 | 235
Determine the following:
The middle 50% of the lengths are from ___ cm to ___ cm
Answer by greenestamps(13203)   (Show Source):
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(NOTE: questions are not solved. Questions are answered; PROBLEMS are solved.)
The middle 50% of the lengths are those between the 25th and 75th percentiles, or between the first and third quartile.
To find the first and third quartiles, first find the median (second quartile).
There are 27 data points. That's one in the middle and 13 on either side of the middle. So the 14th number is the middle one.
There is 1 length in the 20s and 13 in the 30s, so the last length in the 30s is the median. That length is 39cm.
So the median length is 39cm.
There are 13 lengths below the median. Of those 13 lengths, there is one in the middle, with 6 above and 6 below that middle.
So there are 6 lengths below the first quartile, which means the first quartile is the 7th length in the data set. That length is 35cm.
The first quartile length is 35cm.
Similarly, there are 13 lengths above the median. Of those 13 lengths, there is one in the middle, with 6 above and 6 below that middle.
So there are 6 lengths between the median and the third quartile, which means the first quartile is the 7th length in the data set after the median. Note another way of thinking is that there are 6 lengths above the third quartile, so the third quartile length is the 7th LONGEST length. That length is 48cm.
The third quartile length is 48cm.
The middle 50% of the lengths are those above the first quartile length of 35cm and below the third quartile length of 48cm.
ANSWER: Between 35cm and 48cm
In a crude diagram....
Median (second quartile): 13 lengths shorter and 13 lengths longer:
x x x x x x x x x x x x x 39 x x x x x x x x x x x x x
First quartile: the 13 lengths below the median are 6 below the first quartile and 6 above:
x x x x x x 35 x x x x x x
Third quartile: the 13 lengths above the median are 6 below the third quartile and 6 above:
x x x x x x 48 x x x x x x
All together, the numbers of lengths below the first quartile, between the first quartile and the median, between the median and the third quartile, and above the third quartile are all equal (6):
x x x x x x 35 x x x x x x 39 x x x x x x 48 x x x x x x
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