Video Transcript
Calculate the interquartile range
for each data set. What do the interquartile ranges
reveal about the two data sets? Is it (A) the spread of the middle
50 percent of the values is similar for both data sets? (B) The difference between the
minimum and maximum values is similar for both data sets? (C) The median of the two data sets
should be the same? (D) The mean of the two data sets
should be the same? Or (E) the mode of the two data
sets should be the same?
We will begin by clearing some
space to calculate the interquartile range for each data set. Let’s begin by considering data set
one. We begin by writing our seven
values in ascending order, starting with 22 and ending with 51. The median of any data set is the
middle value. In this case, this will be the
fourth value as there are three values on either side of this. The median of data set one is
28.
The lower quartile or 𝑄 one is the
center of the bottom half of our data set. The bottom half of the data set
contains three values, 22, 25, and 26. The middle one of these is 25. This means that the lower quartile
of data set one is 25. The upper quartile or 𝑄 three is
the center of the top half of our data set. This contains the numbers 28, 29,
and 51. The middle one of these is equal to
29. Therefore, the upper quartile is
29.
The interquartile range or IQR is
equal to 𝑄 three minus 𝑄 one. We subtract the lower quartile
value from the upper quartile value. 29 minus 25 is equal to four. The interquartile range of data set
one is equal to four. We will now repeat this method for
data set two.
As there are also seven values in
data set two, the position of the quartiles and median will remain the same. The lowest value of data set two is
19, and the highest value is 28. We can see from our list that the
median is equal to 24; the lower quartile, 21; and the upper quartile, 27. This means that the interquartile
range is equal to 27 minus 21, which is equal to six. The interquartile range of data set
two is six.
We will now move on to the second
part of the question. In the second part of the question,
we are asked to consider what the interquartile ranges reveal about the two date
sets. The interquartile range does not
rely on the median, mean, or mode. Therefore, we know that options
(C), (D), and (E) are all incorrect. The maximum and minimum values also
have no impact on the interquartile range as these are used to calculate the range
of the entire data.
The interquartile range does
contain the middle 50 percent of the values from the lower quartile to the upper
quartile. As our values of four and six are
quite close, we can conclude that the spread of the middle 50 percent of the values
is similar for both data sets. The interquartile range only gives
us information about those middle values.
