Zeta Student Workbook
Copyright 2012
Application & Enrichment
page covering statistics
APPLICATION & ENRICHMENT
28G
In 25G we used the range (the difference between the highest and lowest values) as one
way of describing the spread of a set of data. Another measure of spread used in statistics
is the interquartile range, or IQR, which tells us the range of the middle half of the data.
Follow the steps to see how to find and interpret the IQR.
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Step 1: Find the median (2nd quartile) on the dot plot shown. List all the
data in order. The points are 1, 4, 5, 5, 5, 5, 6, 6, 6, 7, 8, 9, and 13, There
are an odd number of data, so the middle point, or 6, is the median.
Step 2: Find the median of the first half of the data points. This is the 1st
quartile. List all the data to the left of the 2nd quartile: 1, 4, 5, 5, 5, 5. Since
there are an even number of data, use the average of the two middle points.
The median of this new, smaller group of data is 5.
Step 3: Find the median of the second half of the data. This is the 3rd
quartile. List all the data to the right of the 2nd quartile: 6, 6, 7, 8, 9, 13.
The median of the new group is 7.5.
Step 4: To find the interquartile range, or IQR, subtract the 1st quartile
from the 3rd quartile. 7.5 – 5 = 2.5. This means that the middle 50% of the
data—in this case, the points 5, 5, 6, 6, 6, 6, and 7—are clustered in a range
of 2.5. Note that all of these points fall between the numbers 5 (the 1st
quartile) and 7.5 (the 3rd quartile).
Examine the numbers we have found. The mean of all the numbers is 6.15. The
median, as we found in step 1, is 6. The range is 12. The IQR is 2.5. Although the range
of 12 tells us something about the spread, the IQR tells us more by showing that half
of the data cover a range of only 2.5. In this example, the mean and median are quite
similar, but be careful: this is not always the case. The mean is a better measure of the
center of weight, but the median measures the center of the items. If a very large number
were added to the data, the mean would shift dramatically, but the median would not
move much. For example, if the number 102 were added to the set, the mean would
become 13 (the sum of the numbers divided by 14), and the median would become
the average of the two middle numbers. Since both of the middle numbers are six, the
median would stay the same. Adding a huge number to the set doesn’t change the fact
that most of the numbers are either 5s or 6s. This shows how the median is a good
indicator of where there might be a peak in the data. The IQR measures how closely
ZETA APPLICATION & ENRICHMENT 28G
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APPLICATION & ENRICHMENT 28G
the data gathers around the peak. If the IQR is small, more of the numbers are close to
each other. If it is large, the numbers are more spread out. Think of the word “quarter”
to help you remember the word “quartile.” You are dividing the graph into four parts
or quarters and looking at the range of the values in the middle two parts. The IQR is
especially useful in graphs that have one or two pieces of data that are very different from
most other data. A value that is far from other values is called an outlier.
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1. What is the median value on the dot plot above?
2. Find the the 1st quartile (the median of the left half of the data set).
3. Find the 3rd quartile (the median of the right half of the graph).
4. Subtract the 1st quartile from the 3rd quartile to find the interquartile range.
5. Find the range of all the data shown on the dot plot. Compare it to the IQR.
6. Find the mean of the data shown on the dot plot. Compare it to the median.
Notice the differences between this set of data and the one on the previous page. The
IQR remains the same, but the mean and median are farther apart than they were with
the previous example. Notice that the data is clustered near one end with an outlier on
the other end, rather than being more balanced. In a case like this, the IQR gives us a
very different picture of the spread of the data than a simple range between the highest
and lowest values would. In Application & Enrichment 29G, you will learn how to draw
box plots, which are a way to graph the quartiles and the IQR. That activity may help
make the concepts you’ve learned here clearer.
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