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305
7.2
Variation
P
urpose
The role this topic plays in quantitative reasoning
One of the goals of analyzing data is to be able to describe the characteristics of a large data set by using
relatively few numbers. The measures of central tendency (mean, median, and mode) describe important
tendencies for a data set, but they are not sufficient. Measures that represent the degree to which data in
a set is spread out are at least as important as the measures of central tendency. The variance (standard
deviation) is a measure of the spread in a set of data. This gives us a sense of the range for specific values
in a set of data.
There are several measures of variation and the ability to select the best measure for a given situation
is an important quantitative reasoning skill. If an analyst can explain what causes variation in a system
or process, he or she will be able to predict much more accurately how that system will behave in the
future. The behavior of a system is based on the relationship between variables in that system and the
factors that influence them, much as the mean age of a student in your class is influenced by the age of
each individual student.
L
earning Goals
What you should learn while completing this activity
1. Become proficient in computing the variance and standard deviation for variables in a data set.
2. Interpret what the standard deviation represents in different data sets.
3. Construct a visual representation of the spread in a data set.
D
iscovery
Finding out for yourself
Imagine that a certain mathematics class has five exams. Two students compare their results on the
exams. The first student had scores of 60, 70, 80, 90, and 100. The second student had scores of 78, 79,
80, 81, and 82. Both students had an average score of 80 and both had a median score of 80. Which
student was more consistent? How do you measure this kind of consistency? Which is the better student
and why? Which student has greater potential in the future?
W
hat Do You Already Know?
Tapping into your existing knowledge
1. How is the mean calculated?
2. How is the range calculated?
3. How can you find the “deviation from the mean” for a number in a variable in a data set?
4. Have you heard the term
standard deviation
? If so what does it mean to you?
5. Why do many measures of variation square the distances between the variable values and their mean?
6. Examining the scores of the students in the previous section, which student do you expect would
have scores with a higher standard deviation?
7. What does it mean to be within a certain number of units of the mean?