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Unformatted text preview: Chapter 5 The Median and Mode:
Alternatives to the Mean As you learned in the previous chapter, the mean is the balance point
in a distribution—that is, it is the point at which the deviations ow the mean balance (or cancel out) the deviations above the mean. Although the
mean is the most popular average, there are two conditions under which it
should _not be useg (1) when a distribution is glgbnskewed, and (2)
when the data are 11W In this chapter, you will learn about
two alternative averages that can be used when the mean is inappropriate. Maﬁa The median is the average that is deﬁned as the middle score (that is,
the midpoint) in a distribution of ranked scores. As the middle score, it is
the point that has half the scores above it and half the scores below it.
Here’s a simple example from the previous chapter that includes an outlier
score among children’s contributions to a charity expressed in cents: 4, 8, 8, 9,11,14, 80 We saw that the mean contribution is 19.1, which is not very representa
tive of the amounts donated since none of the children gave anything like
19 cents. The median contribution, on the other hand, is 9, which is quite
representative. To determine the median, ﬁrst put the scores in order from low to high.
Then: (1) When the number of scores is odd, the median is the middle score.
(Note that there are three scores below 9 and three scores above 9 in
the above example.) or
(2) When the number of scores is even, sum the middle two scores and
divide by 2. (For example, for the scores 0, 4, 5, and 12, the middle 2? Chapter 5 The Median and Mode: Alternatives to the Mean two scores are 4 and 5. Summing 4 and 5 and dividing by 2, we get
9/2 = 4.5, which is the median.) A minor complication arises when there are ties in the middle (that is, when two or more cases have the same score in the area where the median
lies). Here’s an example: 3, 6, 7, 7, 7, 20 As you can see, when you count to the middle (three scores up or three
scores down), you come to 7. Two of the 75 are in the middle of the distri
bution, but one of them is “above” the middle. What is the median? Well,
we can use the rule we used earlier to get an approximation. Since there is
an even number of scores and the middle two scores are 7, we sum them
and divide by 2: 7 + 7 = 14/2 = 7, which is the approximate median. For
all practical purposes, this approximation is usually more than adequate.
(A method for taking the ties into account is presented in Appendix A. If
you apply the method in Appendix A to the scores in this example, you
will get a median of 6.8, which is very close to the value of 7 we obtained
using a much easier method.) What if there is an odd number of scores with a tie in the middle? For an approximation, we can apply the rule described on page 27. Here’s an
example: 1, 5,6,7, 8, 8, 8,11,50 Since there are 9 scores, we count up 5 or down 5 and come to 8, which is
our approximate median. (Using the more difﬁcult method in Appendix A, we would get 7.7, which rounds to 8, illustrating again that our
approximation method is quite sound.) When should you use the median? Under two circumstances: hen analyzing equal interval data for which the mean is not appropri
ate because the distribution is highly skewed, and (2) when analyzing ordinal data. As you recall from Chapter 1, ordinal
data put cases in rank order, such as teachers’ rankings of a list of ten
discipline problems from 1 (most important) to 10 (least important).
For each type of problem, such as “hitting another child,” we could
calculate the median rank. The medians would allow us to report which problem the average teacher thought was the most important,
which one was the next most important, and so on. 28 Chapter 5 The Median and Mode: Alternatives to the Mean The Mode The last average we will consider is the mode. It is deﬁned as the most ﬁeguently occurring score.JlIere’s an example we looked at earlier irﬁﬁis
chapter, wit—ere we_foun€tr that the median is 8: 1, 5,6,7, 8, 8,8,11,50 The mode is also 8 because 8 occurs more often than any other score. Note
that the mode does not always have the same value as the median and,
thus, does not always have an equal number of cases on each side of it.
Here’s another example we looked at earlier in this chapter, where we
found that the median is 9: 4, 8, 8, 9,11,14,80 For this example, the mode is 8, which does not have an equal number of
cases on both sides of it. A strength of the mode is that it is easy to determine. However, the
mode has several serious weaknesses. First, a distribution may have more
than one mode. Here’s an example, where the modes are 9 and 10: 6, 6, 8, 9, 9, 9,10,10,10,]5 Since we want a single average, the mode is not a good choice here. Se—
cond, for a small population, each score may occur only once, in which
case, all scores are the mode since all occur equally often. For these rea
sons, the mode is seldom used. Instead, almost all researchers use either the ng6 Concluding Comment The nean median, and mode all belong to the “family” of statistics
calle’ They constitute a family because they all are designed
to present one type of information. A more formal name for this family is
measures of central tendency. In the next chapter, we will begin our con
sideration of another family of statistics, measures of variability. ﬂUemagh .
W  RH up f “41. WW 6";H1‘ﬁﬁi‘um aqua] wf t3in mppﬂn
WM‘ mew/(2 Score Much ' Mm «mm ecu/J7 29 Chapter 5 The Median and Mode: Alternatives to the Mean EXERCISE FOR CHAPTER 5 Factual Questions
1. Which average is deﬁned as the middle score?
2. Suppose you read that the median price of a house in Mudsville is
$167,000. What percentage of the houses in Mudsville cost more than
$167,000? 3. What is the median of the scores shown in the box immediately below?
12,14,14, 15, 20, 22, 66 4. What is the median of the scores shown in the box immediately below?
9, 7,10, 4, 8, 6 5. When analyzing equal interval data that is highly skewed, which
average should you use? 6. When analyzing ordinal data, which average should you use? 7. How is the mode deﬁned? 8. What is the mode of the scores for question 3? 9. For a given distribution, is it possible to have more than one mode? 10. What is another name for the family of statistics called averages? 30 Chapter 5 The Median and Mode: Alternatives to the Mean Questions for Discussion 11. 12. 13. In question 3 you computed a median. Compute the mean for the same
scores. Which average is more representative? Why? Suppose you asked clinical psychologists to rank the problems they
treat in their order of difﬁculty. That is, you asked them to give a rank
of 1 to the most difﬁcult problem, a rank of 2 to the next most difﬁcult problem, and so on. Which average would you use when analyzing the
data? Why? Suppose a counselor told you that “the average salary for a beginning
professional in your chosen ﬁeld is $34,000.” Would you be interested in knowing whether the average is the mean, median, or mode? Why?
Why not? 31 ...
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 Spring '08
 Yanovitsky

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