Page 105 - Statistics for Dummies

Chapter 5: Means, Medians, and More
For example, suppose you scored 30 on the Math exam; in Table 5-4 you look
at the row for 30 in the column for Math; you see your score is at the 95th
percentile. In other words 95% of the students scored lower than you, and
only 5% scored higher than you.
Now suppose you also scored a 30 on the Reading exam. Just because a score
of 30 represents the 95th percentile for Math doesn’t necessarily mean a score
of 30 is at the 95th percentile for Reading as well. (It’s probably reasonable
to expect that fewer people score 30 or higher on the Math exam than on the
Reading exam.)
To test my theory, look at column 3 of Table 5-4 in the row for a score of 30. You
see that a score of 30 on the Reading exam puts you at the 91st percentile —
not quite as great as your position on the Math exam, but certainly not a bad
score.
Gathering a five-number summary 89
Beyond reporting a single measure of center and/or a single measure of
spread, you can create a group of statistics and put them together to get
a more detailed description of a data set. The Empirical Rule (as seen in
“Examining the Empirical Rule (68-95-99.7)” earlier in this chapter) uses the
mean and standard deviation in tandem to describe a bell-shaped data set.
In the case where your data are not bell-shaped, you use a different set of
statistics (based on percentiles) to describe the big picture of your data. This
method involves cutting the data into four pieces (with an equal amount of
data in each piece) and reporting the resulting five cutoff points that sepa-
rate these pieces. These cutoff points are represented by a set of five statis-
tics that describe how the data are laid out.
The five-number summary is a set of five descriptive statistics that divide
the data set into four equal sections. The five numbers in a five-number
summary are:
1. The minimum (smallest) number in the data set
2. The 25th percentile (also known as the first quartile, or Q )
1
3. The median (50th percentile)
4. The 75th percentile (also known as the third quartile, or Q )
3
5. The maximum (largest) number in the data set
For example, suppose you want to find the five-number summary of the fol-
lowing 25 (ordered) exam scores: 43, 54, 56, 61, 62, 66, 68, 69, 69, 70, 71, 72,
77, 78, 79, 85, 87, 88, 89, 93, 95, 96, 98, 99, 99. The minimum is 43, the maxi-
mum is 99, and the median is the number directly in the middle, 77.

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