Chapter 8 ♦ Are Your Curves Normal? 213
Here’s another example. In a set of scores with a mean of 100 and
a standard deviation of 10, a raw score of 117 has a corresponding z
score of 1.70. This z score of 1.70 corresponds to an area under the
curve between a z score of 0 and a z score of 1.70 of 45.54%. The
probability of a raw score of 117 occurring is 95.54% (or 50% +
45.54%) or 95.5 out of 100 or .955.
First, even though we are focusing on z scores, there are other types of
standard scores. For example, a t score is a type of standard score that
is computed by multiplying the z score by 10 and adding 50. One
advantage of this type of score is that you rarely have a negative t score.
As with z scores, t scores allow you to compare standard scores from
different distributions.
Second, a standard score is a whole different animal from a standard-
ized score. A standardized score is one that comes from a distribution
with a predefined mean and standard deviation. Standardized scores
from tests such as the SAT and GRE (Graduate Record Exam) are used
so that comparisons can be made easily between scores where the
same mean and standard deviation are being used.
What z Scores Really Represent
The name of the statistics game is being able to estimate the prob-
ability of an outcome. If we take what we have talked about and
done so far in this chapter one step further, statistics is about decid-
ing the probability of some event occurring. Then we use some
criterion to judge whether we think that event is as likely, more
likely, or less likely than what we would expect by chance. The
research hypothesis presents a statement of the expected event, and
we use our statistical tools to evaluate how likely that event is.
That’s the 20-second version of what statistics is, but that’s a lot.
So let’s take everything from the previous paragraph and go through
it again with an example.
Let’s say that your lifelong friend, trusty Lew, gives you a coin
and asks you to determine whether it is a “fair” one—that is, if you
flip it 10 times, you should come up with 5 heads and 5 tails. We
would expect 5 heads (or 5 tails) because the probability is .5 of
either one head or one tail on any one flip. On 10 independent
flips (meaning that one flip does not affect another), we should get
5 heads, and so on. Now the question is, how many heads would
disqualify the coin as being fake or rigged?
