The normal distribution is a distribution that is symmetrical
about a vertical line through the mean. The curve of the distribution is
bell-shaped with the highest point over the mean. The curve will never
cross or touch the horizontal axis. The differences in normal
distributions makes calculating the area under the curve in a specific
interval difficult. Therefore, there is a method to make this easier by
changing raw scores into standardized scores known as "z scores".

The standardization of scores must be done in such a way that we can use
one table for all of the normal distributions. This is done by
considering how many standard deviations a data value lies from the
mean. This makes it to compare a value from one normal distribution to a
value from another normal distribution.

For example, suppose Frank and Tom are taking a history course. Frank is
in the 8 o'clock class and Tom is in the 10 o'clock class. Since each
class has a large number of students, the scores on the final exam each
follow a normal distribution. Frank's section had a class average of 75
and his score was a 83. In Tom's section, the average was a 68 and Tom
scored a 76. When comparing scores they both felt good that they scored 8
points higher than the average. But which one did better with respect
to the rest of the students in the class?

Suppose a graph of the data shows that Frank scored higher than most of
the other students in his class, but Tom's score places him more in the
middle of the upper half. Then you know that Frank's score is clearly
much better with respect to the students in his class than Tom's score
is. But how can you tell if the distributions are not graphically
represented?

You have to change the raw scores to z scores to see how many standard
deviations from the mean each score lies. Suppose the standard deviation
of scores in Frank's section is 5 and the standard deviation of scores
in Tom's section is 6. The z score is the sample value minus the mean
value divided by the standard deviation. Therefore, the z score for
Frank is 8/5 = 1.6, and the z score for Tom is 8/6 = 1.33. It's clear
that Frank had a better score relative to the class since his score fell
1.6 standard deviations about the mean and Tom's score was only 1.33
standard deviations about the mean.

Note that when standardizing the scores, the mean of the original
distribution is always zero, which is logical since the mean lies zero
standard deviations from itself. Any value above the mean has a positive
z score, and any value lower than the mean has a negative z score.

This guide should help anyone understand how to change a raw score to a z
score, which makes comparing scores from different normal distributions
easy.

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