Senior Math
Intro to Z-Score
Before we talk about this new thing, “z-score,” we must first refresh our memories about
Normal Distribution, Standard Deviation, and the 68-95-99.7 Rule.
Empirical (68-95-99.7) Rule
Example 1:
Suppose a Normal model describes the fuel efficiency of cars currently registered in
New Jersey. The mean is 24 mpg, with a standard deviation of 6 mpg. Sketch a normal
curve. Describe the fuel efficiency of the cars using the 68-95-99.7 Rule.
Example 2:
The Postmaster of a city’s Post Office believes that a Normal model is useful in projecting
the number of letters, which will be mailed during the day. They use a mean of 20,000
letters and a standard deviation of 250 letters. Draw and clearly label this model.
What z-score (standard score) allows us to do is to compare apples to oranges. What do I
mean by this? Well consider the following example:
Let’s talk about scoring a decathlon. Silly example, but suppose two competitors tie in each
of the first eight events. In the ninth event, the high jump, one clears the bar 1 in. higher. Then
in the 1500-meter run the other one runs 5 seconds faster. Who wins?
It boils down to knowing whether it is harder to jump an inch higher or run 5 seconds faster.
We have to be able to compare two fundamentally different activities involving different units.
Standard deviations to the rescue!
If we knew the mean performance (by world-class athletes) in each event, and the
standard deviation, we could compute how far each performance was from the mean in SD
units (called z-scores). So consider the three athletes’ performances shown below in a threeevent competition. Note that each placed first, second, and third in an event. Who gets the gold
medal? Who turned in the most remarkable performance of the competition?
Understanding standard deviation is our first step! That’s why we just focused so much on
Normal Models!
Z-score: indicates how many standard deviations an element is from the mean.
We can use the following formula to calculate z-score:
z=
Understanding z-score:

A z-score less than 0 represents an element ________________________

A z-score greater than 0 represents an element ___________________________

A z-score equal to 0 represents an element equal to the mean.

A z-score equal to 1 represents an element that is 1 standard deviation greater than the
mean; a z-score equal to 2, 2 standard deviations greater than the mean; etc.

A z-score equal to ____ represents an element that is 1 standard deviation
________________________ mean; a z-score equal to -2, 2 standard deviations less
than the mean; etc.

About 68% of the elements have a z-score between -1 and 1; about 95% have a z-score
between -2 and 2; and about 99% have a z-score between -3 and 3.
Using what you know about z-score and the SAT score normal curve above, answer the following:
1.) What are the mean and standard deviation?
Mean =
2.) What is the z-score of someone who gets a 690 on the SAT?
3.) What is the z-score of someone who gets a 400 on the SAT?
Stddev =
Let’s go back to the decathlon problem. The table on below shows the three competitors and
their stats for the three events at a track meet.
Z-scores will allow us to compare apples to oranges, or in this case different types of events.
So, let’s find the z-scores. Don’t forget, lower times are better in the 100-m dash!
Competitor
100-m dash
A
10.1 sec
z-scores
Total:
B
z-scores
Total:
C
z-scores
Total:
Shot put
Long jump
Practice
Adult female Dalmatians weigh an average of 50 pounds with a standard deviation of 3.3
pounds. Adult female Boxers weigh an average of 57.5 pounds with a standard deviation of
1.7 pounds. One statistics teacher owns an underweight Dalmatian and an underweight
Boxer. The Dalmatian weighs 45 pounds, and the Boxer weighs 52 pounds. Which dog is
more underweight? Explain.