1. Z Scores and Percentiles
Lesson 3.1 Part 2

2. Histograms What do we remember about histograms? For a data set,
What does the mean tell us?
What does the standard deviation tell us?

3. Normal Distribution
A family of distributions (histograms) with a symmetrical bell shape

4. Standard Normal Distribution
A special case of the normal distribution with a mean of 0 and a standard deviation of 1.

5. Normal vs. Standard Normal
Most normal curves are not standard normal curves
They may be translated along the x axis (different means)
They might be wider or thinner (different standard deviations)
standard
σ > 1
σ < 1

6. Why is this a problem? Imagine there are two MDM4U classes.
Mr. X teaches one section, while Ms. Y teaches the other one.
They both have a quiz
Andy scored 60% on Mr. X’s test, while Mason scored a 70% on Ms. Y’s test. Who did better?

7. Working between distributions
Andy scored 60% on Mr. X’s test, while Mason scored a 70% on Ms. Y’s test. Who did better?
It is hard to compare these two different quizzes… maybe Mr. X’s was tougher, and a 60% on his quiz is better than a 70% on Ms. Y’s quiz
How many standard deviations away from the mean are these scores? This would tell us how we should compare them.

8. Z-scores
The z-score for a given piece of data is how far away it is from the mean – it counts the number of standard deviations
For example: a z-score of 2 means the data is two standard deviations above the mean
A positive z-score indicates that the value lies above the mean.
A negative z-score indicates that the value lies below the mean.

9. Calculating z-scores
The data
The mean
number of standard deviations x is away from the mean
The deviation
The standard deviation
The z-score is the deviation divided by the standard deviation

10. Ex. # 1: Calculating Z-Scores
For the distribution with a mean of 20 and a
standard deviation of 4, determine the
number of standard deviations each piece of
data lies above or below the mean.
a) x = b) x = 25

11. Why z-scores?
convert to
standard
convert to
standard

12. Why z-scores?
Z-Scores allow us to convert any normal distribution to a standard normal distribution
This lets us compare distributions that have different means and standard deviations.

13. Ex. #2
Cayley: got an 84, mean was 74, standard dev was 8
Lauren: got an 83, mean was 70, standard dev was 9.8
Cayley’s z-score:
Lauren’s z score was higher – she did “better” when compared with the rest of her class
Lauren’s z-score:

14. Percentiles Percentiles are another way to talk about z-scores
the kth percentile is the data value that is greater than k% of the population
The table on page 606 and 607 relates z-scores to percentiles.
Lauren:her z-score was 1.32
Looking up 1.32 in the z-score table, you find
Lauren’s mark is at the percentile,
or the 90th percentile (always round down)
She did better than 90% of the students

15. Ex. #3: Calculating Percentiles
Fish in a lake have a mean length of 20 cm and a
standard deviation of 5 cm. Find the percent of
the population that is less than or equal to the
following lengths (the percentile).
a) 23 cm b) 11 cm c) 30 cm

16. Ex. #4: See the textbook question on page 145 (Example 5).c)

17. Practice Questions
Page 148 #4, 5, 9, 14