Transcript The Normal Distribution

Z Scores
Normal vs. Standard Normal

Standard Normal Curve:
00

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)
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
 Ali scored 60% on Mr. X’s test, while
Sandy scored a 70% on Ms. Y’s test.
Who did better?

Working between
distributions


Ali scored 60% on Mr. X’s test, while
Sandy 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 means are these
scores – this would tell us how we should compare them.
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
z  xx
example: a z-score of 2 means the data is
two standard deviations above the mean
Understanding z-scores
The standard deviation
The mean
z  xx
The deviation
The number of standard
deviations is x away
from the mean
The data
Calculating z-scores
The data
number of standard
deviations is x away
from the mean
z
The mean
xx
The z-score is the
deviation divided by the
standard deviation

The deviation
The standard deviation
Why z-scores?
Z-Scores allow us to convert any
normal distribution to a standard
normal distribution
 This lets us compare distributions

Example:
How do two students compare if one
has a mark of 82% in a class with an
average of 72% and a standard
deviation of 6, and the other has a
mark of 81% in a class with an
average of 68% and a standard
deviation of 7.6?
Steps for comparing
Student 1:
Student 2:
X ~ 72,62 
X ~ 68,7.62 
x  x  z
x  x  z
82 = 72 + z(6)
Z = 1.67
81 = 68 + z(7.6)
Z = 1.71
Student 2 is doing better than
student 1 since it is better to be
1.71 sd above the average than
1.67 above.
Percentiles and Z-Score Table
What percent of students have a mark less than or
equal to a student with a mark of 85% in a class with
an average of 80% and a standard deviation of 11.5%?
Find the z-score.
Looking up 0.43 in the z-score table, you find 0.6664
The student with 85% has a mark in the 66th percentile.
She did better than 66% of the students
Student 2: got an 81, mean was 68, standard dev was 7.6
her z-score was 1.71
Looking up 1.71 in the z-score table, you find 0.9564
Student 1’s mark is at the 95.64 percentile,
or the 95th percentile (always round down)
She also did better than 95% of the students