z scores. normal vs. standard normal standard normal curve: most normal curves are not standard...
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Z Scores
Normal vs. Standard Normal Standard Normal Curve:
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)
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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 quizAli 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 thesescores – 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
xxz example: a z-score of 2 means the data is two standard deviations above the mean
Understanding z-scores
xxz
The number of standarddeviations is x away
from the mean
The standard deviationThe mean
The data
The deviation
Calculating z-scores
xx
z
number of standarddeviations is x away
from the mean
The standard deviation
The meanThe data
The deviation
The z-score is thedeviation divided by 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:
26,72~X 26.7,68~X
zxx zxx
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%?
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
Find the z-score.
Student 2: got an 81, mean was 68, standard dev was 7.6her 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