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Unit 4: Normal Distributions
Part 2
Statistics
Focus Points
• Given mean μ and standard deviation σ, convert raw data into z-scores
• Given mean μ and standard deviation σ, convert z-scores into raw data
Standard Score
• A standard score is a score expressed as the deviation from the mean score
of a sample in units of standard deviation.
• The number of standard deviations away from the mean
The Z-Score
• The z-score tells us the number of standard deviations the original
measurement is from the mean.
• The z-score is in standard units.
Z- Values
Guided Exercise # 1
A student has computed that it takes an average of 17 minutes with a standard
deviation of 3 minutes to drive from home, park the car, and walk to an early
morning drive.
a) One day it took the student 21 minutes to get to class. How many standard
deviations from the average is that? Is the z value positive or negative?
x -17
z=
3
21-17 4
z=
= =1.33positive
3
3
Guided Exercise # 1
A student has computed that it takes an average of 17 minutes with a standard
deviation of 3 minutes to drive from home, park the car, and walk to an early
morning drive.
b) Another day it took only 12 minutes to get to class. What is the
measurement in standard units? Is the z value positive or negative?
x -17
z=
3
12 -17 -5
z=
=
= -1.67negative
3
3
Guided Exercise # 1
A student has computed that it takes an average of 17 minutes with a standard
deviation of 3 minutes to drive from home, park the car, and walk to an early
morning drive.
c) Another day it took 17 minutes for the students to go from home to class.
What is the z value?
x -17
z=
3
17 -17 0
z=
= =0
3
3
Going Backwards
• We can reverse the process if we know mean and standard deviation for the
original distribution.
z=
x-m
s
s ´z = x-m
s ´z+m = x
x = z ´s + m
Guided Exercise # 2
The scores of a college entrance exam haw a mean of 480 and a standard
deviation of 70 points.
a) Marissa has a z score of 1.3. What is her raw score?
x = z ´ 70 + 480
x =1.9 ´ 70 + 480 =133+ 480 = 613
Guided Exercise # 2
The scores of a college entrance exam haw a mean of 480 and a standard
deviation of 70 points.
b) Josh has a z score of -1.5. What is his raw score?
x = z ´ 70 + 480
x = -1.5´ 70 + 480 = -105+ 480 = 375
Guided Exercise # 2
The scores of a college entrance exam haw a mean of 480 and a standard
deviation of 70 points.
c) Rachel has a z score of 0. What is her raw score?
x = z ´ 70 + 480
x = 0 ´ 70 + 480 = 0 + 480 = 480