Transcript z-Scores and the Normal Curve

z-Scores: An introduction
I. z-scores and conversions
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What is a z-score?
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A measure of an observation’s distance from the
mean.
The distance is measured in standard deviation
units.
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If a z-score is zero, it’s on the mean.
If a z-score is positive, it’s above the mean.
If a z-score is negative, it’s below the mean.
If a z-score is 1, it’s 1 SD above the mean.
If a z-score is –2, it’s 2 SDs below the mean.
Computing a z-score
z
X 

X X
or z
SD
Examples of computing z-scores
X
X
X X
SD
z
X X
SD
5
3
2
2
1
6
3
3
2
1.5
5
10
-5
4
-1.25
6
3
3
4
.75
4
8
-4
2
-2
Computing raw scores from z scores
X  z   or
z
X X
SD
SD
X  zSD  X
X
zSD
X
1
2
2
3
5
-2
2
-4
2
-2
.5
4
2
10
12
-1
5
-5
10
5
Review Questions 1
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What does it mean if an observation has a z
score of 1?
The mean = 10, the SD = 2, and X = 8. What
is the z-score?
The mean = 8, the SD = 1, and the z-score =
3. What was the value of the original
observation (X)?
Check answers on next slide
Review Questions 1
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What does it mean if an observation has a z
score of 1? It means that, compared to all the
other observations, this score was 1 standard
deviation greater than the average (mean).
The mean = 10, the SD = 2, and X = 8. What
is the z-score? Z = (8-10)/2 = -1. z = -1
The mean = 8, the SD = 1, and the z-score =
3. What was the value of the original
observation (X)? (3 * 1)+8 = 11. X = 11
Review Questions 2
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To move from a raw score to a z score, what
must we know about the raw score
distribution?
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1 mean and standard deviation
2 maximum and minimum
3 median and variance
4 mode and range
Check answers on next slide
Review Questions 2
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To move from a raw score to a z score, what
must we know about the raw score
distribution?
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1 mean and standard deviation
Review Questions 3
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If Judy got a z score of 1.5 on an in-class
exam, what can we say about her score
relative to others who took the exam?
1 it is above average
2 it is average
3 it is below average
4 it is a ‘B’
Check answers on next slide
Review Questions 3
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If Judy got a z score of 1.5 on an in-class
exam, what can we say about her score
relative to others who took the exam?
1 it is above average
Review Questions 4
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If a raw score is 8, the mean is 10 and the
standard deviation is 4, what is the z-score?
1: -1.0
2: -0.5
3: 0.5
4: 2.0
Check answers on next slide
Review Questions 4
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If a raw score is 8, the mean is 10 and the
standard deviation is 4, what is the z-score?
2: -0.5
Why?
Z = (individual score – mean)/ SD
= (
8
- 10 ) / 4
= -0.5