Lecture 5: Z test/score
Where we are going...
- Z-scores requires a knowledge of standard deviation
- Probability theory requires a knowledge of frequency distributions, proportions
- The distribution of sample means requires an understanding of the mean and of the
standard deviation
- Hypothesis testing requires an understanding of z-scores, probability theory, and the
distribution of sample means
Consider the following...
- Imagine you wrote two different tests
TEST 1 TEST 2
X = 60 X = 60
µ = 50 µ = 50
- Are you happier with your mark on the first test, or with your mark on the second test?
- What if I included this information?
TEST 1 TEST 2
X = 60 X = 60
µ = 50 µ = 50
= 20 = 5
- Are you happier with your mark on the first test, or with your mark on the second test?
Why?
Calculate
X−µ
Z= ❑
- Calculate z when X= 60, µ= 50, and = 20
- Calculate z when X= 60, µ= 50, and = 5
What is a z-score?
- It describes the exact location of a score in a distribution
- Az-score of +.5 indicates X was exactly one-half standard deviations above the mean of
the distribution of scores
- Az-score of +2 indicates X was exactly two standard deviations above the mean of the
distribution of scores
Calculate
X−µ
Z= ❑
- Calculate z when X= 50, µ= 60, and = 20
- Calculate z when X= 50, µ= 60, and = 5
What is a z-score?
- Az-score of -.5 indicates X was exactly one-half standard deviations below the mean of
the distribution of scores
- Az-score of -2 indicates X was exactly two standard deviations below the mean of the
distribution of scores
Imagine the test scores were normally distributed - Az-score of “1” indicates that the raw score X is one standard deviation above the mean
of the distribution of scores
- Since the height of this distribution indicates the likelihood of a score occurring, it should
be noted that more extreme z-scores are comparatively rare
- So you should be quite happy with a z-score of +2.0! Why?
50 If =. 5
60 If =.2
60
Try it!
X
1
1
1
2
5 - Transform this distribution of raw scores into a distribution of z-scores
Consider the following distribution of scores
X X – µ (X – µ)2 z
1
1
1
2
5
- What is µ?
- What is ?
X X – µ (X – µ)2 z
1 -1 1 -.65
1 -1 1 -.65
1 -1 1 -.65
2 0 0 0
5 3 9 1.94
- What is µ? (2)
- What is ? (1.55)
Let’s do some more math!
z
-.65
-.65
-.65
0
1.94
- What is the mean of the distribution of z-scores? 0
- What is the standard deviation of the distribution of z-scores? 1
z z – µ
(z – z
µz
)²
-.65 -.65 .42
-.65 -.65 .42
-.65 -.65 .42
0 0 0
1.94 1.94 3.76
Always true!!
- The mean of a distribution of z-scores is ALWAYS 0
• All positive z-scores are above the mean, and all the negative z-scores are below the
mean
- The standard deviation of a distribution of z-scores is ALWAYS 1
Let’s draw graphs!
- First draw a frequency distribution of the raw scores 1, 1, 1, 2, 5
- Then, draw a

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