The Normal Distribution - California State University, Northridge
The Normal Distribution
Cal State Northridge 320 Andrew Ainsworth PhD
The standard deviation
Benefits:
Uses measure of central tendency (i.e. mean) Uses all of the data points Has a special relationship with the normal curve Can be used in further calculations
Psy 320 - Cal State Northridge
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0.025
Normal Distribution
0.02
0.015
0.01
0.005
0
20
40
60
80 100 120 140 160 180
Example: The Mean = 100 and the Standard Deviation = 20
Psy 320 - Cal State Northridge
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1
f(X)
Normal Distribution (Characteristics)
Horizontal Axis = possible X values
Vertical Axis = density (i.e. f(X) related to probability or proportion)
Defined as f ( X ) = 1 (e)-( X -?)2 2 2 2
f (Xi ) = (s)
1
* (2.71828183)-( Xi -X )2 2s2
2 * (3.14159265)
The distribution relies on only the mean and s
Psy 320 - Cal State Northridge
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Normal Distribution (Characteristics)
Bell shaped, symmetrical, unimodal
Mean, median, mode all equal
No real distribution is perfectly normal
But, many distributions are approximately normal, so normal curve statistics apply
Normal curve statistics underlie procedures in most inferential statistics.
Psy 320 - Cal State Northridge
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Normal Distribution
f(X)
? + 4sd ? + 3sd ? + 2sd ? + 1sd
? - 1sd ? - 2sd ? - 3sd ? - 4sd
?
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2
The standard normal distribution
What happens if we subtract the mean from all scores? What happens if we divide all scores by the standard deviation? What happens when we do both???
Psy 320 - Cal State Northridge
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0.025
Normal Distribution
0.02
0.015
f(X)
0.01
0.005
0
20
40
60
80 100 120 140 160 180
-mean -80 -60 -40 -20
0
20 40 60 80
/sd
1
2
3
4
5
6
7
8
9
both -4
-3 -2
-1
0
1
2
3
4
Psy 320 - Cal State Northridge
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The standard normal distribution
A normal distribution with the added properties that the mean = 0 and the s = 1
Converting a distribution into a standard normal means converting raw scores into Z-scores
Psy 320 - Cal State Northridge
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3
Z-Scores
Indicate how many standard deviations a score is away from the mean.
Two components:
Sign: positive (above the mean) or negative (below the mean).
Magnitude: how far from the mean the score falls
Psy 320 - Cal State Northridge
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Z-Score Formula
Raw score Z-score
Zi
=
Xi - X s
= score - mean standard deviation
Z-score Raw score
Xi = Zi (s) + X
Psy 320 - Cal State Northridge
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Properties of Z-Scores
Z-score indicates how many SD's a score falls above or below the mean.
Positive z-scores are above the mean.
Negative z-scores are below the mean.
Area under curve probability
Z is continuous so can only compute probability for range of values
Psy 320 - Cal State Northridge
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4
Properties of Z-Scores
Most z-scores fall between -3 and +3 because scores beyond 3sd from the mean
Z-scores are standardized scores allows for easy comparison of distributions
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The standard normal distribution
Rough estimates of the SND (i.e. Z-scores):
Psy 320 - Cal State Northridge
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The standard normal distribution
Rough estimates of the SND (i.e. Z-scores): 50% above Z = 0, 50% below Z = 0 34% between Z = 0 and Z = 1,
or between Z = 0 and Z = -1 68% between Z = -1 and Z = +1 96% between Z = -2 and Z = +2 99% between Z = -3 and Z = +3
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5
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