Copyright © 2014 by Nelson Education Limited. 4-1 Chapter 4 The Normal Curve
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4-1
Chapter 4The Normal Curve
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4-2
• The Normal Curve
• Z scores
• Use of the Normal Curve table (Appendix A)
• Finding areas above and below Z scores
• Finding probabilities
In this presentation you will learn about:
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4-3
• Bell Shaped• Unimodal• Symmetrical• Unskewed• Mode, Median,
and Mean are same value
Scores
Freq
uenc
y
Theoretical Normal Curve
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• Distances on horizontal axis always cut off the same area. We can use this property to describe areas above or below any point.
Theoretical Normal Curve: Specific Areas
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• To find areas, • first compute a Z score. The formula for computing a Z score is *
This formula changes a “raw” score (Xi ) to a standard deviation or Z score.
• second, use Appendix A to find the area above or below a Z score.
*Converting original scores in a population is done using the same method.
Using the Normal Curve: Z Scores
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Using the Normal Curve: Appendix A
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• Appendix A has three columns.– (a) = Z score– (b) = areas between the mean and the Z score
Using the Normal Curve: Appendix A (continued)
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c c
– ( c) = areas beyond the Z score
Using the Normal Curve: Appendix A (continued)
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•The normal curve table can be used to find the:
1. area between a Z score and the mean. (Section 4.3)2. area either above or below a Z score (4.4) *3. area between two Z scores (4.5)4. probability of randomly selected score (4.6) *
* Only these are demonstrated in this presentation
Using Appendix A to Describe Areas Under
the Normal Curve
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• Find your Z score in column (a).
• To find area below a positive score:– Add column (b) area
to 0.50.
• To find area above a positive score– Look in column (c).
(a) (b) (c)
. . .
1.66 0.4515 0.0485
1.67 0.4525 0.0475
1.68 0.4535 0.0465
. . .
How to Find Area Above or Below a (Positive) Z Score
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4-11
• A person has a body weight of 73 kgs. in a distribution of weights with a = 68 kgs. and s = 3 kgs.
• The person’s weight as a Z score is:
How to Find Area Below a (Positive) Z Score: An Example
67.13
6873
Z
X
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• To find the area below a positive Z score, we consult the
normal curve table (Appendix A) to find the area between the
score and the mean (column b): 0.4525.
Then we add this area to
the area below the mean:
0.5000, or
0.4525 + 0.5000 = 0.9525.
• Areas can be expressed
as percentages: 95.25%.
The area below a Z score
of +1.67 is 95.25%. A person with a body weight of 73 kgs. weighs more than 95.25% of all persons.
How to Find Area Below a (Positive) Z Score: An Example (continued)
Normal curve with Z=+1.67
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• Find your Z score in column (a).
• To find area below a negative score:– Look in column (c).
• To find area above a negative score– Add column (b) area
to 0.50
(a) (b) (c)
. . .
1.66 0.4515 0.0485
1.67 0.4525 0.0475
1.68 0.4535 0.0465
. . .
How to Find Area Above or Below a (Negative) Z Score
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4-14
• On the other hand, the Z score for a person with a body weight of 63 kgs. is: -1.67. • To find the area
below a negative
score we use column
c in Appendix A:
the area below a
Z score of -1.67 is
0.0475, or 4.75%. This person weighs more than 4.75% of all persons.
How to Find Area Below a (Negative) Z Score: An Example
Normal curve with Z=-1.67
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Summary: Finding an Area Above or Below a Z Score
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• Areas under the curve can also be expressed as probabilities.
• Probabilities are proportions and range from 0.00 to 1.00.
• The higher the value, the greater the probability (the more likely the event).
• Probability is essential for understanding inferential statistics in Part II of text.
Finding Probabilities
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If a distribution has: =13 and s = 4,
what is the probability of randomly selecting a score of 19 or more?
1. Use the formula for computing a Z score: For Xi = 19, Z = 1.50
2. Find area above in column (c).
3. Probability is 0.0668 of randomly selecting a score of 19 or more.
(a) (b) (c)
. . .
1.49 0.4319 0.0681
1.50 0.4332 0.0668
1.51 0.4345 0.0655
. . .
X
Finding Probabilities: An Example