copyright © 2012 by nelson education limited. chapter 4 the normal curve 4-1

Copyright © 2012 by Nelson Education Limited.

Chapter 4The Normal Curve

4-1


• The Normal Curve

• Z scores

• The 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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• 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 5.3)2. area either above or below a Z score (5.4) *3. area between two Z scores (5.5)4. probability of randomly selected score (5.6) *

* 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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• A person has a height of 73 inches in a distribution of height where, = 68 inches and s = 3 inches.

• The person’s score 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 height of 73 inches is taller 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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• On the other hand, the Z score for a person with a height of 63 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 is taller 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

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copyright © 2012 by nelson education limited. chapter 4 the normal curve 4-1

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