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Chapter 5
The Normal Curve
In This Presentation
This presentation will introduce The Normal Curve Z scores The use of the Normal Curve table
(Appendix A) Finding areas above and below a
particular score Finding probabilities
Theoretical Normal Curve
Bell Shaped Unimodal Symmetrical Unskewed Mode,
Median, and Mean are same value
Normal Curve
68.26%
95.44%
99.72%
Theoretical Normal Curve
Distances on horizontal axis, expressed in terms of standard deviation units, always cut off the same area. We can use this property to describe areas above or below any point in terms of probability of occuring
Theoretical Normal Curve
General relationships: ±1 s = about 68% ±2 s = about 95% ±3 s = about 99%
Scores
Fre
qu
en
cy
Theoretical Normal Curve
-5 -4 -3 -2 -1 0 1 2 3 4 5
68.26%
95.44%
99.72%
Using the Normal Curve: Z Scores To find areas, first compute Z scores. The formula changes a “raw” score (Xi) to a
standardized score (Z), expressed in terms of standard deviation units above or below the mean
Using Appendix A to Find Areas Below a Score
Appendix A can be used to find the areas above or below a score, provided the distribution is normal
First compute the Z score, taking careful note of the sign of the score.
Draw a picture of the normal curve and shade in the area in which you are interested.
Using Appendix A
b b
Appendix A has three columns. (a) = Z scores. (b) = areas between the score and the mean
Using Appendix A
Appendix A has three columns. ( c) = areas beyond the Z score
c c
Using Appendix A Find your Z score
in Column A. To find area below
a positive score: Add column b area
to .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
. . .
Using Appendix A The area below Z = 1.67 is 0.4525 +
0.5000 or 0.9525. Areas can be expressed as percentages:
0.9525 = 95.25%
95.2
Normal curve w z=1.67
95.25%
Using Appendix A What if the Z score
is negative (–1.67)? To find area below
a negative score: Look in column c.
To find area above a negative score Add column b .50
(a) (b) (c)
. . .
1.66 0.4515 0.0485
1.67 0.4525 0.0475
1.68 0.4535 0.0465
. . .
Using Appendix A The area below Z = - 1.67 is 0.0475. Areas can be expressed as %: 4.75%.
Scores
Fre
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Finding Probabilities
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) that a particular case will fall within that area.
Finding Probabilities
If A distribution has: = 13 s = 4
What is the probability of randomly selecting a score of 19 or more?
Z = (19-13)/4 = 6/4 = 1.5
X
Finding Probabilities
1. Find the Z score.2. For Xi = 19, Z =
1.50.3. Find area above in
column c.4. Probability is
0.0668 or 0.07.
(a) (b) (c)
. . .
1.49 0.4319 0.0681
1.50 0.4332 0.0668
1.51 0.4345 0.0655
. . .