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:
Frequency
 ±1 s = about 68%
 ±2 s = about 95%
 ±3 s = about 99%
Scores
Theoretical Normal Curve
68.26%
95.44%
99.72%
-5
-4
-3
-2
-1
0
1
2
3
4
5
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
 Appendix A has three columns.
 (a) = Z scores.
 (b) = areas between the score and the mean
b
b
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
Frequency
 The area below Z = - 1.67 is 0.0475.
 Areas can be expressed as %: 4.75%.
Scores
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:
 X = 13
 s =4
 What is the probability of randomly
selecting a score of 19 or more?
 Z = (19-13)/4 = 6/4 = 1.5
Finding Probabilities
(a)
.
1.49
1.50
1.51
.
1. Find the Z score.
2. For Xi = 19, Z =
.
.
1.50.
0.4319 0.0681 3. Find area above in
column c.
0.4332 0.0668 4. Probability is
0.0668 or 0.07.
(b)
(c)
0.4345 0.0655
.
.