1. Chapter 5
The Normal Curve

2. Chapter Outline Introduction Computing Z Scores The Normal Curve Table
Finding Total Area Above and Below a Score
Finding Areas Between Two Scores
Using the Normal Curve to Estimate Probabilities

3. 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
Finding probabilities

4. Theoretical Normal Curve
Bell Shaped
Unimodal
Symmetrical
Unskewed
Mode, Median, and Mean are same value

5. Theoretical Normal Curve
Distances on horizontal axis always cut off the same area. We can use this property to describe areas above or below any point

6. Theoretical Normal Curve
General relationships:
±1 s = about 68%
±2 s = about 95%
±3 s = about 99%

7. Theoretical Normal Curve

8. 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).

9. Using Appendix A to Find Areas Below a Score
Appendix A can be used to find the areas above and below a score.
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.

10. Using Appendix A Appendix A has three columns. (a) = Z scores.
(b) = areas between the score and the mean

11. Using Appendix A Appendix A has three columns.
( c) = areas beyond the Z score

12. 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

13. Using Appendix A The area below Z = 1.67 is 0.4525 + 0.5000 or 0.9525.
Areas can be expressed as percentages:
= 95.25%

14. Normal curve w z=1.67

15. 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

16. Using Appendix A The area below Z = - 1.67 is 0.475.
Areas can be expressed as %: 4.75%.

17. 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).

18. Finding Probabilities
If A distribution has:
= 13
s = 4
What is the probability of randomly selecting a score of 19 or more?

19. Finding Probabilities
(c) .
1.49
0.4319
0.0681
1.50
0.4332
0.0668
1.51
0.4345
0.0655
Find the Z score.
For Xi = 19, Z = 1.50.
Find area above in column c.
Probability is or 0.07.