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Understanding the Normal Curve: Computing Z-Scores and Probabilities

Learn about the properties of the theoretical normal curve, computing Z-scores, using the normal curve table, and finding probabilities above and below scores. Discover how to interpret areas under the curve as probabilities in this educational presentation.

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Understanding the Normal Curve: Computing Z-Scores and Probabilities

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  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.

  13. 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%

  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

  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 • Find the Z score. • For Xi = 19, Z = 1.50. • Find area above in column c. • Probability is 0.0668 or 0.07.

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