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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: 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 . .