1. Section 6.1 Introduction to the Normal Distribution Entirely rewritten by D.R.S., University of Cordele.

2. All About the Normal Distribution It is Bell Shaped

3. All About the Normal Distribution It is Symmetric – the Left Side and the Right Side are mirror images of each other with respect to the vertical line at the peak, in the middle. Fold it over at the center line and the curvy parts will match exactly.

4. All About the Normal Distribution It goes on FOREVER, with the horizontal axis as an ASYMPTOTE (forever approaching but never touching nor crossing the axis) The curve gets closer and closer and closer and closer to the horizontal axis but they never touch; they never cross.

5. All About the Normal Distribution The total area under the entire curve – even counting the stretch to ∞ and to – ∞, is EXACTLY 1.0000000000, the same as a 1-by-1 square!!! Total Area = 1.00000 exactly !!! width = 1 length = 1 Area of this square = 1 x 1 = 1.00000 exactly !!!! and this just happens to tie in with the important fact about probability distributions: that the sum of the probabilities in the probability column must equal exactly precisely 1.00000 !!!!

6. The z Axis The z-axis always has 0 in the middle. If you make it go from -3 to +3 with steps of size 1, it fits most problems just fine. z -3 -2 -1 0 1 2 3 It is unusual to see a z value beyond –3 or +3 but it happens. Beyond –4 and +4 is extremely rare! Be suspicious if it happens during your work. It’s possible but extreme.

7. The x Axis The x-axis lines up with the z-axis but it has different numbers because there are many different normal distributions. The numbers depend on the mean and on the standard deviation. x z -3 -2 -1 0 1 2 3

8. The x Axis Suppose it’s the mean test score of 75 and a standard deviation of 8. 75 is in the middle… x 75 z -3 -2 -1 0 1 2 3

9. The x Axis Suppose it’s the mean test score of 75 and a standard deviation of 8. 75 is in the middle; each step up is +8 x 75 83 91 99 z -3 -2 -1 0 1 2 3

10. The x Axis Suppose it’s the mean test score of 75 and a standard deviation of 8. 75 is in the middle; each step up is +8 and each step down is -8 from the mean x 51 59 67 75 83 91 99 z -3 -2 -1 0 1 2 3

11. The x Axis Similarly, if it’s mean life span of 81.4 years with a standard deviation of 4.3 years, we have this x-axis: x 68.5 72.8 77.1 81.4 85.7 90.0 94.3 z -3 -2 -1 0 1 2 3

12. x 68.5 72.8 77.1 81.4 85.7 90.0 94.3 z -3 -2 -1 0 1 2 3

13. x 68.5 72.8 77.1 81.4 85.7 90.0 94.3 z -3 -2 -1 0 1 2 3

14. x 68.5 72.8 77.1 81.4 85.7 90.0 94.3 z -3 -2 -1 0 1 2 3 Going the other way: z to x…

15. x 68.5 72.8 77.1 81.4 85.7 90.0 94.3 z -3 -2 -1 0 1 2 3

16. If you know the x valueTo work backward from z to x 16 These formulas agree with the labeling of the axes you did in the Empirical Rule and Chebyshev’s Theorem problems. In those problems, the z values were always nice integers: -3, -2, -1, 0, 1, 2, 3.

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18. How are the formulas related?. Input x, output z. Input z, output x.

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21. Excel STANDARDIZE function to convert a data value (x) to a standard score (z)