1. M11-Normal Distribution 1 1  Department of ISM, University of Alabama, 1995-2003 Lesson Objective  Understand what the “Normal Distribution” tells you.  Learn the characteristics of the Normal Distribution.  Learn how different values of the mean and standard deviation affect the location and shape of the density curve.

2. M11-Normal Distribution 1 2  Department of ISM, University of Alabama, 1995-2003 The Normal Distributions a.k.a., “The Bell Shaped Curve” Describes the shape for some quantitative, continuous random variables.

3. M11-Normal Distribution 1 3  Department of ISM, University of Alabama, 1995-2003 As the sample size increases, the sample data distribution approaches the shape of the smooth curve shown in the plot. This smooth representation of the population data distribution is called a density function. Observe:

4. M11-Normal Distribution 1 4  Department of ISM, University of Alabama, 1995-2003 Equation of the Normal Distribution density function: 1 f( x ) = 2    ( x –  ) 2 -2  2 e

5. M11-Normal Distribution 1 5  Department of ISM, University of Alabama, 1995-2003  = mean determines the location.  = standard deviation determines spread. Normal Population Distribution has two parameters: These parameters are estimated with the following sample statistics: x is an estimator of  s is an estimator of 

6. M11-Normal Distribution 1 6  Department of ISM, University of Alabama, 1995-2003 Population Census Real Parameter Sample Statistic estimates  x, s

7. M11-Normal Distribution 1 7  Department of ISM, University of Alabama, 1995-2003 Total area = 1.0 or 100% Not concerned about this axis.

8. M11-Normal Distribution 1 8  Department of ISM, University of Alabama, 1995-2003 Standard Z Understand the Std. Normal, you understand all Normals. Standard Normal

9. M11-Normal Distribution 1 9  Department of ISM, University of Alabama, 1995-2003 Observe: For normal density curves, Total area is always 1.0 or 100%. Area == Relative Frequency Symmetric about the mean. Values can range from -  to + 

10. M11-Normal Distribution 1 10  Department of ISM, University of Alabama, 1995-2003 X ~ N(  = 66,  = 9) or N(66, 9) Z = the number of standard deviations that an X - value is from the mean. Notation: X -   Z = Z ~ N(  = 0,  = 1 ) or N(0,1) Z follows the “Standard Normal Distribution”

11. M11-Normal Distribution 1 11  Department of ISM, University of Alabama, 1995-2003 66 X ~ N(  = 66,  = 9 ) 0Z X

12. M11-Normal Distribution 1 12  Department of ISM, University of Alabama, 1995-2003 ____, ±1  ______, ±2  ______, ±3  Empirical Rule of the Normal Distribution

13. M11-Normal Distribution 1 13  Department of ISM, University of Alabama, 1995-2003 Three Normals AA B C

14. M11-Normal Distribution 1 14  Department of ISM, University of Alabama, 1995-2003 1. Largest mean? 2. Smallest mean? 3. Largest standard deviation? 4. Smallest standard deviation? Estimate the value for each of the above. Which population has the:

15. M11-Normal Distribution 1 15  Department of ISM, University of Alabama, 1995-2003 Target Lower Specification Limit Upper Specification Limit Distribution is ________, The mean is ____, the standard deviation is ____. Specification Limits: What the customer requires!

16. M11-Normal Distribution 1 16  Department of ISM, University of Alabama, 1995-2003 Lower Specification Limit Upper Specification Limit Actual Waste cost us $$$! Process has no margin for error; Items are produced that the customer cannot use and will not buy! Waste cost us $$$! The mean has drifted off target. Proportion of Defects Target

17. M11-Normal Distribution 1 17  Department of ISM, University of Alabama, 1995-2003 Lower Specification Limit Upper Specification Limit The mean is on target, but the variation has increased. Proportion of Defects Target Actual  is ______.

18. M11-Normal Distribution 1 18  Department of ISM, University of Alabama, 1995-2003 Lower Specification Limit Upper Specification Limit The mean is off target AND the variation has increased. Actual Target Proportion of Defects

19. M11-Normal Distribution 1 19  Department of ISM, University of Alabama, 1995-2003 Lower Specification Limit Upper Specification Limit The mean is on target and the variation has been reduced! Target Actual  is ______.

20. M11-Normal Distribution 1 20  Department of ISM, University of Alabama, 1995-2003 Lower Specification Limit Upper Specification Limit The mean is off target, but the variation is still reduced! Actual Target

21. M11-Normal Distribution 1 21  Department of ISM, University of Alabama, 1995-2003 Continuous Quality Improvement It’s just good business!  Reduce variation  Reduce costs (i.e., waste)  Increased profits  Increased market share  Better return on investment

22. M11-Normal Distribution 1 22  Department of ISM, University of Alabama, 1995-2003 If the mean and variance do not change over time, then our process is stable (“in statistical control”) “Control charts” are used to check for this stability in production and service processes.

23. M11-Normal Distribution 1 23  Department of ISM, University of Alabama, 1995-2003