1. 2005 Updated 10/19/09 Normal Distribution Tripthi M. Mathew, MD, MPH, MBA

2. Objectives  Learning Objective - To understand the topic on Normal Distribution and its importance in different disciplines.  Performance Objectives At the end of this lecture the student will be able to:  Draw normal distribution curves and calculate the standard score (z score)  Apply the basic knowledge of normal distribution to solve problems.  Interpret the results of the problems.

3. Tripthi M. Mathew, MD, MPH, MBA Types of Distribution  Frequency Distribution  Normal (Gaussian) Distribution  Probability Distribution  Poisson Distribution  Binomial Distribution  Sampling Distribution  t distribution  F distribution

4. Tripthi M. Mathew, MD, MPH, MBA What is Normal (Gaussian) Distribution?  The normal distribution is a descriptive model that describes real world situations.  It is defined as a continuous frequency distribution of infinite range* (can take any values not just integers as in the case of binomial and Poisson distribution).  This is the most important probability distribution in statistics and important tool in analysis of epidemiological data and management science. * Used by Permission of Oxford University Press

5. Tripthi M. Mathew, MD, MPH, MBA Characteristics of Normal Distribution  It links frequency distribution to probability distribution  Has a Bell Shape Curve and is Symmetric  It is Symmetric around the mean: Two halves of the curve are the same (mirror images)

6. Tripthi M. Mathew, MD, MPH, MBA Characteristics of Normal Distribution Cont’d  Hence Mean = Median  The total area under the curve is 1 (or 100%)  Normal Distribution has the same shape as Standard Normal Distribution.

7. Tripthi M. Mathew, MD, MPH, MBA Characteristics of Normal Distribution Cont’d  In a Standard Normal Distribution: The mean (μ ) = 0 and Standard deviation (σ) =1

8. Tripthi M. Mathew, MD, MPH, MBA Z Score (Standard Score) 3  Z = X - μ  Z indicates how many standard deviations away from the mean the point x lies.  Z score is calculated to 2 decimal places. σ

9. Tripthi M. Mathew, MD, MPH, MBA Tables  Areas under the standard normal curve (Appendices of the textbook)

10. Tripthi M. Mathew, MD, MPH, MBA 13.6% 2.2% 0.15 -3 -2 -1 μ 1 2 3 Diagram of Normal Distribution Curve (z distribution) 33.35% Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw -Hill Companies.

11. Tripthi M. Mathew, MD, MPH, MBA Distinguishing Features  The mean ± 1 standard deviation covers 66.7% of the area under the curve  The mean ± 2 standard deviation covers 95% of the area under the curve  The mean ± 3 standard deviation covers 99.7% of the area under the curve

12. Tripthi M. Mathew, MD, MPH, MBA Skewness  Positive Skewness: Mean ≥ Median  Negative Skewness: Median ≥ Mean  Pearson’s Coefficient of Skewness 3 : = 3 (Mean –Median) Standard deviation

13. Tripthi M. Mathew, MD, MPH, MBA Positive Skewness (Tail to Right)

14. Tripthi M. Mathew, MD, MPH, MBA Negative Skewness (Tail to Left)

15. Tripthi M. Mathew, MD, MPH, MBA Exercises  Assuming the normal heart rate (H.R) in normal healthy individuals is normally distributed with Mean = 70 and Standard Deviation =10 beats/min The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw - Hill Companies.

16. Tripthi M. Mathew, MD, MPH, MBA Exercise # 1 Then: 1) What area under the curve is above 80 beats/min? Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw -Hill Companies

17. Tripthi M. Mathew, MD, MPH, MBA 13.6% 2.2% 0.15 -3 -2 -1 μ 1 2 3 Diagram of Exercise # 1 0.159 33.35% The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw -Hill Companies

18. Tripthi M. Mathew, MD, MPH, MBA Exercise # 2 Then: 2) What area of the curve is above 90 beats/min? The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw - Hill Companies

19. Tripthi M. Mathew, MD, MPH, MBA 13.6% 2.2% 0.15 -3 -2 -1 μ 1 2 3 Diagram of Exercise # 2 0.023 33.35% The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw - Hill Companies.

20. Tripthi M. Mathew, MD, MPH, MBA Exercise # 3 Then: 3) What area of the curve is between 50-90 beats/min? The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw - Hill Companies.

21. Tripthi M. Mathew, MD, MPH, MBA 13.6% 2.2% 0.15 -3 -2 -1 μ 1 2 3 Diagram of Exercise # 3 0.954 33.35% The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw - Hill Companies.

22. Tripthi M. Mathew, MD, MPH, MBA Exercise # 4 Then: 4) What area of the curve is above 100 beats/min? The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw - Hill Companies.

23. Tripthi M. Mathew, MD, MPH, MBA 13.6% 2.2% 0.15 -3 -2 -1 μ 1 2 3 Diagram of Exercise # 4 0.015 33.35% The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw -Hill Companies.

24. Tripthi M. Mathew, MD, MPH, MBA Exercise # 5 5) What area of the curve is below 40 beats per min or above 100 beats per min? The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw -Hill Companies.

25. Tripthi M. Mathew, MD, MPH, MBA 13.6% 2.2% 0.15 -3 -2 -1 μ 1 2 3 Diagram of Exercise # 5 0.015 33.35% The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw -Hill Companies.

26. Tripthi M. Mathew, MD, MPH, MBA Solution/Answers 1) 15.9% or 0.159 2) 2.3% or 0.023 3) 95.4% or 0.954 The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw - Hill Companies.

27. Tripthi M. Mathew, MD, MPH, MBA Solution/Answers Cont’d 4) 0.15 % or 0.015 5) 0.3 % or 0.015 (for each tail) The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.©McGraw - Hill Companies.

28. Tripthi M. Mathew, MD, MPH, MBA Application/Uses of Normal Distribution  It’s application goes beyond describing distributions  It is used by researchers and modelers.  The major use of normal distribution is the role it plays in statistical inference.  The z score along with the t –score, chi-square and F- statistics is important in hypothesis testing.  It helps managers/management make decisions.

29. Tripthi M. Mathew, MD, MPH, MBA References/Further Reading 1) Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2 nd edition, 1994. McGraw -Hill Companies. 2) Last, J. A Dictionary of Epidemiology. 3 rd edition,1995. Oxford University Press. 3) Wisniewski, M. Quantitative Methods For Decision Makers, 3 rd edition, 2002. Pearson Education. 4) Pidd, M. Tools For Thinking. Modelling in Management Science. 2 nd edition, 2003. John Wiley & Sons Ltd.