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Normal Distribution

Normal Distribution. Tripthi M. Mathew, MD, MPH. 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:

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Normal Distribution

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  1. Normal Distribution Tripthi M. Mathew, MD, MPH

  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. Tripthi M. Mathew, MD, MPH

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

  4. 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. Tripthi M. Mathew, MD, MPH

  5. 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) Tripthi M. Mathew, MD, MPH

  6. 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. Tripthi M. Mathew, MD, MPH

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

  8. 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 Tripthi M. Mathew, MD, MPH

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

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

  11. DistinguishingFeatures • 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 Tripthi M. Mathew, MD, MPH

  12. Skewness • Positive Skewness: Mean≥ Median • Negative Skewness: Median ≥ Mean • Pearson’s Coefficient of Skewness3: = 3 (Mean –Median) Standard deviation Tripthi M. Mathew, MD, MPH

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

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

  15. 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, 2ndedition, 1994. Tripthi M. Mathew, MD, MPH

  16. 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, 2ndedition, 1994. Tripthi M. Mathew, MD, MPH

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

  18. 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, 2ndedition, 1994. Tripthi M. Mathew, MD, MPH

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

  20. 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, 2ndedition, 1994. Tripthi M. Mathew, MD, MPH

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

  22. 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, 2ndedition, 1994. Tripthi M. Mathew, MD, MPH

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

  24. 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, 2ndedition, 1994. Tripthi M. Mathew, MD, MPH

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

  26. 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, 2ndedition, 1994. Tripthi M. Mathew, MD, MPH

  27. 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, 2ndedition, 1994. Tripthi M. Mathew, MD, MPH

  28. 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. Tripthi M. Mathew, MD, MPH

  29. References/Further Reading 1)Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2ndedition, 1994. 2) Last, J. A Dictionary of Epidemiology. 3rd edition,1995. 3) Wisniewski, M. Quantitative Methods For Decision Makers, 3rd edition, 2002. 4) Pidd, M. Tools For Thinking. Modelling in Management Science. 2nd edition, 2003. Tripthi M. Mathew, MD, MPH

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