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2005 Normal Distribution

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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: 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.

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

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Tripthi M. Mathew, MD, MPH 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.

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Tripthi M. Mathew, MD, MPH 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)

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

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

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Tripthi M. Mathew, MD, MPH 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

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Tripthi M. Mathew, MD, MPH Tables Areas under the standard normal curve (See Normal Table)

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

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Tripthi M. Mathew, MD, MPH 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

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Tripthi M. Mathew, MD, MPH Skewness Positive Skewness: Mean Median Negative Skewness: Median Mean Pearsons Coefficient of Skewness 3 : = 3 (Mean –Median) Standard deviation

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

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

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Tripthi M. Mathew, MD, MPH 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.

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Tripthi M. Mathew, MD, MPH 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.

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Tripthi M. Mathew, MD, MPH 13.6% 2.2% μ Diagram of Exercise # % The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

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Tripthi M. Mathew, MD, MPH 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.

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Tripthi M. Mathew, MD, MPH 13.6% 2.2% μ Diagram of Exercise # % The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

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

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Tripthi M. Mathew, MD, MPH 13.6% 2.2% μ Diagram of Exercise # % The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

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Tripthi M. Mathew, MD, MPH 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.

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Tripthi M. Mathew, MD, MPH 13.6% 2.2% μ Diagram of Exercise # % The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

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Tripthi M. Mathew, MD, MPH 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.

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Tripthi M. Mathew, MD, MPH 13.6% 2.2% μ Diagram of Exercise # % The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

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Tripthi M. Mathew, MD, MPH Solution/Answers 1) 15.9% or ) 2.3% or ) 95.4% or The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

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Tripthi M. Mathew, MD, MPH Solution/Answers Contd 4) 0.15 % or ) 0.3 % or (for each tail) The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

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Tripthi M. Mathew, MD, MPH Application/Uses of Normal Distribution Its 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.

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