2005 normal distribution. tripthi m. mathew, md, mph objectives learning objective - to understand...
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2005
Normal Distribution
Tripthi M. Mathew, MD, MPH
Objectives
Learning Objective- To understand the topic on Normal Distribution and
its importance in different disciplines.
Performance ObjectivesAt 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
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
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
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
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
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
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
Tables
Areas under the standard normal curve
(See Normal Table)
Tripthi M. Mathew, MD, MPH
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.
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
Tripthi M. Mathew, MD, MPH
Skewness
Positive Skewness: Mean ≥ Median
Negative Skewness: Median ≥ Mean
Pearson’s Coefficient of Skewness3:
= 3 (Mean –Median)
Standard deviation
Tripthi M. Mathew, MD, MPH
Positive Skewness (Tail to Right)
Tripthi M. Mathew, MD, MPH
Negative Skewness (Tail to Left)
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.
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.
Tripthi M. Mathew, MD, MPH
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.
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.
Tripthi M. Mathew, MD, MPH
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.
Tripthi M. Mathew, MD, MPH
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.
Tripthi M. Mathew, MD, MPH
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.
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.
Tripthi M. Mathew, MD, MPH
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.
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.
Tripthi M. Mathew, MD, MPH
13.6%
2.2%
0.15
-3 -2 -1 μ 1 2 3
Diagram of Exercise # 5
0.0150.015
33.35%
The exercises are modified from examples in Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Tripthi M. Mathew, MD, MPH
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.
Tripthi M. Mathew, MD, MPH
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.
Tripthi M. Mathew, MD, MPH
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.