Normal Distribution

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.

3
Types of Distribution

• Frequency Distribution
• Normal (Gaussian) Distribution
• Probability Distribution
• Poisson Distribution
• Binomial Distribution
• Sampling Distribution
• t distribution
• F distribution

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.

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)

6
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.

7
Characteristics of Normal Distribution Contd

• In a Standard Normal Distribution
• The mean (µ ) 0 and
• Standard deviation (s) 1

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.

s
9
Tables

• Areas under the standard normal curve
• (Appendices of the textbook)

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, 2nd edition,
  1994.

11
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
Skewness

• Positive Skewness Mean Median
• Negative Skewness Median Mean
• Pearsons Coefficient of Skewness3
• 3 (Mean Median)
• Standard deviation

13
Positive Skewness (Tail to Right)
14
Negative Skewness (Tail to Left)
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, 2nd edition, 1994.

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, 2nd edition,
  1994.

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, 2nd edition, 1994.

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, 2nd edition, 1994.

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, 2nd edition, 1994.

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, 2nd edition, 1994.

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, 2nd edition, 1994.

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, 2nd edition, 1994.

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, 2nd edition, 1994.

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, 2nd edition, 1994.

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, 2nd edition, 1994.

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, 2nd edition, 1994.

27
Solution/Answers Contd

• 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.

28
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.

29
References/Further Reading

• 1) Dawson-Saunders, B Trapp, RG. Basic and
• Clinical Biostatistics, 2nd edition, 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.