Normal Distribution
Tripthi M. Mathew, MD, MPH
2005
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
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
σ
 Z indicates how many standard
deviations away from the mean the point
x lies.
 Z score is calculated to 2 decimal
places.
Tripthi M. Mathew, MD, MPH
Tables
 Areas under the standard normal curve
(Appendices of the textbook)
Tripthi M. Mathew, MD, MPH
Diagram of Normal Distribution Curve
(z distribution)
33.35%
13.6%
2.2%
0.15
-3
-2
-1
μ
1
2
Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical
Biostatistics, 2nd edition, 1994.
Tripthi M. Mathew, MD, MPH
3
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
Diagram of Exercise # 1
33.35%
13.6%
2.2%
0.15
0.159
-3
-2
-1
μ
1
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Tripthi M. Mathew, MD, MPH
2
3
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
Diagram of Exercise # 2
33.35%
13.6%
2.2%
0.15
0.023
-3
-2
-1
μ
1
2
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Tripthi M. Mathew, MD, MPH
3
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
Diagram of Exercise # 3
33.35%
13.6%
2.2%
0.954
0.15
-3
-2
-1
μ
1
2
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Tripthi M. Mathew, MD, MPH
3
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
Diagram of Exercise # 4
33.35%
13.6%
2.2%
0.15
0.015
-3
-2
-1
μ
1
2
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Tripthi M. Mathew, MD, MPH
3
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
Diagram of Exercise # 5
33.35%
13.6%
2.2%
0.15
0.015
0.015
-3
-2
-1
μ
1
2
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Tripthi M. Mathew, MD, MPH
3
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.
Tripthi M. Mathew, MD, MPH
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.
Tripthi M. Mathew, MD, MPH