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Normal Distribution
Tripthi M. Mathew, MD, MPH, MBA
2005
Updated 10/19/09
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, MBA
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, MBA
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.
* Used by Permission of Oxford University Press
Tripthi M. Mathew, MD, MPH, MBA
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, MBA
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, MBA
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, MBA
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, MBA
Tables
 Areas under the standard normal curve
(Appendices of the textbook)
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw -Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
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, MBA
Skewness
 Positive Skewness:
Mean ≥ Median
 Negative Skewness:
Median ≥ Mean
 Pearson’s Coefficient of Skewness3:
= 3 (Mean –Median)
Standard deviation
Tripthi M. Mathew, MD, MPH, MBA
Positive Skewness (Tail to Right)
Tripthi M. Mathew, MD, MPH, MBA
Negative Skewness (Tail to Left)
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw
- Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw -Hill Companies
Tripthi M. Mathew, MD, MPH, MBA
Diagram of Exercise # 1
33.35%
13.6%
2.2%
0.15
0.159
-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. ©McGraw -Hill
Companies
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw
- Hill Companies
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw
- Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw
- Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw
- Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw
- Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw
-Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw
-Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
Diagram of Exercise # 5
33.35%
13.6%
2.2%
0.15
0.015
0.015
-3
-2
-1
μ
1
2
3
The exercises are modified from examples in Dawson-Saunders, B &
Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994. ©McGraw
-Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
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. ©McGraw
- Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
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.©McGraw
- Hill Companies.
Tripthi M. Mathew, MD, MPH, MBA
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, MBA
References/Further Reading
1) Dawson-Saunders, B & Trapp, RG. Basic and
Clinical Biostatistics, 2nd edition, 1994. McGraw -Hill
Companies.
2) Last, J. A Dictionary of Epidemiology. 3rd edition,1995.
Oxford University Press.
3) Wisniewski, M. Quantitative Methods For
Decision Makers, 3rd edition, 2002. Pearson Education.
4) Pidd, M. Tools For Thinking. Modelling in Management
Science. 2nd edition, 2003. John Wiley & Sons Ltd.
Tripthi M. Mathew, MD, MPH, MBA