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## Normal Distribution

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### Diagram of Normal Distribution Curve (z distribution) 33.35 ... Assuming the normal heart rate (H.R) in normal healthy individuals is normally ... – PowerPoint PPT presentation

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Title: 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
• 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
• 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