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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
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.
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