2005 normal distribution. tripthi m. mathew, md, mph objectives learning objective - to understand...

2005

Normal Distribution

Tripthi M. Mathew, MD, MPH

Objectives

Learning Objective- To understand the topic on Normal Distribution and

its importance in different disciplines.

Performance ObjectivesAt 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

Z indicates how many standard deviations away from the mean the point x lies.

Z score is calculated to 2 decimal places.

σ

Tripthi M. Mathew, MD, MPH

Tables

Areas under the standard normal curve

(See Normal Table)

Tripthi M. Mathew, MD, MPH

13.6%

2.2%

0.15

-3 -2 -1 μ 1 2 3

Diagram of Normal Distribution Curve (z distribution)

33.35%

Modified from Dawson-Saunders, B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.

Tripthi M. Mathew, MD, MPH

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

13.6%

2.2%

0.15

-3 -2 -1 μ 1 2 3

Diagram of Exercise # 1

0.159

33.35%

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 # 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

13.6%

2.2%

0.15

-3 -2 -1 μ 1 2 3

Diagram of Exercise # 2

0.023

33.35%

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 # 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

13.6%

2.2%

0.15

-3 -2 -1 μ 1 2 3

Diagram of Exercise # 3

0.954

33.35%

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 # 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

13.6%

2.2%

0.15

-3 -2 -1 μ 1 2 3

Diagram of Exercise # 4

0.015

33.35%

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 # 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

13.6%

2.2%

0.15

-3 -2 -1 μ 1 2 3

Diagram of Exercise # 5

0.0150.015

33.35%

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

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.