Normal distribution.pdf- Normal distribution.pdf- Normal distribution.pdf

PUBH 601 Concepts and Methods of Biostatistics
4-Normal distribution
Manar Elhassan, PhD
Mohamed Sherbash, MPH
Department of Public Health
College of Health Sciences Qatar
University
Fall 2022
PUBH 601 Concepts and Methods of Biostatistics. Fall 2022 1

Recap from last week
Graphical Summaries for
1. Nominal data
a) Bar Chart
b) Pie Chart
2. Ordinal Data
a) Frequency distribution
b) Stem and Leaf Plot
3. Continuousdata
a) Histogram
b) Frequency polygon
c) Box and whisker Plot(Box plot)
d) Scatter plot

Objectives of today’s class
• Importance of the Normal distribution
• Properties of the Normal distribution
• Standard normal curve
• Assessing normality
Normal distribution

Student Learning Outcomes of the Course
At the end of this class, students will be able to:
• Discuss the importance of the normal distribution
• Define the properties of the normal curve
• Describe the concept of the standard normal curve
• Compute and interpret z scores
• Assess normality
• Perform analysis using Stata

Introduction
Department of PublicHealth

Introduction
PUBH 601 Concepts and Methods of Biostatistics. Fall 2022
Department of Public Health 6
Body weights among adults in a population
Bell-shaped
A single peak in the center
Symmetrical, Median= Mode

Introduction
Bell-shaped
Many quantitative
variables in medicine and
public health sciences have
similar characteristics

Introduction
Bell-shaped
Many quantitative
variables in medicine and
public health sciences have
similar characteristics
These are characteristic of so-
called Normal or Gaussian
distributions

Approximately Normal Distributed
a sympthatic
data go to infiniy

Importance of the
Normal distribution

Importance of the Normal distribution
• The Normal distribution occupies a central role in statistical
analysis
• Many inference ideas are based on the Normal distribution
• Data which are not normal can often be transformed into
normal
• The normal distribution is the most important and most widely
used distribution in statistics

Properties of the
Normal distribution

Properties
• Symmetric around their mean
• Mean, median, and mode are equal
• Normal distributions are denser in
the center and less dense in the
tails
• Area under the normal curve is
equal to 1.0

Properties
Normal distributions are defined by two parameters:
• mean (µ) and the standard deviation (σ)
mean = population

Properties
Same mean - Different variance

Properties
Same variance - Different mean

The beauty of the normal distribution
No matter what the mean µ and standard deviation σ are,
• 68% of the area of a normal distribution is within
one standard deviation of the mean
• 95% of the area of a normal distribution is within
two standard deviations of the mean
• 99.7% (almost all values) fall within
3 standard deviations of the mean

68-95-99.7 Rule

Standard normal curve

The standard normal curve has
mean zero and standard deviation 1

• If a variable is normally distributed then a change of
units does not affect this
• Any normally distributed variable can be related to
the standard normal distribution whose mean is
zero and whose standard deviation is 1

z-scores

z-score is the number of standard deviations from the mean
𝒛 =
𝒙−𝝁
𝝈
Percentiles of a normal distribution is
x = µ + zσ
z-scores

Activity 4-1: Learn through interactive applet
Learning outcome:
• Relate z-scores to x values (percentiles)
• Understand area under the normal curve
http://www.rossmanchance.com/applets/NormCalc.html
Generatingthe normal distribution and exploring
probabilities

Activity 4-1: Learn through interactive applet
Results from a study collected on weights of 800 students found
the mean weight was 50 ± 5 kg.
Questions:
• What is the probability that a student has weight greater than 60kg?
• What is the probability that a student has weight less than 40kg?
• What is the probability that a student has weight less than 55kg and
greater that 45?
• What is the z score for a weightof 55?
• What weightis exceededby 2.5% or 0.025 of the students?
• What weightis exceededby 97.5% or 0.975 of the students?

Example

Example
What weight is exceeded by 2.5% or 0.025
of the population, if weight of runners has
µ= 127.8 cm and σ= 15.5 cm?
x = µ + zσ
Weight = 127.8+15.5*z = 127.8+15.5*1.96
= 158.2
158.2 is the 97.5th percentile

Example
What weight is exceeded by 97.5% or 0.975
of the population, if weight of runners has
µ = 127.8 cm and σ = 15.5 cm?
x = µ + zσ
Weight=127.8-15.5*z == 127.8+ (-.96)*15.5
= 97.42
97.4 is the 2.5th percentile

z values for commonly used percentiles

Assessing normality

(1) Draw a histogram
(2) Draw a box-plot
(3) Normal Q-Q plot
Assessing normality
Subjectively Objectively
(4) Kolmogrov-Smirnov test
(5) Shapiro-Wilk test
(6) Descriptive methods

(1) Draw a histogram
If the distribution is bell-shaped and symmetrical, then it is
approximately Normal.
Normal Not Normal

(2) Draw a box-plot
If the median in a boxplot cuts the rectangle in approximately
half, and the two whiskers are of equal length, then we can
assume that the distribution is Normal.
Normal Not Normal
Department of Public Health PUBH 601 Concepts and Methods of Biostatistics. Fall 2022 41

(3) Normal Q-Q plot
• Produce a Normal plot (using Stata) which plots the cumulative
frequency distribution of the data (on the horizontal axis) against
that of the Normal distribution.
• If the resulting plot is a straight line, the distribution isNormal.
• If the plot is curved, the distribution is Not Normal.

(3) Normal Q-Q plot
Normal plot Not normal
Straight line
Curved
Height in centimeters Weight in kilograms

(3) Normal Q-Q plot
Negative skewness shows
downward curve on Q-Q plot
Positive skewness shows
upward curve on Q-Q plot

.,.
.,.
Kolmogorov-Smirnov Shapiro-Wilks
(4) Kolmogorov-Smirnov (5) Shapiro-Wilks
Statistic df p-value Statistic df p-value
Weight in Kg 0.104 2400 < 0.001 0.912 2400 < 0.001
Height in cm 0.018 2400 0.69 0.999 2400 0.62

.,.
.,.
Weight in Kg 0.104 2400 < 0.001 0.912 2400 < 0.001
Height in cm 0.018 2400 0.69 0.999 2400 0.62
Use p-value to see if variable
significantly depart from Normality
Use a cut-off value of p-value <0.05

.,.
.,.
p-value ≤ 0.05 ⇒ Not Normal (Weight)
Weight in Kg 0.104 2400 < 0.001 0.912 2400 < 0.001
Height in cm 0.018 2400 0.69 0.999 2400 0.62
Not normal

.,.
.,.
p-value > 0.05 ⇒ Normal (Height)
Weight in Kg 0.104 2400 < 0.001 0.912 2400 < 0.001
Height in cm 0.018 2400 0.69 0.999 2400 0.62
Normal

(6) Descriptive methods
Use the relationship between mean, median and SD
Normal variable
• Mean nearly equal to the median and mode
• Standard deviation < 1/3 of the mean

Example Height in cm
• Mean, median mode are almost equal
• SD (=11.3) < 1/3 of mean (=45.7)
Normal variable

Example Skinfold thickness
• Mean > Median > Mode
• SD (=15.6) > 1/3 of mean (= 8.2)
Not normal variable

Tools to assess the normality of a variable

Note
Normality tests are too sensitive to
even small deviation from normality so
don’t always rely on them given that
most bivariate tests are robust to small
deviations

Activity 4-2
Assessment using Team based learning

Activity 4-3
Data analysis using Stata

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