Sampling Distribution
Tripthi M. Mathew, MD, MPH
Objectives
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Learning Objective
To understand the topic on Sampling
Distribution and its importance in different
disciplines.
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Performance Objectives
At the end of this lecture the student will be able to:
 Apply the basic knowledge of sampling
distribution to solve problems.
 Interpret the results of the problems.
Types of Distribution
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Frequency Distribution
Normal (Gaussian) Distribution
Probability Distribution
Poisson Distribution
Binomial Distribution
Sampling Distribution
t distribution
F distribution
What is Sampling Distribution?
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Sampling is defined as the process of
selecting a number of observations (subjects)
from all the observations (subjects) from a
particular group or population.
Sampling distribution is defined as the
frequency distribution of the statistic for
many samples.
It is the distribution of means and is also
called the sampling distribution of the mean.
Features of Sampling Distribution
The 4 features of sampling distribution include:
1) The statistic of interest (Proportion, SD, or
Mean)
2) Random selection of sample
3) Size of the random sample (very important)
4) The characteristics of the population being
sampled.
Characteristics of Sampling Distribution
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Central Limit Theorem
When random samples of size is taken
from a population, the distribution of
sample means will approach the normal
distribution.
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When the Sampling distribution of the
mean has sample sizes of 30 or more then
it is said to be normally distributed.
Statistical Characteristics of Sampling
Distribution
The major statistics are:
Mean
Standard deviation
Standard error
The standard error (SE or SEM) of the sampling
distribution is given by the formula:
s
√n
Where, n - sample size
s- standard deviation of the sample
x – sample mean
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Statistical Characteristics of Sampling
Distribution Cont’d
a) SE of a proportion = √ p (1-p)/n
Where, p is the sample proportion
b) SE of a percentage =√ p (100-p)/n
Where, p is the sample percentage
Statistical Characteristics of Sampling
Distribution Cont’d
Confidence Interval
a) CI = p ± z α/2 √ p (1-p)/n
b) CI= p ± z α/2 √ p (100-p)/n
Statistical Characteristics of Sampling
Distribution Cont’d
Z Score (Standard Score)
Z=
x- μ
σ /√n
Where, X is the sample mean
μ is the mean of the sampling distribution
σ is the SE of the sampling distribution
√n
Exercises
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An Epidemiologist studied a randomly
selected group of 25 individuals (men
and women) between 30- 49 years of age
and finds that their mean heart rate is 70
beats per minute.
Exercises are modified from examples in Dawson-Saunders,
B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Exercise # 1
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How frequently will the sample of 25
individuals have a mean heart rate of 74
beats per minute or higher?
or in other words
What proportion of samples will have mean
values of 74 beats per minute or greater, if
repeated samples of 25 individuals are
randomly selected from the population?
Exercises are modified from examples in Dawson-Saunders,
B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Exercise # 2
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Further investigation revealed that the 25
individuals appeared to have used a drug for
treatment and now the epidemiologist (Epi)
wants to detect the adverse effects of the drug
on the heart rate. The Epidemiologist assumes
that a mean heart rate in the upper 5% of the
distribution will be cause for concern.
Determine the value that divides the upper 5%
from the lower 95% of the sampling distribution.
Exercises are modified from examples in Dawson-Saunders,
B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
The Use of Normal Curve to solve problems
95%
5%
73.29
μ
1 2
Exercises are modified from examples in Dawson-Saunders,
B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Exercise # 3
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The “disease detective” (Epi) wants to know how
many patients should be included in the study to
determine the drug’s effect. The Epi assumes that the
mean heart rate must not rise above 72 beats per
minute, 90% of the time.
or in other words
To include individuals in the study, what should the
random sample size be so that 90% of the mean
samples of this size will be 72 beats per minute or
less?
Exercises are modified from examples in Dawson-Saunders,
B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Solution/Answers
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1) 2.3%
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2) 73.29
Exercises are modified from examples in Dawson-Saunders,
B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Solution/Answers
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3) 40.96
Exercises are modified from examples in Dawson-Saunders,
B & Trapp, RG. Basic and Clinical Biostatistics, 2nd edition, 1994.
Other Types of Sampling Distribution
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F distribution
This is a sampling distribution of the
mean with an estimated standard
deviation.
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t Distribution
This is the sampling distribution of two
variances (squared standard deviations).
Application of Sampling distribution
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The sampling distribution like the normal
distribution is a descriptive model, so it is
used to describe real world situations.
It is very useful to make statements about
the probability of specific observations
occurring.
Investigators/researchers/modelers use it
for estimation and hypothesis testing.
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.