1. FINDING THE MEAN AND VARIANCE OF
THE SAMPLING DISTRIBUTION OF
MEANS
2. Lesson Objectives
At the end of this lesson, you are expected to:
• find the mean and variance of the sampling
distribution of the sample means; and
• state and explain the Central Limit Theorem.
4. Lesson Introduction
Statisticians do not just describe the variation
of the individual data values about the mean of
the population.
They are also interested to know how the
means of the samples of the same size taken
from the same population vary about the
population mean.
6. Discussion Points
The Central Limit Theorem
If random samples of size n are drawn from a
population, then as n becomes larger, the
sampling distribution of the mean approaches
the normal distribution, regardless of the
shape of the population distribution.
7. Example 1
Consider a population consisting of 1, 2, 3, 4,
and 5. Suppose samples of size 2 are drawn from
this population.
Describe the sampling distribution of the sample
means.
8. Solution to Example 1
Step 1. Determine the number of possible
samples of size n = 2.
Use the formula NCn. Here N = 5 and n = 2.
5C2 = 10
So, there are 10 possible samples of size 2 that
can be drawn.
9. Solution to Example 1
Step 2. List all possible samples and their
corresponding means.
10. Solution to Example 1
Step 3.
Construct the
sampling
distribution of the
sample means.
11. Solution to Example 1
Step 4.
Compute the mean of
the sampling
distribution of the
sample means (μX).
Follow these steps:
• Multiply the
sample mean by
the corresponding
probability.
• Add the results.
12. Example 2
Consider a population consisting of 1, 2, 3, 4,
and 5. Suppose samples of size 2 are drawn from
this population.
Compute the variance the sampling distribution
of the sample means.
14. Example 3
A population has a mean of 60 and a standard
deviation of 5. A random sample of 16
measurements is drawn from this population.
Describe the sampling distribution of the
sample means by computing its mean and
standard deviation.
Assume that the population is infinite.
16. Exercise 1
Consider all samples of size 5 from this population:
2, 5, 6, 8, 10, 12, 13
a. Compute the mean (μ) and standard deviation (σ) of the
population.
b. List all samples of size 5 and compute the mean for each
sample.
c. Construct the sampling distribution of the sample means.
d. Calculate the mean of the sampling distribution of the
sample means. Compare this to the mean of the population.
e. Calculate the standard deviation of the sampling distribution
of the sample means. Compare this to the standard deviation
of the population.
17. Exercise 2
The scores of individual students on a national
test have a normal distribution with mean 18.6
and standard deviation 5.9.
At Federico Ramos Rural High School, 76 students
took the test. If the scores at this school have the
same distribution as national scores, what are the
mean and standard deviation of the sample mean
for 76 students?
18. Exercise 3
In 2015, the mean return of all common stocks on
the Philippine Stock Exchange was 3.5%. The
standard deviation of the returns was about 26%.
A student of finance forms all possible portfolios
that invested equal amounts in 5 of these stocks
and records the return for each portfolio. This
return is the average of the returns of the 5 stocks
chosen. What are the mean and standard
deviation of the portfolio returns?
19. Summary
The mean of the sampling distribution of the
sample means is equal to the population mean
μ.
The variance of the sampling distribution of the
sample means σ is given by:
2
x
2
n
g
N n
n 1
for finite population
2
x
2
n
for infinite population
20. Summary
The standard deviation of the sampling
distribution of the sample means σ is given by:
x
2
n
g
N n
n 1
for finite population
x
n
for infinite population
21. Summary
The Central Limit Theorem
If random samples of size n are drawn from a
population, then as n becomes larger, the
sampling distribution of the mean approaches
the normal distribution, regardless of the
shape of the population distribution.
