Statistics and Probability
Pauline Misty M. Panganiban
SAMPLING DISTRIBUTION
OF SAMPLE MEANS

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
illustrate random sampling;
distinguish between parameter
and statistic; and
construct sampling distribution of
sample means.
Objectives

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
Constructing Frequency Distribution
The following are the blood types of a group of
individuals in a government office. Construct a
frequency distribution for the different blood types

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
Constructing Frequency Distribution
x f
A 5
B 7
O 9
AB 4

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
•You have learned in your previous
lessons how to construct frequency
distribution and probability
distribution. In this lesson, you will
learn how to construct sampling
distribution of the sample means.

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
Computing for the mean of a Sample
Find the mean of the following sets numbers

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Combination
A combination is a mathematical technique
that determines the number of possible
arrangements in a collection of items where the
order of the selection does not matter. In
combinations, you can select the items in any
order.

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Combination of N objects
Taken r at a time.
The number of samples of size r that can be
drawn from a population of size N is given by NCn.
𝑛𝐶𝑟 =
𝑛!
𝑟!(𝑛−𝑟)!
Where:
n = size of the population
r = size of the sample

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
Evaluating Combination of N objects
Taken n at a time.
1. 4C2
2. 6C4
3. 6C3
4. 10C4
=
𝑁!
𝑛!(𝑁−𝑛)!
=
4!
2!(4−2)!
=
4!
2!(2)!
=
4 3 2!
2!2!
=
12
2
= 𝟔
=
𝑁!
𝑛!(𝑁−𝑛)!
=
6!
4!(6−4)!
=
6!
4!(2)!
=
6(5)(4!)
4!2!
=
30
2
= 𝟏𝟓
=
𝑁!
𝑛!(𝑁−𝑛)!
=
6!
3!(6−3)!
=
6!
3!(3)!
=
6(5)(4)(3!)
3!3!
=
6 5 (4)
3(2)(1)
= 𝟐𝟎
=
𝑁!
𝑛!(𝑁−𝑛)!
=
10!
4!(10−4)!
=
10!
4!(6)!
=
10(9)(8)(7)(6!)
4!6!
=
10(9)(8)(7)
4(3)(2)(1)
= 𝟐𝟏𝟎

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
Evaluate the following.
1. 5𝐶3
2. 8𝐶4
3. 9𝐶6
4. 10𝐶3
5. 12𝐶8

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
•Researchers use sampling if taking
a census of the entire population is
impractical. Data from the sample
are used to calculate statistics,
which are estimates of the
corresponding population
parameters.

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
•Descriptive measures computed
from a population.

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
•Descriptive measures computed
from a sample.

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
Sampling Distribution of Sample Means
• The number of samples of size r that can be
drawn from a population of size N is given by NCr.
• A sampling distribution of sample means is a
frequency distribution using the means computed
from all possible random samples of a specific
size taken from a population.
• The probability distribution of the sample means
is also called the sampling distribution of the
sample means.

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
Steps in Constructing the Sampling
Distribution of the Means
1. Determine the number of possible samples that can be
drawn from the population using the formula: NCr
where N = size of the population r = size of the
sample
2. List all the possible samples and compute the mean of
each sample.
3. Construct a frequency distribution of the sample
means obtained in Step 2.

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
Example
A population consists of the numbers 2, 4,
9, 10, and 5.
a. List all possible samples of size 3 from
this population.
b.Compute the mean of each sample.
c. Prepare a sampling distribution of the
sample means.

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Solution to Example 1a
The possible
samples of size
3 from 2, 4, 9,
10, and 5 are…

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Solution to Example 1b
The mean of each sample are as follows:

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Solution to Example 1c
The sampling distribution of the sample
means

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
Exercise
A group of students got the following scores in
a test: 6, 9, 12, 15, 18, and 21. Consider
samples of size 3 that can be drawn from this
population.
a. List all the possible samples and the
corresponding mean.
b. Construct the sampling distribution of the
sample means.

L i b e r a t i n g N e t w o r k i n E d u c a t i o n
Exercise 1
Samples of three cards are drawn at
random from a population of eight cards
numbered from 1 to 8.
a. How many possible samples can be
drawn?
b.Construct the sampling distribution of
sample means.

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Exercise 3
A finite population consists of 8 elements.
10, 10, 10, 10, 10, 12, 18, 40
a. How many samples of size n = 2 can be drawn
from this population?
b. List all the possible samples and the
corresponding means.
c. Construct the sampling distribution of the
sample means.

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Thank you!