This document discusses sampling distributions and their importance in inferential statistics. It provides examples of constructing sampling distributions of sample means both with and without replacement from a population. Key steps include determining possible sample sizes, listing samples and computing their means, constructing the sampling distribution as a frequency distribution of sample means, and displaying it as a histogram. Sampling distributions are important as they allow making inferences about large populations based on analyzing sample data.

1. SAMPLING DISTRIBUTION
ED 202 – INFERENTIAL STATISTICS
Mark John P. Munda
Mamlad High School
Rommel C. Velasco
Zeferino Arroyo High School
REPORTERS :

2. Opening Prayer
ALMIGHTY, OUR FATHER, WE PRAISE AND THANK YOU FOR THIS DAY. WE COME
TO YOU THIS HOUR ASKING FOR YOUR BLESSING AND HELP US AS WE ARE
VIRTUALLY GATHERED TOGETHER. WE PRAY FOR GUIDANCE IN THE MATTERS AT
HAND AND ASK THAT YOU WOULD CLEARLY SHOW US HOW TO CONDUCT OUR
VIRTUAL CLASS WITH A SPIRIT OF JOY AND ENTHUSIASM. GIVE US THE
DESIRE TO FIND WAYS AND EXCEL IN OUR CLASS. HELP US TO WORK TOGETHER
AND ENCOURAGE EACH OTHER TO EXCELLENCE. WE ASK THAT WE WOULD
CHALLENGE EACH OTHER AND GAIN MORE KNOWLEDGE TO REACH HIGHER AND
FARTHER TO BE THE BEST WE CAN BE. WE ASK THIS IN THE NAME OF THE LORD
JESUS CHRIST, AMEN.

3. SAMPLING
DISTRIBUTION

4. What is sampling distribution of sample means?
• 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 means of the samples are less than or greater than the mean of the
population.
• Using a sampling distribution simplifies the process of making inferences about large
amounts of data. For this reason, it is used often as a statistical resource in data
science.

5. Sampling Distribution
(without and with replacement)
• In sampling without replacement, each sample unit of the population has only one
chance to be selected in the sample. For example, if one draws a simple random
sample such that no unit occurs more than one time in the sample, the sample is
drawn without replacement.
• If a unit can occur one or more times in the sample, then the sample is drawn with
replacement.

6. Population Size and Sample Size
• A population is the entire group that you want
to draw conclusions about.
• A sample is the specific group that you will
collect data from. The size of the sample is
always less than the total size of the
population.
• In research, a population doesn’t always refer
to people. It can mean a group containing
elements of anything you want to study, such
as objects, events, organizations, countries,
species, organisms, etc.

7. Steps in Constructing the Sampling Distribution of the Means
1. Determine the number of sets of all possible random samples that can be drawn
from the given population.
2.List all the possible samples and compute the mean of each sample.
3.Construct the sampling distribution.
4.Construct a histogram of the sampling distribution of the means.

8. SAMPLING DISTRIBUTION OF
THE MEANS
WITHOUT REPLACEMENT

9. = 1 +
𝑛𝑥
1!
+
𝑛 𝑛 − 1 𝑥2
2!
+ ⋯
Example: A population consists of the numbers 2, 4, 9, 10, and 5. Let us list all possible sample size
of 3 from this population and compute the mean of each sample.
1. Determine the number of sets of all possible random samples that can be drawn from the
given population by using the formula, nCr, where n is the population size and r is the sample
size.
nCr =
n!
r!(n-r)!
n = 5 r = 3
There are 10 possible samples of size 3 that can be drawn from the given data.
nCr =
n!
r!(n-r)! =
5!
3!(5-3)! =
5 · 4 · 3!
3!2! =
5 · 4
2 · 1 =
20
2 = 10

10. = 1 +
𝑛𝑥
1!
+
𝑛 𝑛 − 1 𝑥2
2!
+ ⋯
Example: A population consists of the numbers 2, 4, 9, 10, and 5. Let us list all possible sample size
of 3 from this population and compute the mean of each sample.
2. List all the possible samples and compute the mean of each sample.
2 4 9 10 5
nCr = 10
X = 2 + 4 + 9
3
_
= 5
Sample Mean
2,4,9 5
2,4,10 5.33
2,4,5 3.67
2,9,10 7
2,9,5 5.33
2,10,5 5.67
4,9,10 7.67
4,9,5 6
4,10,5 6.33
9,10,5 8

11. = 1 +
𝑛𝑥
1!
+
𝑛 𝑛 − 1 𝑥2
2!
+ ⋯
Example: A population consists of the numbers 2, 4, 9, 10, and 5. Let us list all possible sample size
of 3 from this population and compute the mean of each sample.
3. Construct the sampling distribution.
Sample Mean
2,4,9 5
2,4,10 5.33
2,4,5 3.67
2,9,10 7
2,9,5 5.33
2,10,5 5.67
4,9,10 7.67
4,9,5 6
4,10,5 6.33
9,10,5 8
Sample Mean (x) Frequency Probability
3.67 1 1/10 = 0.10
5 1 1/10 = 0.10
5.33 2 2/10 = 0.20
5.67 1 1/10 = 0.10
6 1 1/10 = 0.10
6.33 1 1/10 = 0.10
7 1 1/10 = 0.10
7.67 1 1/10 = 0.10
8 1 1/10 = 0.10
Total 10 1.00

12. = 1 +
𝑛𝑥
1!
+
𝑛 𝑛 − 1 𝑥2
2!
+ ⋯
Example: A population consists of the numbers 2, 4, 9, 10, and 5. Let us list all possible sample size
of 3 from this population and compute the mean of each sample.
4. Construct a histogram of the sampling distribution of
the means.
Sample
Mean (x)
Frequency P(x)
3.67 1 1/10 = 0.10
5 1 1/10 = 0.10
5.33 2 2/10 = 0.20
5.67 1 1/10 = 0.10
6 1 1/10 = 0.10
6.33 1 1/10 = 0.10
7 1 1/10 = 0.10
7.67 1 1/10 = 0.10
8 1 1/10 = 0.10
Total 10 1.00
0.40
0.30
0.20
0.10
0
3.67 5 5.33 5.67 6 6.33 7 7.67 8

13. SAMPLING DISTRIBUTION OF
THE MEANS
WITH REPLACEMENT

14. = 1 +
𝑛𝑥
1!
+
𝑛 𝑛 − 1 𝑥2
2!
+ ⋯
Example: A population consists of three pool balls with the numbers 1, 2, and 3. Two of the balls
are selected randomly with replacement. Let us list all possible sample size of 2 from this
population and compute the mean of each sample.
1. Determine the number of sets of all possible random samples that can be drawn from the
given population by using the formula, P (n, r) = n , where n is the population size and r is the
sample size.
P (n, r) = n n = 3 r = 2
There are 9 possible samples of size 2 that can be drawn from the given data.
R r
R r
P (n, r) = P (3,2) = 3 = 9
R R 2

15. = 1 +
𝑛𝑥
1!
+
𝑛 𝑛 − 1 𝑥2
2!
+ ⋯
2. List all the possible samples and compute the mean of each sample.
1 2 3
X = 2 + 4 + 9
3
_
= 5
P (n, r) = n = 9
R r
Sample Mean
1,1 1.0
1,2 1.5
1,3 2.0
2,1 1.5
2,2 2.0
2,3 2.5
3,1 2.0
3,2 2.5
3,3 3.0
X = 1 + 1
2
= 1
Example: A population consists of three pool balls with the numbers 1, 2, and 3. Two of the balls are
selected randomly with replacement. Let us list all possible sample size of 2 from this population and
compute the mean of each sample.

16. = 1 +
𝑛𝑥
1!
+
𝑛 𝑛 − 1 𝑥2
2!
+ ⋯
Example: A population consists of three pool balls with the numbers 1, 2, and 3. Two of the balls
are selected randomly with replacement. Let us list all possible sample size of 2 from this
population and compute the mean of each sample.
3. Construct the sampling distribution.
Sample Mean (x) Frequency Probability
1.0 1 1/9 = 0.11
1.5 2 2/9 = 0.22
2.0 3 3/9 = 0.33
2.5 2 2/9 = 0.22
3.0 1 1/9 = 0.11
Total 9 .99 or 1.00
Sample Mean
1,1 1.0
1,2 1.5
1,3 2.0
2,1 1.5
2,2 2.0
2,3 2.5
3,1 2.0
3,2 2.5
3,3 3.0

17. = 1 +
𝑛𝑥
1!
+
𝑛 𝑛 − 1 𝑥2
2!
+ ⋯
Example: A population consists of three pool balls with the numbers 1, 2, and 3. Two of the balls
are selected randomly with replacement. Let us list all possible sample size of 2 from this
population and compute the mean of each sample.
4. Construct a histogram of the sampling distribution of
the means.
Sample
Mean (x)
Frequency P(x)
1.0 1 1/9 = 0.11
1.5 2 2/9 = 0.22
2.0 3 3/9 = 0.33
2.5 2 2/9 = 0.22
3.0 1 1/9 = 0.11
Total 9 0.99 or 1.00
0.40
0.30
0.20
0.10
0 1.0 1.5 2.0 2.5 3.0
0.45
0.35
0.25
0.15
0.05

18. Generalization
Sampling is a statistical procedure that is concerned with the
selection of the individual observation; it helps us to make statistical
inferences about the population. In sampling, we assume that samples are
drawn from the population, and sample means and population means are
equal.
Sampling distributions are important for inferential statistics. In
practice, one will collect sample data and, from these data, estimate
parameters of the population distribution. Thus, knowledge of the sampling
distribution can be very useful in making inferences about the overall
population.

19. “One of the most sincere forms of respect is
actually listening to what another has to say.”
-Bryant McGill