This document discusses sampling distributions and constructing the sampling distribution of the sample mean. It has three main objectives: correctly identifying sampling distributions of statistics like the sample mean, constructing sampling distributions of the sample mean using probability tables, and solving problems involving sampling distributions. It provides examples of constructing sampling distributions of the sample mean and other statistics from populations. Key points are that a sampling distribution is the probability distribution of a statistic from repeated random samples, and there are three ways to determine a sampling distribution: using probability laws, simulation, or statistical theorems. The document includes practice problems for students to construct different sampling distributions.

1. Lesson 2
Sampling
Distributions

2. Objectives
At the end of this lesson, the learner should be able to
● correctly identify sampling distributions of statistics
(sample mean);
● properly construct sampling distribution of the sample
mean using a probability distribution table; and
● accurately solve real-life problems involving sampling
distribution of the sample mean.

3. Essential Questions
● Why is it important to know the total number of samples
in creating a sampling distribution of the sample means?
● What do you think is the shape of the histogram of the
sampling distribution of the sample means?

4. Warm Up!
This lesson will tackle about sampling distribution of a
statistic. Before we start the discussion, let us review some of
the most common statistics that we use. Let us play the
Mean, Median & Mode Game of a collection of buildings.
(Click the link to access the website.)
“Mean, Median & Mode Game.” Kids Math Games. Retrieved
14 July 2019 from
http://www.kidsmathgamesonline.com/numbers/meanmedia
nmode.html

5. Guide Questions
● How do you solve for the mean of a collection of data?
● How do you get the median of a set of data?
● Aside from the mean, median, and mode, what other
statistics can you get from the data in the activity?

6. Learn about It!
Sampling Distribution
a probability distribution of a statistic obtained from all possible samples of a
particular size from a population
1
Example:
Population: {1, 2, 3}
Sample size: 𝑛 = 2
Statistic: sample mean
1

7. Learn about It!
Information From Repeated Random Sampling
1. What values of the statistic that can occur?
2. How often each of these values occurs?
1
Example:
From the previous example,
• The values of the sample mean are {1.5, 2, 2.5}.
• Each of the value of the sample mean occurs once only.
2

8. Learn about It!
Three Ways of Determining the Sampling
Distribution of a Statistic
To determine the sampling distribution of a statistic, we can use (1) laws of
probability; (2) simulation; or (3) statistical theorems.
3
1. Derive the distribution mathematically using the laws of
probability.

9. Learn about It!
Three Ways of Determining the Sampling
Distribution of a Statistic
To determine the sampling distribution of a statistic, we can use (1) laws of
probability; (2) simulation; or (3) statistical theorems.
3
2. Use a simulation to approximate the distribution. Draw a
large number of samples of size 𝑛, calculate the value of
the statistic for each sample, and tabulate the results in a
relative frequency histogram. When the number of
samples is large, the histogram will be very close to the
theoretical sampling distribution.

10. Learn about It!
Three Ways of Determining the Sampling
Distribution of a Statistic
To determine the sampling distribution of a statistic, we can use (1) laws of
probability; (2) simulation; or (3) statistical theorems.
3
3. Use statistical theorems to derive exact or approximate
sampling distribution.

11. Learn about It!
Sampling Distribution of the Sample Mean
the distribution of the infinite number of sample means
1
Example:
The larger the sample size, the closer the sampling
distribution of the mean would be to a normal distribution.
4

12. Try It!
Example 1: A population consists of the numbers 1, 2, 3, 4,
and 5. List all the possible samples of size 2 and construct
the sampling distribution of the sample mean.

13. Try It!
Example 1: A population consists of the numbers 1, 2, 3, 4,
and 5. List all the possible samples of size 2 and construct
the sampling distribution of the sample mean.
Solution:
1. List all the possible samples of size 2
and compute for the sample mean.

14. Try It!
Solution:
2. Construct the sampling distribution of
the sample mean using a probability
distribution table.
Recall that the probability of each sample
mean is computed by dividing its
frequency by the total frequency of all
possible sample means.

15. Try It!
Example 2: Using the same experiment in Example 1,
construct the sampling distribution of the sample range.

16. Try It!
Example 2: Using the same experiment in Example 1,
construct the sampling distribution of the sample range.
Solution:
1. List all the possible samples of size 2
and compute for the sample range.

17. Try It!
Solution:
2. Construct the sampling distribution of the sample range
using a probability distribution table.

18. Let’s Practice!
Individual Practice:
1. Samples of three cards are drawn at random from a
population of six cards numbered from 1 to 6. Construct
the sampling distribution of the sample mean.
2. Let “max” be the highest value in a set. Using the above
experiment, construct the sampling distribution of the
sample max.

19. Let’s Practice!
Group Practice: To be done by 2-5 groups
A box contains six paper bills: ₱20, ₱50, ₱100, ₱200, ₱500,
and ₱1 000. Two paper bills are picked at random as
sample. The amount of the chosen paper bills are checked
and the mean amount is solved. What is the probability the
mean amount of the chosen paper bills is at least ₱500?

20. Key Points
Sampling Distribution
a probability distribution of a statistic obtained from all possible samples of a
particular size from a population
1
1
Information From Repeated Random Sampling
1. What values of the statistic that can occur?
2. How often each of these values occurs?
2

21. Key Points
Three Ways of Determining the Sampling
Distribution of a Statistic
1. Derive the distribution mathematically using the laws of probability.
2. Use a simulation to approximate the distribution.
3. Use statistical theorems to derive exact or approximate sampling distribution.
3
Sampling Distribution of the Sample Mean
the distribution of the infinite number of sample means
4

22. Synthesis
● How do you construct the sampling distribution of the
sample mean?
● Rate your performance in the last exam you had. Is it
below average, average, or above average? Explain.
● What are the characteristics of a random sample?