Finding the Mean
and Variance of the
Sampling
Distribution of the
OBJECTI
At the end of the lesson the students should be able
to:
VES
1. identify the sampling distribution of sample
means,
2. calculate the mean, variance and constructing
the sampling distribution of the sample mean and
3. participate actively in class and group activity.
= =
Where:
= population variance
= population standard deviation
N = the size of the population
= each value from the population
= population mean.
Solutions:
Example: A population
consists of the numbers Step 1: Compute the population
2,4,9,10 and 5. Let us list all mean.
possible sample size variance
of the sampling distribution
of the sample means.
Step 2: Compute the population
variance
X X-
2 -4 16
4 -2 4
= 9 3 9
= 10 4 16
5 5 1
= 9.2 (the variance of the population)
Step 3: Determine the number of
Step 5: Construct the sampling
N=5 n=3
possible samples.
distribution of the sample means.
there are 10 possible samples can be drawn SAMPLE FREQUENCY PROBABILITY
Step 4: List all possible samples and MEAN (x)
their corresponding means.
3.67 1 1/10 = 0.10
SAMPLE MEAN 5.00 1 1/10 = 0.10
2,4,9 5.00
5.33 2 2/10 = 0.20
2,4,10 5.33
5.67 1 1/10 = 0.10
2,4,5 3.67
6.00 1 1/10 = 0.10
2,9,10 7.00
6.33 1 1/10 = 0.10
2,9,5 5.33
7.00 1 1/10 = 0.10
2,10,5 5.67
7.67 1 1/10 = 0.10
4,9,10 7.67
8.00 1 1/10 = 0.10
4,9,5 6.00
TOTAL 10 1
4,10,5 6.33
9,10,5 8
Step 6: Compute the mean of the sampling distribution of the sample
means
SAMPLE PROBABILITY
MEAN (x)
3.67 1/10 = 0.10 0.367
5.00 1/10 = 0.10 0.5
5.33 2/10 = 0.20 1.066
5.67 1/10 = 0.10 0.567
6.00 1/10 = 0.10 0.6
6.33 1/10 = 0.10 0.633
7.00 1/10 = 0.10 0.7
7.67 1/10 = 0.10 0.767
8.00 1/10 = 0.10 0.8
TOTAL 1 6
Step 7: Compute the variance and standard deviation of the sampling
distribution of the sample means.
- =
3.67 0.10 -2.33 5.43 0.543 = 1.53
so, the variance of the
5.00 0.10 -1 1 0.1
sampling distribution of
5.33 0.20 -0.67 0.45 0.09
5.67 0.10 -0.33 0.11 0.011 the sample mean is 1.53.
=
6.00 0.10 0 0 0
= 1.24
6.33 0.10 0.33 0.11 0.011
7.00 0.10 1 1 0.1
7.67 0.10 1.67 2.79 0.279
8.00 0.10 2 4 0.4
TOTAL 1 1.533
Step 8. Construct a histogram of the sampling distribution of the means.
2/10
1/10
3.67 5 5.5 5.33 5.67 6 6.33 7 7.67 8
Example 2: The following table Solutions:
gives tutorial rate of six teachers Step 1: Compute the population
in Central Luzon per month. mean.
Suppose that random samples of
size 4 are taken from this
population of six teachers, do
the following task.
X-
Step 2: Compute the population
Teache Tutorial Rate (in
variance
X
r thousand pesos) X
=
8 -10 100
A 8 12 -6 36
= 46.67
=
B 12
16 -2 4
C 16
D 20 20 2 4
100
E 24 24 6 36
F 28
28 10
Example 2: The following table Solutions:
gives tutorial rate of six teachers Step 1: Compute the population
in Central Luzon per month. mean.
Suppose that random samples of
size 4 are taken from this
population of six teachers, do
the following task.
X-
Step 2: Compute the population
variance
X
Step 3: Determine the 2 -10 100
N=6 n=3
number of possible samples. 4 -6 36
9 2 4
there are 15 possible samples 10 2 36
can be drawn 5 6 100
Step 4: SAMPLE MEAN
8,12,16,20 14 Step 5: Construct the sampling
8,12,16,24 15 distribution of the sample means.
8,12,16,28 16 SAMPLE FREQUENCY PROBABILITY
8,12,20,24 16 MEAN (x)
8,12,20,28 17 14 1 1/15 = 0.07
8,12,24,28 18 15 1 1/15 = 0.07
8,16,20,24 17 16 2 2/15= 0.07
8,16,20,28 18 17 2 2/15 = 0.13
8,16,24,28 19
18 3 3/15 = 0.2
8,20,24,28 20
19 2 2/15 = 0.13
12,16,20,24 18
20 2 2/10 = 0.13
12,16,20,28 19
12,16,24,28 20 21 1 1/15 = 0.07
12,20,24,28 21 22 1 1/15 = 0.07
16,20,24,28 22 TOTAL 15 1
Step 7: Compute the variance and standard deviation of the sampling
-
distribution of the sample means.
= 4.8
14 0.07 -4 16 1.12
15 0.07 -3 9 0.63
16 0.13 -2 4 0.52 so, the variance of the
17 0.13 -1 1 0.13 sampling distribution of
18 0.2 0 0 0 the sample mean is 4.67.
=
19 0.13 1 1 0.13
= 2.19
20 0.13 2 4 0.52
21 0.07 3 9 0.63
22 0.07 4 16 1.12
TOTAL 1 4.8
In - Class Activity 1 In - Class Activity 2
Population Sampling Population Sampling
(N=5) Distribution (N =5) Distribution
of the of the
Sample Sample
Means (n=2) Means (n=3)
Mean 3.00 3.00 3.00 3.00
Variance 2.00 0.75 2.00 0.33
Standard 1.41 0.87 1.41 0.57
Properties of the Sampling
Distributions of the Means
1. The mean of the population μ is also called the mean
of the sample
= ∑[∙P(X)]
2. The variance of the sampling distribution of the
sample means σ is given by
=
Alteranative:
= For finite population without replacement.
For infinite population with replacement.
3. The standard deviation of the sampling distribution of the sample
mean is given by:
= for finite where is the finite population correction factor.
for infinite population.
GROUP
Instructions:
ACTIVITY
1. Divide the class into 4 groups.
2. Each group must choose a leader (facilitates the activity), a
secretary (records and writes the output), and a presenter (shares
the results).
3. Solve the assigned problem as a group.
4. You have 10 minutes to complete the activity:
• 1 minute: Gather materials
• 6 minutes: Discuss and solve the problem
• 2 minutes: Present the output
5. Active participation from all members is required
Direction: On your answer sheets, copy and complete the table below by
indicating the desired data and values, simple interpretations, and
implications related to the means and variance of the sampling distribution
of the sample mean.
G-1.Survey G-2. G-3. Average G-4. Amount
on Study Estimating on score in a spent on
Hours average of math class school snacks
N=5 height of N=5 per day.
n=2 students in a n=3 N=6
school. n=2
N=4
n=2
Illustrative Example
Population Mean
Population Variance
Standard deviation
Variance of the
sampling
distribution of the
sample means
Real-life
Rubric for the Group Activity
Criteria Excellent (4 points) Good (3 points) Satisfactory (2 points) Needs Improvement (1
point)
Completion of Table All sections completed Most sections are correct Some sections incomplete Few sections correct; many
accurately with clear mean, with minor data or or with multiple errors. incomplete answers.
variance, and standard calculation errors.
deviation.
Interpretation of Results Insightful interpretations Interpretations are clear Unclear interpretations No interpretations
clearly connect statistics to but could be more with weak real-life links. provided or interpretations
real life. detailed. Real-life links are are confusing and
partially explained. irrelevant.
Accuracy of Calculations Accurate calculations with Mostly correct calculations Multiple errors in Most or all calculations are
proper units and with minor, non-critical calculations that affect incorrect.
presentation. errors. understanding.
Real-Life Implications Thoughtful implications Meaningful implications Implications are unclear No real-life implications
reflect real-world but lack specificity. with limited real-world provided or entirely
understanding. insight. irrelevant.
Timelines Done within the given time. 1 minute elapsed time. 2 minute elapsed time. 3 minute elapsed time.
Participations/ Consistently collaborates, Works well with others; Works with others but Rarely cooperates; often
cooperation listens actively, and listens and helps struggles to collaborate. disruption.
supports team success. occasionally.
THANK
