Download Distribution of the Sample Means

Distribution of the
Sample Means
Topics:
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Essentials
Distributions
Sampling Error
Distribution of the Sample Means
Properties of the Distribution:
• 1) Mean 
• 2) Std. Dev. 
• 3) Central Limit Theorem
A large sample in statistics
Example: Calculating  and 
Additional Topic
x
x
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Essentials: Distribution of Sample Means
(A distribution unlike others)

Be able to explain what the Distribution of Sample Means represents.

Know the three characteristics of this distribution.

Be able to use a set of data demonstrate the calculation of the mean and
standard deviation of this distribution.

What is a statistically large sample?
Some Common Distribution Shapes
Distribution of the Sample Means

Sampling Error:
– the difference between the sample
measure and the population measure due
to the fact that a sample is not a perfect
representation of the population.
– the error resulting from using a sample to
estimate a population characteristic.
Distribution of the Sample Means
Distribution of the Sample Means – is a
distribution obtained by using the means
computed from random samples of a specific
size taken from a population.
 Distribution of the Sample Mean, x – the
distribution of all possible sample means for a
variable x, and for a given sample size.

Properties of the Distribution of
Sample Means
1.
2.
The mean of the sample means will be
the same as the population mean.
The standard deviation of the sample
means will be smaller than the
standard deviation of the population,
and it will be equal to the population
standard deviation divided by the
square root of the sample size.
A Third Property of the
Distribution of Sample Means

A third property of the distribution of the
sample means concerns the shape of
the distribution, and is explained by the
Central Limit Theorem.
The Central Limit Theorem

As the sample size n increases, the
shape of the distribution of the sample
means taken from a population with
mean  and standard deviation 
will approach a normal distribution. This
distribution will have mean  and
standard deviation  n
Two Important Things to Remember
When Using The Central Limit
Theorem
1.
2.
When the original variable is normally
distributed, the distribution of the sample
means will be normally distributed, for any
sample size n.
When the distribution of the original variable
departs from normality, a sample size of 30
or more is needed to use the normal
distribution to approximate the distribution
of the sample means. The larger the
sample, the better the approximation will be.
An Example
Suppose I give an 8-point quiz to a
small class of four students. The results
of the quiz were 2, 6, 4, and 8.
 We will assume that the four students
constitute the population.

The Mean and Standard Deviation of
the Population (the four scores)

The mean of the population is:
2648

5
4

The standard deviation of the population
is:
(2  5)2  (6  5)2  (4  5)2  (8  5)2

 2.236
4
Distribution of Quiz Scores
Frequency
1.0
0.5
0.0
2
4
6
Score
8
All Possible Samples of Size 2
Taken With Replacement
SAMPLE
MEAN
2,2
2,4
2,6
2,8
4,2
4,4
4,6
4,8
2
3
4
5
3
4
5
6
SAMPLE
6,2
6,4
6,6
6,8
8,2
8,4
8,6
8,8
MEAN
4
5
6
7
5
6
7
8
Frequency Distribution of the
Sample Means
MEAN
2
3
4
5
6
7
8
f
1
2
3
4
3
2
1
Distribution of the Sample Means
4
Frequency
3
2
1
0
2
3
4
5
6
Sample Mean
7
8
The Mean of the Sample Means
Denoted  x
 In our example:

23 453 45 6 45 6 7 5 6 7 8
x 
5
16

So,
x   ,
which in this case = 5
The Standard Deviation of the
Sample Means
Denoted  x
 In our example:

(2  5)2  (3  5)2  (4  5)2  (5  5)2  ...  (8  5)2
x 
 1.581
16
Which is the same as the population
standard deviation divided by 2

2.236
 1.581
2
Additional Topic:
Demonstration showing increasing sample size
yielding better estimations of the population value.
Heights of Five Starting Players on a
Men’s Basketball Team (inches)
Possible Samples of Size n = 2 From a
Population of Size N = 5
Possible Samples of Size n = 4
From a Population of Size N = 5
Dot plots of the Sampling Distributions
for Various Sample Sizes (N = 5)