Distribution of the Sample Means

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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.

Sampling Error – 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, – the distribution of all possible sample means for a variable x, and for a given sample size.

x

Properties of the Distribution of Sample Means

1. The mean of the sample means will be the same as the population mean.

2. 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.

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:

The standard deviation of the population is:

54

8462

236.24

)58()54()56()52( 2222

Distribution of Quiz Scores

2 4 6 8

0.0

0.5

1.0

Score

Fre

qu

ency

All Possible Samples of Size 2 Taken With Replacement

SAMPLE MEAN SAMPLE MEAN

2,2 2 6,2 4

2,4 3 6,4 5

2,6 4 6,6 6

2,8 5 6,8 7

4,2 3 8,2 5

4,4 4 8,4 6

4,6 5 8,6 7

4,8 6 8,8 8

Frequency Distribution of the Sample Means

MEAN f 2 1

3 2 4 3 5 4 6 3 7 2 8 1

Distribution of the Sample Means

2 3 4 5 6 7 8

0

1

2

3

4

Sample Mean

Fre

qu

ency

The Mean of the Sample Means

Denoted In our example:

So, , which in this case = 5

x

516

8765765465435432

x

x

x

nx

x

The Standard Deviation of the Sample Means

Denoted In our example:

Which is the same as the population standard deviation divided by

x

581.116

)58(...)55()54()53()52( 22222

x

2

581.12

236.2

nx

nx

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

Theorem1. When the original variable is normally

distributed, the distribution of the sample means will be normally distributed, for any sample size n.

2. 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.