Chapter 7 * Sampling Distributions

 

The Normal Distribution

Scores (x) of many different variables approximate a normal distribution

 

If we know the mean and standard deviation of a distribution of scores we can use z-scores to look up the probability associated with observing any particular score.

 

For example:

IQ scores are normally distributed with a mean = 100 and s = 16. If we randomly sample from the population what is the probability that has an IQ of 124 or greater?

 

This is the foundation of inferential statistics. In inferential statistics we use a sample to draw inferences or test our hypotheses about a population of scores.

Typically we use a sample size that is greater than 1 and compare our sample mean to what we believe to be the case about a population. To do this we need to consider how sample means are distributed.

Sampling Distribution of the Mean

probability distribution of sample means

With large sample sizes the probability of sample means always approximates a normal distribution

Consider the following example:

If a population consists of N = 4 scores (1,2,3,4) and is uniform (i.e., all scores have the same frequency, 1) then

m = (1 + 2 + 3 + 4) / 4 = 2.5

 

= 1.118

 

Histogram of the population:

 

How many samples of n = 2 can be drawn with replacement?

4 x 4 = 16
Sample #
Sample Values
Sample Mean
Sample #
Sample Values
Sample Mean
1
1,1
1
9
3,1
2
2
1,2
1.5
10
3,2
2.5
3
1,3
2
11
3,3
3
4
1,4
2.5
12
3,4
3.5
5
2,1
1.5
13
4,1
2.5
6
2,2
2
14
4,2
3
7
2,3
2.5
15
4,3
3.5
8
2,4
3
16
4,4
4

Now consider how the population of sample means (n = 2) are distributed.

mmeans = 2.5 Notice that this is the same value as the mean of scores.

smeans = .791 Notice that this value is less than the value as the s of scores

and actually equals:

= 1.118/1.41 = .791

Sampling Distribution of the Sample Means (n =2)

 

 

What happens as n is increased?

If the mean of IQ scores is 100 and s = 16 what is that probability that if we sampled 2 people their mean IQ would be 124 or greater?