Topic 8. Sampling and Sampling Distributions

Topic 8. Sampling and Sampling Distributions

The Sampling Distribution

Let’s estimate the mean education for Kunovichberg (a small island in the South Pacific).

There is a population of 5...the variable is years of education:

Case 1: 20

Case 2: 12

Case 3: 14

Case 4: 12

Case 5: 8

Y=13.2

Y=4.4

How many possible samples of 3 are there from the population? There are 10 possible samples:

1 (20), 2 (12), 3 (14) – mean=15.3

1 (20), 2 (12), 4 (12) – mean=14.7

1 (20), 2 (12), 5 (8) – mean=13.3

1 (20), 3 (14), 4 (12) – mean=15.3

1 (20), 3 (14), 5 (8) – mean=14

1 (20), 4 (12), 5 (8) – mean=13.3

2 (12), 3 (14), 4 (12) – mean=12.7

2 (12), 3 (14), 5 (8) – mean=11.3

2 (12), 4 (12), 5 (8) – mean=10.7

3 (14), 4 (12), 5 (8) – mean=11.3

Here is the frequency distribution for our sampling distribution:

Mean / Frequency / Percent
10.7 / 1 / 10
11.3 / 2 / 20
12.7 / 1 / 10
13.3 / 2 / 20
14 / 1 / 10
14.7 / 1 / 10
15.3 / 2 / 20
Sum / 10 / 100

The standard deviation of the sampling distribution is called the standard error of the mean. This describes how much variability there is in the value of (the mean education) from sample to sample. The bigger the number, the more different are the means from sample to sample. Here is the formula:

Samples of size 3:

Sample / Case 1 / Case 2 / Case 3 / Sample mean
1 / 20 / 12 / 14 / 15.3
2 / 20 / 12 / 12 / 14.7
3 / 20 / 12 / 8 / 13.3
4 / 20 / 14 / 12 / 15.3
5 / 20 / 14 / 8 / 14.0
6 / 20 / 12 / 8 / 13.3
7 / 12 / 14 / 12 / 12.7
8 / 12 / 14 / 8 / 11.3
9 / 12 / 12 / 8 / 10.7
10 / 14 / 12 / 8 / 11.3
Mean of means / Standard error
13.2 / 2.5

Sample of size 4:

Sample / Case 1 / Case 2 / Case 3 / Case 4 / Sample mean
1 / 20 / 12 / 14 / 12 / 14.5
2 / 20 / 12 / 14 / 8 / 13.5
3 / 20 / 12 / 12 / 8 / 13.0
4 / 20 / 14 / 12 / 8 / 13.5
5 / 12 / 14 / 12 / 8 / 11.5
Mean of means / St dev of sampling distrib
13.2 / 2.2

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