with the following returns:
Stock.
.......................
Return
A.................................
7%
B................................
12%
C.................................
-3%
D................................
21%
E..................................
3%
In this example, the population mean is equal to 8%, and the population standard
deviation is equal to 8.15%. Now, suppose that we decide to take a random sample of
three stocks. Assuming that the order is not important and sampling is done without
replacement, applying combination equation (n=5, and x=3) there are ten possibilities:
Sample Stocks.
..............
Returns.
............
Mean
1) A, B, C.
.....................
7%.
.12%.
.-3%.
.....
5.33%
2) A, B, D.
.....................
7%.
.12%.
.21%.
...13.33%
3) A, B, E.
.....................
7%.
.12%.
.3%.
.......
7.33%
4) A, C, D.
.....................
7%.
.-3%.
.21%.
.....
8.33%
5) A, C, E.
.....................
7%.
.-3%.
.3%.
.......
2.33%
6) A, D, E.
.....................
7%.
.21%.
.3%.
.....
10.33%
7) B, C, D.
....................
12%.
.-3%.
.21%.
....10.00%
8) B, C, E.
....................
12%.
.-3%.
.3%.
.......
4.00%
9) B, D, E.
....................
12%.
.21%.
.3%.
.....
12.00%
0) C, D, E.
....................
-3%.
.21%.
.3%.
.......
7.00%
As the above example shows, two (or more) samples from the same population will
likely have different sample values (mean values ranges from 2.33% to 13.33%), and
therefore possibly lead to different decisions. Thus, the sample mean reported to the
decision maker in the company will depend on the sample selected, i.e., sample 1, 2,
3,.
....or 10. Note that the sample means (column 3 in the above table) also are different
from the population mean, i.e., 8. For example, if sample 4 is selected, the sampling
error
(the difference between a sample statistic and its corresponding population
parameter)
is fairly small (8.33 - 8.0 = 0.33), but if the selected sample is sample 2,
the error is quite large (13.33 - 8.0 = 5.33).
Because the decision maker cannot know
how large the sampling error will be before selecting the sample, he/she should know
how the possible sample means are distributed.
Defination: