Insight to the meaning of the standard deviation will be gained by experience in finding standard deviations. The sets of data should be selected so as to demonstrate key ideas. Two points to investigate are what happens to the mean and standard deviation when a constant is added to all the numbers in a set of data and what happens when all the data are multiplied by a constant. This is an opportunity to give experience in coordinate transformations. In Algebra when the teacher moves a graph of an equation just by looking at its formula and the previous example many students find this to be sleight of hand. If the students are non-Algebra students they will be gaining experience for later. If the students are Algebra students not only are we providing an example to be studied but also we are integrating a statistics concept into the standard curriculum.
It is to be hoped that calculators will be provided to do these calculations. I believe some learning is just doing the calculations oneself. If it is done by computer much of the point is lost. Once the work gets boring and we have seen the point then the computer can be used.
Here are some sets and some questions to be asked with each one.
Draw a histogram for each set and find the mean and standard deviation.
|
Example
|
|
Mean
|
Standard dev.
|
|
1
|
19,20,21
|
20
|
0.81649
|
Compare each of the following to example one. How is example one changed to get the new case? How did the mean and standard deviation change? Make another pair to show the same point using more numbers.
|
2
|
-1, 0, 1
|
0
|
0.81648
|
|
3
|
19,20,20,21
|
20
|
0.70711
|
|
4
|
38,40,42
|
40
|
1.63299
|
|
5
|
57,60,63
|
60
|
2.44949
|
|
6
|
19,19,20,20,21,21
|
20
|
0.81648
|
Example 2 is 20 subtracted from each score in example 1. The mean has decreased by 20 but the dispersion has not changed so the standard deviation stay the same. Example 3 shows how adjoining the mean to a set of scores does not change the mean but does decrease the dispersion and the standard deviation. Examples 4 and 5 are example 1 multiplied by 2 and 3 respectively and the means and standard deviations have likewise been multiplied by 2 and 3. Example 6 shows that multiplying the frequencies by a constant changes neither the mean nor the standard deviation.
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Example
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|
Mean
|
Standard dev.
|
|
7
|
3,5,8
|
5.33 . . .
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2.05480
|
|
8
|
4,7,9
|
6.66 . . .
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2.05480
|
Tell how examples 7 and 8 were combined to make examples 9 and 10. How did the means change? How did the standard deviations change?
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9
|
7,12,17
|
12.0
|
4.08248
|
|
10
|
12,20,21,27,32,35,
|
|
|
45,45,56,72
|
35.55 . . .
|
18.04384
|
Example 9 is the term by term sum of 7 and 8 the new mean is the sum of the given means. There does not seem to be a simple relationship for the standard deviations. Example 10 is the set of all possible products formed by taking one number from each set. The mean is the product of the two given means, but again there is no obvious relationship for the standard deviations.
After the students can find standard deviations tell them we may expect certain percentages of the data to fall within fixed numbers of standard deviations from the mean. Here would be the place to show the class a normal curve telling them the area of an interval under the curve represents the probability of a score falling in that interval. The probability is 68% that a score lies within plus or minus one standard deviation of the mean, 95% that a score is within plus or minus two standard deviations of the mean, and 99% that a score is within three standard deviations of the mean. I would stress that these percentages are dependent upon the distribution that is being considered.
