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Chapter 9-B: Statistics |
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As we have seen, every list of numbers has an average. Also as we have seen, the numbers of the list do not need to actually be anywhere near the actual average. In fact, most numbers in the list differ, or deviate from the average by some degree. The standard deviation tells us generally how the numbers will deviate from the average. The greater the deviation, the higher the standard deviation.
Let's use a sample list of numbers to understand the standard deviation formula: 3, 8, 6, -4, 2 The average of this list is 3. The formula for Standard Deviation is as follows: where x is a term in the list, is the average of the list, and n is the number of numbers in the list. You will therefore follow these steps: 1. Find the average of the list of numbers 2. Subtract each term from the average 3. Square each of the differences 4. Add up the squared values 5. Divide the sum by the number of numbers 6. Take the square root of that number Apply that with the list above (average is 3):
Step 5: 84/5 = 16.8Step 6: ˜ 4.1 So the Standard Deviation of {3, 8, 6, -4, 2} is 4.1. What does it all mean? It is an indication of how different, or how varied , the numbers in the list are compared with the average. In our current example, the average is 3, but the SD is 4.1. That must mean that the numbers vary widely from the average. And, by looking, they do. That is the key. You want to be in a place where you can figure out relative SD just by looking and by logic. ExampleWhich of the following lists has the highest standard deviation? I. 8, 8, 8, 8, 8 II. 6, 7, 8, 9, 10 III. 4, 6, 8, 10, 12 Answer: List IIIYou can, of course, figure out each list's standard deviation by using the formula. But it is more effective if you just look at the numbers and think about it for a second. You can probably see quickly that all three of the above lists have the same average, 8. So which list has the most deviation from 8? In List I, all the numbers are the same as the average. There is no deviation at all. It is List III that shows the greatest deviation. ExampleWhich of the following lists has the highest standard deviation? I. 6, 7, 8, 9, 10 II. 4, 6, 8, 10, 12 III. 18, 20, 22, 24, 26 Answer: II and III have the same Standard DeviationThis is a trick question. While the average of List III is higher than List II, the numbers of List III are the same distance from its average of 22 as the numbers of List II are from its average of 8. They are both lists of even numbers, and as such, have the same dispersion. Therefore, they have the same standard deviation.
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