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Counting by Units of Standard Deviation

Though we don't need to know how to calculate standard deviation, we do need to know how to use it. The standard deviation of a certain list is always expressed along with the average of the same list.

For example, one might say that the average gambler's winning in Las Vegas is $200, with a standard deviation of $30. It is then possible to count in units of standard deviation. One standard deviation would be $30, two standard deviations would be 2 x 30 = $60, and three standard deviations would be 3 x 30 = $90.

Example

Farmers in the United States grew an average (arithmetic mean) of 80 tons of corn each, with a standard deviation of 10 tons. What value is two standard deviations away from the mean?

A)  90

B) 82

C) 78

D) 75

E) 60

Answer: 60

Since the standard deviation is 10, two standard deviations is 2x10=20. Standard deviations can be counted above and below the average, which, in this case, is 80. So two standard deviations away from 80 would be 20 tons less or more. 80 ­ 20 = 60.

Standard Deviation in Practical Use

Standard Deviation is called “standard” because the results are standardized. This means that one standard deviation in a given list has the same statistical meaning as one standard deviation in another list. The percentage of the population at any standard deviation is always the same:

Between + 1 and ­ 1: 68%

Between + 2 and ­ 2: 95%

Between + 3 and ­ 3: 99%

Imagine a situation in which a college professor is bragging to his colleagues about a recent exam he gave his class. The exam was difficult, he tells them, but the average grade was still 85%. He interprets this to mean that he is a good teacher and his students have learned well. Hearing this, his suspicious colleague asks him what the standard deviation of the grades was. Let's look at and graph two possible scenarios.

Average Grade: 85 Average Grade: 85

Standard Deviation: 1 Standard Deviation: 5

Because the results of the students' grades are standardized, we can use standard deviation to determine the percent of the class that falls into a specific range of test scores, and that range always corresponds to the standard deviation. Therefore, for each scenario, we can see in the graphs above that 99% of the professor's class falls between ­3 and +3 standard deviations. The corresponding range of test scores in the case where the standard deviation is 1 is from 82-88, or 6 points, while the range of scores when the standard deviation is 5 is from 70-100, or 30 points.

Which one corroborates the professor's claim that his students have learned the material well? The graph with the lower standard deviation, the first one, tells us that his students all generally scored in the same range, while the second one, the one with higher standard deviation, tells us that his students' scores were more erratic. Yes, some students scored very well, but others scored very poorly. The first one supports the professor, while the second proves his colleague's suspicions correct.

Now let's take a look at our original example, the one with the politician. How erratic are her numbers, the ones that so neatly averaged out to $80,000? Let's see what the standard deviation is of our three scenarios.

So we can see that when there is no deviation at all, as in Group 1, the standard deviation is necessarily 0. When there is slight deviation, as in Group 2, the standard deviation is also slight. When the numbers vary wildly from one another, as in Group 3, the standard deviation is very high.

The politician is caught. When pressed about her statistic, she tells the audience that the average salary in her state is $80,000 with a standard deviation of $78,294. The reporter sees through her empty average, and slams her in the papers.

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