7 pages Table of contents

Chapter 9: Standard Deviation

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Statistical Measurements

Imagine that a politician in your state is up for re-election. Trying to impress the public with the work she has done, the politician announces at a press conference that at the end of her term the average yearly salary per person in the state is $80,000.

Would you support her based on that number? It sounds good ­ it sounds  like everyone in her state is making $80,000. But can we be sure? A reporter wants to know if her statistic is valid, and he picks three groups of five people randomly from the state. Each group has the same average, $80,000. Let's look at the different groups now.

What can we learn from this chart?
Group 1 represents what the politician wants her state to believe: that everyone earns $80,000.
Group 2 represents what most people would see as an acceptable situation: some people earn more, some earn less, but they all earn about the same and together they average $80,000.
Group 3 represents what is probably most likely: that most people don't earn very much at all but a few people earn so much as to sway the average in their favor.

Statistical measurements are used to condense the behavior of multiple elements into a single number that we can work with and comprehend. Unfortunately, the average is a particularly poor statistical measure, as the same average can represent an unlimited number of situations.

How do we avoid this pitfall then? There are several other important statistical measurements that help us understand the average and paint a more complete picture of what is happening. Those measurements are:

• Range
• Median
• Mode
• Standard Deviation

By understanding these four terms on top of the average, you will be able to evaluate a situation more sophisticatedly than ever before. You'll also do well on the GRE. We will start with Range, Median, and Mode , then continue with Standard Deviation .

Range

The range of any list of numbers is the difference of the highest and lowest numbers.

Example

Find the range of the lists below:

•  8, 8, 8, 8, 8

•  1, 5, 3, 14, 17

•  6, 8, -14, 7, -2

Answer

•  Range = 8 ­ 8 = 0

•  Range = 17 ­ 1 = 16

•  Range = 8 ­ (-14) = 22

Median

The median of any list of numbers is the number in the middle when the list is arranged in numeric order. If there is no middle number, the median is the average of the two middle numbers.

Example

Find the median of the lists below:

•  3, 4, 5, 6, 7

•  8, 14, 5, 6, 2, 21, 21

•  8, 14, 5, 6, 2, 21

Answer

•  Median = 5

•  First arrange the list in order: 2, 5, 6, 8, 14, 21, 21. Median = 8

•  First arrange the list in order: 2, 5, 6, 8, 14, 21. The median is the average of the two in the middle, 6 and 8, or 7.

Mode

The mode of any list of numbers is the number that appears the most often. If two or more numbers appear the same number of times, they are all considered the mode. Having two modes is referred to as being bi-modal.

Example

Find the mode of the lists below:

•  3, 5, 4, 5, 7, 5, 2

•  4, 2, 9, 4, 2

Answer

•  Mode = 5

•  Mode = 2 and 4

So, now that we understand Range, Median, and Mode, let's see them applied to our politician's claim from above:

Chapter 9-B:: Statistics

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.

	800Score Tip: While this chapter explores the intricate Standard Deviation formula, the GRE does not require you to know it. Your goal in learning Standard Deviation should be to understand what it is, how it works, and what it represents, rather than how to calculate its value.

Formula:

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 = 16.8

Step 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.

Example

Which 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 III

You 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.

Example

Which 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 Deviation

This 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.

Chapter 9-C: Standard Deviation

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 2x30=$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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