Download Section

Print out chapter

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:

Statistics Part B