Understanding measures of central tendency

In the GRE Quantitative Section, a topic that often confuses students is the different ways to measure the “average” of data sets. The three most common measures are mean, median, and mode.

This post will go over the different ways to calculate these averages, and tips to make your life easier when encountering them on the GRE. 

MODE

Mode is the “average,” or “measure of central tendency,” which is the easiest to calculate. The mode of a data set is simply the number that is most often repeated. So, for example in the following set: 

{1,1,1,2,2,4,6,7,8,8,9,10} 

The mode is 1, because there are three 1s in the set, and the most any other number is present is only twice. 

MEDIAN

The next measure of central tendency I’ll cover is the median. The median is simply the middle number of the set, after the data has been arranged in ascending (or descending) order. 

For example, let’s take the same set as above. To calculate the median of a set already in order, you can simply remove numbers from each endpoint at the same time. So, 

{1,1,1,2,2,4,6,7,8,8,9,10} 

{1,1,2,2,4,6,7,8,8,9} 

{1,2,2,4,6,7,8,8} 

{2,2,4,6,7,8} 

{2,4,6,7} 

{4,6} 

{} 

Oops, we went too far! When we are only left with one number, that number is the median (this occurs when there are an odd number of values in the set). When there are only two numbers left, to calculate the median, we take the average of the two numbers left. In this case, the two numbers left are 4 and 6, so the median is (4 + 6)/2 = 5. 

One piece of advice for finding the median is to first arrange your data set in ascending order. It is tempting to try to cross out the highest and lowest numbers by looking at the unordered list, but not only is it easier to make mistakes in this manner, this “shortcut” often takes longer! 

MEAN

The last “average” I will cover is the mean. It is more common to see the mean used as an average in everyday life, in fact, another term for the mean is “arithmetic average”. To calculate the mean, you simply add up all the numbers in the set, and divide this sum by the number of numbers in the set. 

For example, take: 

{5,3,10,8,4} 

To calculate the mean of this data, we add up our numbers, 5 + 3 + 10 + 8 + 4 = 8 + 10 + 12 = 30. 

Then, we divide this sum of 30, by the size of our data set. Our data set consisted of 5 numbers, so we calculate 30/5 = 6, and get a mean of 6. Notice how the mean is not present as one of the numbers in the set, unlike the mode, and unlike the medians of sets with odd amounts of numbers. 

BONUS! Measuring Standard Deviation 

A more advanced topic that gets covered occasionally on the GRE is standard deviation. The standard deviation measures a kind of average, the average distance a random number in the data set is away from the mean. Obviously then, in order to calculate standard deviation, you will need to first calculate the mean. Once you have the mean of a data set, to calculate the standard deviation, you take each number in the data, subtract it from the mean, square the difference, find the average of all these squares, and then finally square root this average. 

That was a lot of words, let’s get more practical and try to find the standard deviation of a set. 

Let’s use the same example as before. We’ve already calculated that the mean of the following set: 

{5,3,10,8,4}  

is 6. 

So, to calculate the standard deviation, first, we calculate the difference for each of these numbers from 6. 

5 - 6 = -1 

3 - 6 = -3 

10 - 6 = 4 

8 - 6 = 2 

4 - 6 = -2 

Then, we square each of these results. 

(-1)² = 1 

(-3)² = 9 

(4)² = 16 

(2)² = 4 

(-2)² = 4 

Next, to find the average, we add up all these squares, and get 1 + 9 + 16 + 4 + 4 = 34. Dividing by 5 to find our average, we get 6.8. Finally, we square root this average to get 2.6 as our standard deviation.