MEASURES OF CENTRAL TENDENCY

Measures of central tendency describe how close something is to "the middle". The three most common measures of central tendency in statistics are: MEAN, MEDIAN and MODE. First, let us discuss what each is, and then when and how each can best be used.

MEAN 

Add up all the observations and divide by the total number of observations. This is also called the "arithmetic average".

EXAMPLE: 

5 students are asked their shoe size. Their shoe sizes are as follows: 6, 8, 9, 9 and 10. The average is: 6 + 8 + 9 + 9 + 10 = 42 divided by the number of students (5). So the MEAN is 42/5 = 8.4.

MEDIAN

Line up all of the observations in order and find the value that is in the middle: 

EXAMPLE: 

Using the same example on shoe size, the median is: 6, 8, 9, 9, 10 .

So here the median is clearly 9, but it is not always that simple. What if we have two middle numbers as follows?

6, 7, 8, 9, 9, 10

8 and 9 are both in the middle, so which is the median? If there are two numbers in the middle, take the average = (8 + 9)/2 = 8.5. 

So in this second case, the median is 8.5...right between 8 and 9, the two middle numbers. 

It can also be tricky if you are using ordinal level data rather than ratio level data. For example, consider the following table showing the number of responses for different numbers of TV hours watched per week:

TV hours watched this week	Number of people
0-5	4
6-9	2
10+	1

Let's line up everyone in a row: 

Four people watched "0-5", two watched "6-9" and one watched "10+":  0-5, 0-5, 0-5, 0-5, 6-9, 6-9, 6-9, 10+.

REMEMBER: that the median is the value that corresponds to the middle observation, not the number of the middle observation itself! Some students are tempted to say that the median is "4", because the 4th observation is right in the middle. This is not the case! The median here is "0-5", not "4".  

FOR MORE ON THIS SEE THE FREQUENCY TABLE PAGE.

MODE

The mode is simply the most common (most frequent) response. 

EXAMPLE:

In the example above on TV hours, the mode is "0-5 hours". More people watched 0-5 hours than any other category. 

There can also be more than one mode in a given dataset. Consider the following table of ages of season 19 contestants on The Bachelor:

Name	Age
Mackenzie	21
Ashley Salter	24
Megan	24
Alissa Giambrone	24
Jordan Branch	24
Amanda Goerlitz	24
Jillian Anderson	25
Bo Stanley	25
Kara Wilson	25
Michelle	25
Ashley Iaconetti	26
Becca Tilley	26
Nikki Delventhal	26
Tara Eddings	26
Brittany Fetkin	26
Britt Nilsson	27
Samantha Steffen	27
Jade Roper	28
Kelsey Poe	28
Kimberly Sherbach	28
Reegan Cornwell	28
Carly Waddell	29
Kaitlyn Bristowe	29
Whitney Bischoff	29
Amber James	29
Tracy Darakis	29
Juelia Kinney	30
Tandra Steiner	30
Nicole	31
Trina Scherenberg	33

Five were 24, five were 26 and five were 29. So there are 3 modes: 24, 26 and 29. This dataset is said to be "multi-modal".

WHEN TO USE MEAN, MEDIAN AND MODE:

Mean, median and mode are all "averages"; however, when people say "average" they are often talking about the mean, or "arithmetic average".  So how do you know when to use each one. 

• The mean is the most sensitive to outliers, so if you have a dataset with very extreme cases (or observations) you may be better of using the median. 

INCOME is a classic example. If Bill Gates walks into a room with 6 other people who make $50,000 a year, the mean income would be over $50 million. But that is not a very average picture of what the typical person makes! The median and mode would still be $50,000 with or without Bill Gates, and either is a better representation of what is typical!

• The mean is not an option for nominal and ordinal level data (see the page of level of measurement for more on this). 

How can you compute the mean of 3 cats, 4 dogs and 5 chickens (nominal) or even the mean TV hours in the example above (the categories are not equal)? 

• Median and mode can be used with ordinal level data

• Mode can be used with nominal data

TABLE: WHICH MEASURES OF CENTRAL TENDENCY CAN BE USED WITH WHICH LEVEL OF MEASUREMENT

Measure can be used?
	Mean	Median	Mode
Nominal	No	No	Yes
Ordinal	No	Yes	Yes
Ratio	Yes	Yes	Yes

SEE MORE ON THE PAGE ON LEVELS OF MEASUREMENT.