Measures of Central Tendency: Mean, Median, and Mode
Introduction: Using Measures of Central Tendency to Describe Variables
In almost every Introductory Statistics course, you will begin by learning how to calculate the mean, median, and mode. You will often hear the mean, median, and mode referred to as measures of central tendency. You may find yourself asking, what is the meaning of this term? How can it be defined?
A measure of central tendency is a value that describes a data set. It is a measure that tells us where the data tends to be clustered. It allows us to locate the "center of gravity" of a distribution.
Got it? Great. Let's move on.
At this point you may find yourself asking, why do we need three measures of central tendency? Can't we just choose one? This is an excellent question! However, we do indeed need all three measures because the measure(s) we are able to use depends on the nature of the data being analyzed. Specifically, the decision whether to find the mean, median, or mode (or some combination of the three) depends on how the specific variable we are examining is measured.
Alright then, what is a variable?
A variable is a characteristic or numerical quantity that can take on different values, meaning, it is a piece of information that can vary. This may seem somewhat obscure. Let's look at a few examples for clarification.
Examples of Variables
 Age  Age is a variable because it can take on a range of numerical values (0100) that describe how old an individual is, typically measured in years.
 Highest Degree Completed  Highest degree is a variable because it includes several categories pertaining to educational attainment (Less Than High School, High School Diploma, Associate's Degree, Bachelor's Degree, Graduate Degree).
 Gender  Gender is a variable because can take on more than one value (male or female).
While "Age," "Highest Degree Obtained," and "Gender" are examples of variables, the specific numerical quantities or categories assigned to each variable are called values. Hence, age is variable, while male and female are values.
In order to determine the appropriate measure(s) of central tendency, we focus primarily on variables and the values assigned to them. Specifically, we need to ask, how is a given variable being measured? Once we have determined this, we will know which measures of central tendency can be calculated. How to identify the level of measurement for a variable will be covered in greater depth in the next section.
Level of Measurement: Determining Whether a Variable is Measured at the Nominal, Ordinal, or IntervalRatio Level
Levels of measurement are often described as "scales of measure." To put it simply, the level of measurement for a given variable is a way of classifying how a variable is quantified or described. There are three levels of measurement:
 The nominal level of measurement  A nominallevel variable is comprised of values that can be namedbut not ranked or quantified.
 The ordinal level of measurement  An ordinallevel variable is comprised of values that can rankedbut not quantified.
 The intervalratio level of measurement  An intervalratio level variable is comprised of values that can be quantified (described by numbers).
*Sometimes a distinction is made between interval and ratio levels of measurement. While both ordinal and intervallevel variables have numerical values, ordinallevel variables have values with an absolute zero, whereas intervallevel variables do not. However, for our purposes, it is not necessary to make this distinction.
Take a look at the examples provided below to enhance your familiarity with the three levels of measurement.
Examples of Nominal, Ordinal, and IntervalRatio Level Variables and Values
Level of Measurement
 Variable
 Values


IntervalRatio
 Age
 0100 (years)

IntervalRatio
 Number of Siblings
 0, 1, 2, 3, 4, 5, 6, 7, 8

Ordinal
 Highest Degree Completed
 Less Than High School, High School Diploma, Associate's Degree, Bachelor's Degree, Graduate Degree (Masters/Ph.D./Doctorate)

Ordinal
 Overall Happiness
 Very Happy, Somewhat Happy, Somewhat Unhappy, Very Unhappy

Nominal
 Gender
 Male, Female

Nominal
 Marital Status
 Single, Married, Divorced, Widowed

Using a Variable's Level of Measurement to Determine Appropriate Measures of Central Tendency
Once you identify a variable's level of measurement you are able to determine the measure(s) of central tendency that can be computed a given variable.
For intervalratio level variables, we can find the mean, median, and mode. For ordinal level variables, we can find the median and mode (but not the mean). For nominal level variables, we can find the mode (but not the mean or median).
It is important to follow these guidelines when identifying the measures of central tendency that are suitable to calculate for a given variable, because as you will see in the sections that follow, finding an inappropriate measure of central tendency simply does not make sense, and moreover, is incorrect.
Available Measures of Central Tendency for Each Level of Measurement
IntervalRatio
 Ordinal
 Nominal
 

Mean
 ✔
 
Median
 ✔
 ✔
 
Mode
 ✔
 ✔
 ✔

The Mean: A Distribution's Numerical Average
The mean is simply a numerical average. It can be found by adding up each value assigned to an intervalratio variable and dividing the sum by the total number of cases.
Example 1: We surveyed 5 people, asking each respondent their age (in years). The ages reported in our survey were: 21, 45, 24, 78, 45. Find the mean.
 (21 + 45 + 24 + 78 + 45) / (5) = 42.6
To find the mean, we added the ages of each respondent, then divided by the total number of people in our study (5). The mean for age is 42.6 years.
Example 2: We surveyed 8 people, asking each respondent how many siblings they have. The number of siblings reported in our survey were: 4, 0, 2, 1, 3, 1, 1, 2
 (4 + 0 + 2 + 1 + 3 + 1 + 1 + 2) / (8) = 1.75
To find the mean, we added the number of siblings of each respondent, then divided by the total number of people in our study (8). The mean for number of siblings is 1.75.
The Median: The Center Value
The median is the value that lies in the center of the distribution. When the data are ordered from least to greatest, the median is located in the middle of the list. The median can be found for both numbers and ranked categories. It is first necessary to order your values from least to greatest. If there is only one center value (there are an equal number of cases above and below), great, you've found the median! If there are two center values (this will happen when there is an odd number of cases), the median is found by taking the average of the two center values.
Example 1: We surveyed 5 people, asking each respondent their age (in years). The ages reported in our survey were: 21, 45, 24, 78, 45. Find the median.
 We must first rearrange the values for age from least to greatest: 21, 24, 45, 45, 78
 We then identify the value(s) in the center: 21, 24, 45, 45, 78
 Answer: The median is 45
Example 2: We surveyed 8 people, asking each respondent how many siblings they have. The number of siblings reported in our survey were: 4, 0, 2, 1, 3, 1, 1, 2. Find the median.
 We must first rearrange the values for number of siblings from least to greatest: 0, 1, 1, 1, 2, 2, 3, 4
 We then identify the value(s) in the center: 0, 1, 1, 1, 2, 2, 3, 4
 Since there are two center values, we must take the average of them: (1+2)/(2)=1.5
 Answer: The median is 1.5
Example 3: We surveyed 7 people, asking each respondent to report their overall level of happiness. The levels of happiness reported in our survey were: very happy, somewhat happy, very happy, somewhat unhappy, very unhappy, somewhat unhappy, somewhat happy. Find the median.
 We must first rearrange the values for level of happiness from least to greatest: very unhappy, somewhat unhappy, somewhat unhappy, somewhat happy, somewhat happy, very happy, very happy
 We then identify the value(s) in the center: very unhappy, somewhat unhappy, somewhat unhappy, somewhat happy, somewhat happy, very happy, very happy
 Answer: The median is somewhat happy.
The Mode: The Most Frequently Occurring Value
The mode is the value that occurs most frequently. It is found by determining the number or category that appears most often. If no value occurs more than once, there is no mode. If there are two values that occur most often, report both of themthis type of distribution is bimodal.
Example 1: We surveyed 5 people, asking each respondent their age (in years). The ages reported in our survey were: 21, 45, 24, 78, 45. Find the mode.
 We see in the following distribution (21, 45, 24, 78, 45) that 45 occurs twice, whereas the other ages occur only once. Therefore, 25 is the mode for age.
Example 2: We surveyed 7 people, asking each respondent to report their gender. The genders reported in our survey were: male, female, female, female, male, male, female. Find the mode.
 We see in the following distribution (male, female, female, female, male, male, female) that "female" occurs four times, whereas "male" only occurs three times. Therefore, female is the mode for gender.
Measures of Central Tendency: In Review
Conclusion
Now that you are familiar with how to calculate measures of central tendency, you should possess the knowledge to compute them for any variable (based on its level of measurement). Best of luck to all of you in your statistical endeavors!
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