Measures of Central Tendency

Learning Outcomes

  • Calculate the mean, median, and mode of a set of data

In statistics, we are interested in collecting and describing data. We look at  characteristics, such as where the data appears to be centered, and its variability. Certain numerical averages called mean, median, and mode help us describe the central tendency of a data set.

Recall for success

Look for red boxes like this one throughout the text. They’ll show up just in time to give helpful reminders of the math you’ll need, right where you’ll need it.

Mean, Median, and Mode

Silver sphere which has red smaller spheres clustered on its left side, with magnetic attraction

Let’s begin by trying to find the most “typical” value of a data set.

Note that we just used the word “typical” although in many cases you might think of using the word “average.” We need to be careful with the word “average” as it means different things to different people in different contexts.  One of the most common uses of the word “average” is what mathematicians and statisticians call the arithmetic mean, or just plain old mean for short.  “Arithmetic mean” sounds rather fancy, but you have likely calculated a mean many times without realizing it; the mean is what most people think of when they use the word “average.”

Mean

The mean of a set of data is the sum of the data values divided by the number of values.

examples

Marci’s exam scores for her last math class were 79, 86, 82, and 94. What would the mean of these values be?

The number of touchdown (TD) passes thrown by each of the 31 teams in the National Football League in the 2000 season are shown below.

37 33 33 32 29 28 28 23 22 22 22 21 21 21 20

20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6

What is the mean number of TD passes?

Both examples are described further in the following video.

Try It

The price of a jar of peanut butter at 5 stores was $3.29, $3.59, $3.79, $3.75, and $3.99. Find the mean price.

examples

The one hundred families in a particular neighborhood are asked their annual household income, to the nearest $5 thousand dollars. The results are summarized in a frequency table below.

Income (thousands of dollars) Frequency
15 6
20 8
25 11
30 17
35 19
40 20
45 12
50 7

What is the mean average income in this neighborhood?

Extending off the last example, suppose a new family moves into the neighborhood example that has a household income of $5 million ($5000 thousand).

What is the new mean of this neighborhood’s income?

Both situations are explained further in this video.

While 83.1 thousand dollars ($83,069) is the correct mean household income, it no longer represents a “typical” value.

Imagine the data values on a see-saw or balance scale. The mean is the value that keeps the data in balance, like in the picture below.

Drawing of a balance bar. A large blue block is on left end, and two smaller blue rectangles are on right end of balance point. One is close to the balance, one is further away.

If we graph our household data, the $5 million data value is so far out to the right that the mean has to adjust up to keep things in balance.

Drawing of a balance bar. A large blue block is on left end and two smaller blue rectangles are on also on left of balance point. On right, a small blue rectangle is significantly far away from the balance point.

For this reason, when working with data that have outliers – values far outside the primary grouping – it is common to use a different measure of center, the median.

Median

The median of a set of data is the value in the middle when the data is in order.

  • To find the median, begin by listing the data in order from smallest to largest, or largest to smallest.
  • If the number of data values, N, is odd, then the median is the middle data value. This value can be found by rounding N/2 up to the next whole number.
  • If the number of data values is even, there is no one middle value, so we find the mean of the two middle values (values N/2 and N/2 + 1)

example

Returning to the football touchdown data, we would start by listing the data in order. Luckily, it was already in decreasing order, so we can work with it without needing to reorder it first.

37 33 33 32 29 28 28 23 22 22 22 21 21 21 20

20 19 19 18 18 18 18 16 15 14 14 14 12 12 9 6

What is the median TD value?

Find the median of these quiz scores: 5 10 8 6 4 8 2 5 7 7

Learn more about these median examples in this video.

Try It

The price of a jar of peanut butter at 5 stores was $3.29, $3.59, $3.79, $3.75, and $3.99. Find the median price.

Example

Let us return now to our original household income data

Income (thousands of dollars) Frequency
15 6
20 8
25 11
30 17
35 19
40 20
45 12
50 7

What is the mean of this neighborhood’s household income?

If we add in the new neighbor with a $5 million household income, then there will be 101 data values, and the 51st value will be the median. As we discovered in the last example, the 51st value is $35 thousand. Notice that the new neighbor did not affect the median in this case. The median is not swayed as much by outliers as the mean is.

View more about the median of this neighborhood’s household incomes here.

 

In addition to the mean and the median, there is one other common measurement of the “typical” value of a data set: the mode.

Mode

The mode is the element of the data set that occurs most frequently.

The mode is fairly useless with data like weights or heights where there are a large number of possible values. The mode is most commonly used for categorical data, for which median and mean cannot be computed.

Example

In our vehicle color survey earlier in this section, we collected the data

Color Frequency
Blue 3
Green 5
Red 4
White 3
Black 2
Grey 3

Which color is the mode?

Mode in this example is explained by the video here.

It is possible for a data set to have more than one mode if several categories have the same frequency, or no modes if each every category occurs only once.

Try It

Reviewers were asked to rate a product on a scale of 1 to 5. Find

  1. The mean rating
  2. The median rating
  3. The mode rating
Rating Frequency
1 4
2 8
3 7
4 3
5 1

Summary Key Concepts

  • Calculate the mean of a set of numbers.
    1. Write the formula for the mean [latex]\text{mean}={\Large\frac{\text{sum of values in data set}}{n}}[/latex]
    2. Find the sum of all the values in the set. Write the sum in the numerator.
    3. Count the number, n, of values in the set. Write this number in the denominator.
    4. Simplify the fraction.
    5. Check to see that the mean is reasonable. It should be greater than the least number and less than the greatest number in the set.
  • Find the median of a set of numbers.
    1. List the numbers from least to greatest.
    2. Count how many numbers are in the set. Call this [latex]n[/latex] .
    3. Is [latex]n[/latex] odd or even? If [latex]n[/latex] is an odd number, the median is the middle value. If [latex]n[/latex] is an even number, the median is the mean of the two middle values
  • Identify the mode of a set of numbers.
    1. List the data values in numerical order.
    2. Count the number of times each value appears.
    3. The mode is the value with the highest frequency.

Glossary

mean
The mean of a set of [latex]n[/latex] numbers is the arithmetic average of the numbers. The formula is [latex]\text{mean}={\Large\frac{\text{sum of values in data set}}{n}}[/latex]
median
The median of a set of data values is the middle value.

  • Half the data values are less than or equal to the median.
  • Half the data values are greater than or equal to the median.
mode
The mode of a set of numbers is the number with the highest frequency. If no numbers are repeated, there in no mode. If there are multiple numbers repeated at the highest frequency, there are multiple modes.