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Mean
Common Core: 6.SP.5

Mean

Description:

This lesson will explain finding the mean of a data set.

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Tutorial

What's Covered

In this tutorial, you're going to learn about the mean of a data set. You will learn:

  1. Calculating Mean
  2. Outliers
  3. Notation
  4. Weighted Averages

1. Calculating Mean

The word “mean” is often used interchangeably with the word “average.” You may see that in this tutorial. However, there are several different things that can be called an average. So the mean is the most common of those.

Term to Know

Mean

The "average" value of a data set. It is obtained by dividing the sum of the values by the number of values in the set.

So in this tutorial, “mean” will be used interchangeably with the word “average,” whereas other concepts, such as median, will not be implied.

Concept to Know

The mean of a data set is found by adding up all the values together and dividing by how many there are. Notationally, it looks like this:

The 1s, 2s, and 3s in the X₁ X₂ X₃ indicate that they're the first number in the list and then the second number in the list, all the way up until the last number in the list, marked by the Xn. And the “n” value in the denominator is the total number of values.

Try It

This data set shows how tall the players on the Chicago Bulls basketball team are.

What height would be considered an average height for basketball players on this basketball team?

To find the answer, add all the values and divide by however many values there are. So In this case, we're going to add up all of the players' heights and divide by 15 because there are 15 players. The result is 78.33 inches. That's the average height of a player on the Bulls.

ExampleTo calculate mean using technology, you can use a spreadsheet. Create your list of values and select, “equals average.” The spreadsheet will suggest some formulas for you.

Highlight all of the fields, close the parentheses, and hit “Enter.”

Here, the spreadsheet returns with 78.33, which is the same number as you got in your notation above.


2. Outliers

Here's an example of when the mean is a poor representation of where the center really is:

Try It

Suppose that you have 12 employees. Eight of them are shift workers. Three of them are managers. And there’s one boss. Here is a chart of their salaries:

Boss Shift Worker Manager
$200,000 $42,000 $55,000


What is the mean?

Taken together, the mean of the eight shift workers, the three managers, and the boss is over $58,000.

However, how many of the employees make more than $58,000? And how many make less than $58,000?

11 of the 12 employees make less than $58,000, and only one makes more than that. And the one who does make above $58,000 makes substantially above that amount. Therefore, it doesn't really make a whole lot of sense to measure center. The boss’s $200,000 salary is an outlier in this data set.

In the presence of outliers, which are very few high or very few very low values, the mean won't give an accurate representation of center.


3. Notation

There are a couple of accepted notations for expressing average:

  1. Using the Greek letter mu, which is pronounced “mew”: ᘈ
    1. You will see this quite a lot as a notation for mean.
  2. Using the “x-bar”: x̅
    1. This is simply an x with a bar over it; you can also use a y-bar, or whatever value you're using.

Sometimes, you may use a different special notation to shorten up all of this summation. It's called "summation notation" or "sigma notation."

Term to Know

Summation Notation

A notation that uses the Greek letter sigma to state that values should be added together.

This notation uses the Greek letter, sigma: Σ When you use summation notation, it will look like this:

The Xᵢ (read as “x subscript i”) is just like the X₁ and X₂ and X₃ in the original summation (from the first section). So this notation means that the value of Xᵢ will be the sum of all the Xs, starting from the first one (where the i value is 1) and finishing at the “nth,” or last, one. When that's all done, you divide by n.

So the compact formula of the summation notation is the same as the large, lengthy to write out formula.



4. Weighted Averages

Sometimes, all of the values in your data set will not be weighted the same.

Term to Know

Weighted Mean/Average

A way of calculating a mean when not all the values count for the same amount. Each value should be multiplied by its weight and added together, then divide the sum by the sum of the weights.

If you are calculating the mean of a data set where there are weighted values, you will need to account for these differences.

ExampleSometimes, in an academic course, exams are weighted for some percentage of the grade and the final is weighted for a greater percentage.

Try It

How do you do you find the mean?

Count each of the first three tests as each one test. Count the final exam as three tests because it’s weighed three times as much. Then multiply each of these tests by their weights. That would look like this:

You are counting these not as four tests, but essentially more like six tests, because the final one counts for three of the others, so you divide by 6 in the end.

This weighted average, or the weighted mean, is 87½.


Summary

The mean is one measure of center that we can use, and it's what is meant by the word “average.” When measuring mean, it is important to consider outliers. Sometimes, summation notation can be used as a shortcut instead of writing the whole long string of added values. And then finally, weighted averages can be found by multiplying each value times its weight and counting it, essentially, that many times.

Thank you and good luck!

Source: THIS WORK IS ADAPTED FROM SOPHIA AUTHOR JONATHAN OSTERS

TERMS TO KNOW
  • Mean

    The "average" value of a data set. It is obtained by dividing the sum of the values by the number of values in the set.

  • Summation Notation

    A notation that uses the Greek letter sigma to state that values should be added together.

  • Weighted Mean/Average

    A way of calculating a mean when not all the values count for the same amount. Each value should be multiplied by its weight and added together, then divide the sum by the sum of the weights.