Explainer: The Mean of a Data Set

In this explainer, we will learn how to find the mean of a set of data and how to use it to find missing data.

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

Statistics like the mean are called measures of center; they describe a typical value of the data set. The median is another example of a measure of center.

Definition: The Mean

The mean is a measure of center of a data set.

It is calculated by adding all of the data values together and then dividing by the number of data values.

So, the mean ๐‘€ is given by ๐‘€=.thesumofthedatavaluesthetotalnumberofdatavalues

For example, the data set with values 1, 2, 3, 4, and 5 has mean ๐‘€=1+2+3+4+55=3.

We will start with an example to recap how to find the mean.

Example 1: Finding the Mean Using Summaries of the Data Values

Amelia bought 5 boats for $23,694. Later, she bought another boat which was on sale for $1,716. Calculate the mean cost of all the boats that she bought.

Answer

The mean is equal to thesumofthedatavaluesthetotalnumberofdatavalues.

In this example, the data values are the prices of each of the boats. So, we have to find the total cost of the boats รท the total number of boats.

The number of boats is 5+1=6.

The total cost of the 6 boats is equal to the sum of the total cost of the first five boats and the cost of the sixth boat. This is, 23,694+1,716=25,410.

Hence, the mean is 25,410รท6=4,235.

We conclude that the mean cost of the boats is $4โ€‰235.

In some questions, you may know the mean and the number of data values but not the total. In others you may know the mean and all but one of the data values. In both these cases we can use the formula for calculating the mean to help us find the missing values.

First, we will see an example where we can write and solve an equation to find the total number of data values.

Example 2: Using the Formula for the Mean to Find the Number of Data Values

In the soccer world cup, a team scored 32 total goals with a mean of 2 goals per game. How many games did they play in total?

Answer

The mean of a set of data values is equal to thesumofthedatavaluesthetotalnumberofdatavalues.

In this example, the data values are the number of goals scored in each game. Therefore, meanthetotalgoalsscoredthenumberofgames=.

From the question, we know that the mean is 2 and the total number of goals is 32. We will write ๐‘ฅ for the unknown number of games. Substituting these values into the equation for the mean gives 2=32๐‘ฅ2=32รท๐‘ฅ.or

We can solve either of these equations to get the number of games played: ๐‘ฅ=32รท2=16.

Then, the team played 16 games in total.

Next, we will see two examples of finding an unknown data value.

Example 3: Using the Formula for the Mean to Find a Missing Data Value

Given the mean of the values 8, 22, 4, 12, and ๐‘‹ is 15, find the value of ๐‘‹.

Answer

We are told that the mean of the 5 data values is 15, and we know that meanthesumofthedatavaluesthetotalnumberofdatavalues=.

The sum of the data values is 8+22+4+12+๐‘‹=46+๐‘‹.

When we substitute what we know into the equation for the mean, we get 15=46+๐‘‹515=(46+๐‘‹)รท5.or

Using inverse operations, we get that 15ร—5=46+๐‘‹.

Then, after multiplying we get 75=46+๐‘‹ which tells us that ๐‘‹=75โˆ’46=29.

Hence, the unknown value is 29.

Example 4: Using the Formula for the Mean to Find a Missing Data Value

The ages of the people at a gathering were 49, 27, 37, 44, 34, 36, 19, 24, 23, 40, 20, 21, and 43. When one more person joined the gathering, the mean age became 31. How old was the person who joined?

Answer

Let ๐‘ฅ be the unknown age of the person who joined. Then, the data set is the list of ages of all the people at the gathering: 49,27,37,44,34,36,19,24,23,40,20,21,43,๐‘ฅ.

We are told that the mean of their ages is 31, which means that 31=.thesumoftheagesthetotalnumberofpeople

There are 14 people (or 14 data values in the data set) and the sum of their ages is 49+27+37+44+34+36+19+24+23+40+20+21+43+๐‘ฅ=417+๐‘ฅ.

Substituting these values into the formula for the mean tells us that 31=417+๐‘ฅ1431=(417+๐‘ฅ)รท14.or

Using that multiplication is the inverse of division, we get that 31ร—14=417+๐‘ฅ434=417+๐‘ฅ.

Solving this, we find ๐‘ฅ=434โˆ’417=17.

So, the person who joined was 17 years old.

Finally, we will summarize the methods we have learned for using the formula for the mean to answer questions.

How to Use the Formula for the Mean to Find Missing Data Values

When given a question which either asks you to calculate the mean, or asks you to use the mean to calculate other missing data, you can:

  1. Use a variable like ๐‘ฅ or ๐‘ฆ to represent the piece of information you are trying to find.
  2. Put all the information you know into the formula for the mean: meanthesumofthedatavaluesthetotalnumberofdatavalues=.
  3. Solve the equation you have written to find the value of the unknown variable.

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