Bill coaches a baseball team. He has been keeping track all season of the total runs scored each game that his team plays. This is the list of data he has collected. Each number represents runs scored by his team in each game.

2, 8, 8, 14, 9, 12, 14, 20, 19, 14

Bill wants to find the average amount of runs scored each game over the entire season.

In this concept, you will learn how to find the mean, or average, of a set of number data.

### Mean

One way to analyze a set of number data is to find the **mean**. Another word for finding the mean is to find the average. An average combines numbers in the data set into one number that best represents the whole set.

There are two steps to finding the mean.

- Add up all of the numbers in the data set.
- Divide the total by the number of items (numbers) in the set.

10, 7, 3, 8, 2

First, add all the numbers together.

\begin{align*}10+3+7+8+2=30\end{align*}

Next, divide the total, 30, by the number of items in the set. There are 5 numbers in the set, so divide 30 by 5.

\begin{align*}30\div 5=6\end{align*}

The mean, or average, of the set is 6.

### Examples

#### Example 1

Earlier, you were given a problem about Bill and his baseball team's scores.

He wants to find the average number of runs his team scored throughout the season.

2, 8, 8, 14, 9, 12, 14, 20, 19, 14

First, he adds up the values in the data set.

\begin{align*}2 + 8 + 8 + 14 + 9 + 12 + 14 + 20 + 19 + 14 &= 120\end{align*}

Next, Bill divides the total by the number of games his team played this season. They played 10 games.

Then, decide if the answer can be kept actual or needs to be rounded. In this case, the answer comes out as a whole number.

The answer is that Bill's team scored an average (or mean) of 12 runs each game.

#### Example 2

Find the average of the data set below.

Jacob has the following quiz scores in his chemistry class for first quarter.

78, 90, 83, 88, 67, 90, 84, 69

What is Jacob's average for the quarter?

First, add up all of the scores.

\begin{align*}78 + 90 + 83 + 88 + 67 + 90 + 84 + 69 = 649\end{align*}

Next, divide by the number of scores (in other words, the number of quizzes Jacob took).

\begin{align*}649 \div 8 = 81.1\end{align*}

Then, look at the quotient and decide if the answer should be a whole number rounded answer, or if it can be left as the exact decimal answer. It depends on the context of the data. In this case, all of Jacob's quiz scores were whole numbers so you can round the answer to a whole number as well.

The answer is that Jacob's average is an 81.

#### Example 3

Find the mean of the following data set.

3, 4, 5, 6, 2, 5, 6, 12, 2

First, add the numbers in the data set together.

\begin{align*}3+4+5+6+2+5+6+12+2=45\end{align*}

Next, divide by the total number of items in the data set.

\begin{align*}45\div 9= 5\end{align*}

Then, write your answer in either whole number or exact decimal form.

The answer is 5.

#### Example 4

Find the mean of the following data set.

22, 11, 33, 44, 66, 76, 88, 86, 4

First, add the numbers in the data set together.

\begin{align*}22+11+33+44+66+76+88+86+4=430\end{align*}

Next, divide by the total number of items in the data set.

\begin{align*}430\div 9= 47.77\end{align*}

Then, write your answer in either whole number or exact decimal form.

The answer is 47.8 or round up to 48.

#### Example 5

Find the mean of the following data set.

37, 123, 234, 567, 321, 909, 909, 900

First, add the numbers in the data set together.

\begin{align*}37+123+234+567+321+909+909+900=4000\end{align*}

Next, divide by the total number of items in the data set.

Then, write your answer in either whole number or exact decimal form.

The answer is 500.

### Review

Find the mean for each set of data. You may round to the nearest tenth when necessary.

- 4, 5, 4, 5, 3, 3
- 6, 7, 8, 3, 2, 4
- 11, 10, 9, 13, 14, 16
- 21, 23, 25, 22, 22, 27
- 27, 29, 29, 32, 30, 32, 31
- 34, 35, 34, 37, 38, 39, 39
- 43, 44, 43, 46, 39, 50
- 122, 100, 134, 156, 144, 110
- 224, 222, 220, 222, 224, 224
- 540, 542, 544, 550, 548, 547
- 762, 890, 900, 789, 780, 645, 700
- 300, 400, 342, 345, 403, 302
- 200, 199, 203, 255, 245, 230, 211
- 1009, 1000, 1200, 1209, 1208, 1217
- 2300, 2456, 2341, 2400, 2541, 2321

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.18.