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# Mean

## Sum of the values divided by the number of values

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Mean

### Let's Think About It

License: CC BY-NC 3.0

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.

### Guidance

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.

1. Add up all of the numbers in the data set.
2. Divide the total by the number of items (numbers) in the set.
Here is an example. This is a small data set to find the average.

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.

### Guided Practice

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.

### Examples

#### Example 1

Find the mean of the following data:

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 2

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 3

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.

\begin{align*}4000\div 8=500\end{align*}

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

The answer is 500.

### Follow Up

License: CC BY-NC 3.0

Remember 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.

\begin{align*}120\div 10= 12\end{align*}

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.

### Explore More

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

1. 4, 5, 4, 5, 3, 3

2. 6, 7, 8, 3, 2, 4

3. 11, 10, 9, 13, 14, 16

4. 21, 23, 25, 22, 22, 27

5. 27, 29, 29, 32, 30, 32, 31

6. 34, 35, 34, 37, 38, 39, 39

7. 43, 44, 43, 46, 39, 50

8. 122, 100, 134, 156, 144, 110

9. 224, 222, 220, 222, 224, 224

10. 540, 542, 544, 550, 548, 547

11. 762, 890, 900, 789, 780, 645, 700

12. 300, 400, 342, 345, 403, 302

13. 200, 199, 203, 255, 245, 230, 211

14. 1009, 1000, 1200, 1209, 1208, 1217

15. 2300, 2456, 2341, 2400, 2541, 2321

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 2.18.

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### Vocabulary Language: English Spanish

mean

The mean, often called the average, of a numerical set of data is simply the sum of the data values divided by the number of values.

measures of central tendency

The mean, median, and mode are known as the measures of central tendency.

Average

The arithmetic mean is often called the average.

Geometric mean

The geometric mean is a method of finding the ‘middle’ value in a set that contains some values that are intrinsically more influential than others.

Harmonic mean

A harmonic mean is calculated by dividing the number of values in the set by the sum of the inverses of the values in the set.

Population Mean

The population mean is the mean of all of the members of an entire population.

Sample Mean

A sample mean is the mean only of the members of a sample or subset of a population.

weighted

A weighted value or set of values takes into account varying levels of importance among members of the set.

weighted average

A weighted average is an average that multiplies each component by a factor representing its frequency or probability.

weighted harmonic mean

A weighted harmonic mean is a harmonic mean of values with varying frequencies or weights.

### Image Attributions

1. [1]^ License: CC BY-NC 3.0
2. [2]^ License: CC BY-NC 3.0

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