# 5.5 Averages and probability

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By the end of this section, you will be able to:
• Calculate the mean of a set of numbers
• Find the median of a set of numbers
• Find the mode of a set of numbers
• Apply the basic definition of probability

Before you get started, take this readiness quiz.

1. Simplify: $\frac{4+9+2}{3}.$
If you missed this problem, review Multiply and Divide Mixed Numbers and Complex Fractions .
2. Simplify: $4\left(8\right)+6\left(3\right).$
If you missed this problem, review Use the Language of Algebra .
3. Convert $\frac{5}{2}$ to a decimal.
If you missed this problem, review Multiply and Divide Mixed Numbers and Complex Fractions .

One application of decimals that arises often is finding the average of a set of numbers. What do you think of when you hear the word average ? Is it your grade point average, the average rent for an apartment in your city, the batting average of a player on your favorite baseball team? The average is a typical value in a set of numerical data. Calculating an average sometimes involves working with decimal numbers. In this section, we will look at three different ways to calculate an average.

## Calculate the mean of a set of numbers

The mean    is often called the arithmetic average. It is computed by dividing the sum of the values by the number of values. Students want to know the mean of their test scores. Climatologists report that the mean temperature has, or has not, changed. City planners are interested in the mean household size.

Suppose Ethan’s first three test scores were $85,88,\text{and}\phantom{\rule{0.2em}{0ex}}94.$ To find the mean score, he would add them and divide by $3.$

$\begin{array}{}\\ \frac{85+88+94}{3}\\ \frac{267}{3}\\ 89\end{array}$

His mean test score is $89$ points.

## The mean

The mean    of a set of $n$ numbers is the arithmetic average of the numbers.

$\text{mean}=\frac{\text{sum of values in data set}}{n}$

## Calculate the mean of a set of numbers.

1. Write the formula for the mean
$\text{mean}=\frac{\text{sum of values in data set}}{n}$
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 mean of the numbers $8,12,15,9,\text{and}\phantom{\rule{0.2em}{0ex}}6.$

## Solution

 Write the formula for the mean: $\text{mean}=\frac{\text{sum of all the numbers}}{n}$ Write the sum of the numbers in the numerator. $\text{mean}=\frac{8+12+15+9+6}{n}$ Count how many numbers are in the set. There are 5 numbers in the set, so $n=5$ . $\text{mean}=\frac{8+12+15+9+6}{5}$ Add the numbers in the numerator. $\text{mean}=\frac{50}{5}$ Then divide. $\text{mean}=10$ Check to see that the mean is 'typical': 10 is neither less than 6 nor greater than 15. The mean is 10.

Find the mean of the numbers: $8,9,7,12,10,5.$

8.5

Find the mean of the numbers: $9,13,11,7,5.$

9

The ages of the members of a family who got together for a birthday celebration were $16,26,53,56,65,70,93,\text{and}\phantom{\rule{0.2em}{0ex}}97$ years. Find the mean age.

## Solution

 Write the formula for the mean: $\text{mean}=\frac{\text{sum of all the numbers}}{n}$ Write the sum of the numbers in the numerator. $\text{mean}=\frac{16+26+53+56+65+70+93+97}{n}$ Count how many numbers are in the set. Call this $n$ and write it in the denominator. $\text{mean}=\frac{16+26+53+56+65+70+93+97}{8}$ Simplify the fraction. $\text{mean}=\frac{476}{8}$ $\text{mean}=59.5$

Is $59.5$ ‘typical’? Yes, it is neither less than $16$ nor greater than $97.$ The mean age is $59.5$ years.

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