# Decimals

## Averages and Probability

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  To find the mean score, he would add them and divide by 
His mean test score is  points.
Did you notice that in the last example, while all the numbers were whole numbers, the mean was  a number with one decimal place? It is customary to report the mean to one more decimal place than the original numbers. In the next example, all the numbers represent money, and it will make sense to report the mean in dollars and cents.

### Find the Median of a Set of Numbers

When Ann, Bianca, Dora, Eve, and Francine sing together on stage, they line up in order of their heights. Their heights, in inches, are shown in .
Dora is in the middle of the group. Her height,  is the median of the girls’ heights. Half of the heights are less than or equal to Dora’s height, and half are greater than or equal. The median is the middle value.
What if Carmen, the pianist, joins the singing group on stage? Carmen is  inches tall, so she fits in the height order between Bianca and Dora. Now the data set looks like this:
There is no single middle value. The heights of the six girls can be divided into two equal parts.
Statisticians have agreed that in cases like this the median is the mean of the two values closest to the middle. So the median is the mean of  The median height is  inches.
Notice that when the number of girls was  the median was the third height, but when the number of girls was  the median was the mean of the third and fourth heights. In general, when the number of values is odd, the median will be the one value in the middle, but when the number is even, the median is the mean of the two middle values.

### Identify the Mode of a Set of Numbers

The average is one number in a set of numbers that is somehow typical of the whole set of numbers. The mean and median are both often called the average.  Yes, it can be confusing when the word average refers to two different numbers, the mean and the median!  In fact, there is a third number that is also an average.  This average is the mode. The mode of a set of numbers is the number that occurs the most. The frequency, is the number of times a number occurs. So the mode of a set of numbers is the number with the highest frequency.
Suppose Jolene kept track of the number of miles she ran since the start of the month, as shown in .
If we list the numbers in order it is easier to identify the one with the highest frequency.
Jolene ran  miles three times, and every other distance is listed only once. So the mode of the data is  miles.
Some data sets do not have a mode because no value appears more than any other. And some data sets have more than one mode. In a given set, if two or more data values have the same highest frequency, we say they are all modes.

### Use the Basic Definition of Probability

The probability of an event tells us how likely that event is to occur. We usually write probabilities as fractions or decimals.
For example, picture a fruit bowl that contains five pieces of fruit - three bananas and two apples.
If you want to choose one piece of fruit to eat for a snack and don’t care what it is, there is a  probability you will choose a banana, because there are three bananas out of the total of five pieces of fruit. The probability of an event is the number of favorable outcomes divided by the total number of outcomes.
Converting the fraction  to a decimal, we would say there is a  probability of choosing a banana.
This basic definition of probability assumes that all the outcomes are equally likely to occur. If you study probabilities in a later math class, you’ll learn about several other ways to calculate probabilities.

### Key Concepts

1. Calculate the mean of a set of numbers.

2. Find the median of a set of numbers.

3. Identify the mode of a set of numbers.


### Practice Makes Perfect

Calculate the Mean of a Set of Numbers 
In the following exercises, find the mean.
Find the Median of a Set of Numbers 
In the following exercises, find the median.
Identify the Mode of a Set of Numbers 
In the following exercises, identify the mode.
Use the Basic Definition of Probability 
In the following exercises, express the probability as both a fraction and a decimal. (Round to three decimal places, if necessary.)

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ After looking at the checklist, do you think you are well prepared for the next section? Why or why not?
