5.5 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 $85, 88, and 94.$ To find the mean score, he would add them and divide by $3 .$

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

His mean test score is $89$ points.

Example 1

Find the mean of the numbers $8, 12, 15, 9, and 6.$

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

Example 2

The ages of the members of a family who got together for a birthday celebration were $16, 26, 53, 56, 65, 70, 93, and 97$ years. Find the mean age.

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

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

Did you notice that in the last example, while all the numbers were whole numbers, the mean was $59.5,$ 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.

Example 3

For the past four months, Daisy’s cell phone bills were $$42.75, $50.12, $41.54, $48.15.$ Find the mean cost of Daisy’s cell phone bills.

Write the formula for the mean.	$mean = \frac{sum of all the numbers}{n}$
Count how many numbers are in the set. Call this $n$ and write it in the denominator.	$mean = \frac{sum of all the numbers}{4}$
Write the sum of all the numbers in the numerator.	$mean = \frac{42.75 + 50.12 + 41.54 + 48.15}{4}$
Simplify the fraction.	$mean = \frac{182.56}{4}$
	$mean equals 45.64$

Does $$45.64$ seem ‘typical’ of this set of numbers? Yes, it is neither less than $$41.54$ nor greater than $$50.12.$

The mean cost of her cell phone bill was $$45.64$

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 Table 5.7.

**Table 5.7**
Ann	Bianca	Dora	Eve	Francine
$59$	$60$	$65$	$68$	$70$

Dora is in the middle of the group. Her height, $65 ″,$ 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.

The numbers 59, 60, 65, 68, and 70 are listed. 59 and 60 have a brace beneath them and in red are labeled “2 below.” 68 and 70 have a brace beneath them and in red are labeled “2 above.” 65 has an arrow pointing to it and is labeled as the median.

What if Carmen, the pianist, joins the singing group on stage? Carmen is $62$ inches tall, so she fits in the height order between Bianca and Dora. Now the data set looks like this:

$59, 60, 62, 65, 68, 70$

There is no single middle value. The heights of the six girls can be divided into two equal parts.

The numbers 59, 60, and 62 are listed, followed by a blank space, then 65, 68, and 70.

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 $62 and 65, \frac{62 + 65}{2} .$ The median height is $63.5$ inches.

The numbers 9, 11, 12, 13, 15, 18, and 19 are listed. 9, 11, and 12 have a brace beneath them and are labeled “3 below.” 15, 18, and 19 have a brace beneath them and are labeled “3 above.” 13 has an arrow pointing to it and is labeled as the median.

Notice that when the number of girls was $5,$ the median was the third height, but when the number of girls was $6,$ 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.

Example 4

Find the median of $12, 13, 19, 9, 11, 15, and 18.$

List the numbers in order from smallest to largest.	9, 11, 12, 13, 15, 18, 19
Count how many numbers are in the set. Call this $n$ .	$n equals 7$
Is $n$ odd or even?	odd
The median is the middle value.
The middle is the number in the 4th position.	So the median of the data is 13.

Example 5

Kristen received the following scores on her weekly math quizzes:

$83, 79, 85, 86, 92, 100, 76, 90, 88, and 64.$ Find her median score.

Find the median of 83, 79, 85, 86, 92, 100, 76, 90, 88, and 64.
List the numbers in order from smallest to largest.	64, 76, 79, 83, 85, 86, 88, 90, 92, 100
Count the number of data values in the set. Call this $n.$	$n equals 10$
Is $n$ odd or even?	even
The median is the mean of the two middle values, the 5th and 6th numbers.
Find the mean of 85 and 86.	$mean = \frac{85 + 86}{2}$
	$mean equals 85.5$
	Kristen's median score is 85.5.

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 Figure 5.11.

An image of a calendar is shown. On Thursday the first, labeled New Year's Day, is written 2 mi. On Saturday the third is written 15 mi. On the 4th, 8 mi. On the 6th, 3 mi. On the 7th, 8 mi. On the 9th, 5 mi. On the 10th, 8 mi. — **Figure 5.11**

If we list the numbers in order it is easier to identify the one with the highest frequency.

$2, 3, 5, 8, 8, 8, 15$

Jolene ran $8$ miles three times, and every other distance is listed only once. So the mode of the data is $8$ miles.

Example 7

The data lists the heights (in inches) of students in a statistics class. Identify the mode.

$56$	$61$	$63$	$64$	$65$	$66$	$67$	$67$
$60$	$62$	$63$	$64$	$65$	$66$	$67$	$70$
$60$	$63$	$63$	$64$	$66$	$66$	$67$	$74$
$61$	$63$	$64$	$65$	$66$	$67$	$67$

List each number with its frequency.

Figure 5.13

Now look for the highest frequency. The highest frequency is $6,$ which corresponds to the height $67$ inches. So the mode of this set of heights is $67$ inches.

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 $\frac{3}{5}$ 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.

Two equations are shown. The top equation says the probability of an event equals the number of favorable outcomes over the total number of outcomes. The bottom equation says the probability of choosing a banana equals 3 over 5. There is a blue arrow pointing to the 3 with the text, 'There are 3 bananas.' There is a blue arrow pointing to the 5 with the text, 'There are 5 pieces of fruit.'

Converting the fraction $\frac{3}{5}$ to a decimal, we would say there is a $0.6$ probability of choosing a banana.

$\begin{matrix} Probability of choosing a banana = \frac{3}{5} \\ Probability of choosing a banana = 0.6 \end{matrix}$

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.

Example 8

The ski club is holding a raffle to raise money. They sold $100$ tickets. All of the tickets are placed in a jar. One ticket will be pulled out of the jar at random, and the winner will receive a prize. Cherie bought one raffle ticket.

ⓐ Find the probability she will win the prize.

ⓑ Convert the fraction to a decimal.

ⓐ
What are you asked to find?	The probability Cherie wins the prize.
What is the number of favorable outcomes?	1, because Cherie has 1 ticket.
Use the definition of probability.	$Probability of an event = \frac{number of favorable outcomes}{total number of outcomes}$
Substitute into the numerator and denominator.	$Probability Cherie wins = \frac{1}{100}$

ⓑ
Convert the fraction to a decimal.
Write the probability as a fraction.	$Probability = \frac{1}{100}$
Convert the fraction to a decimal.	$Probability equals 0.01$

Example 9

Three women and five men interviewed for a job. One of the candidates will be offered the job.

ⓐ Find the probability the job is offered to a woman.

ⓑ Convert the fraction to a decimal.

ⓐ
What are you asked to find?	The probability the job is offered to a woman.
What is the number of favorable outcomes?	3, because there are three women.
What are the total number of outcomes?	8, because 8 people interviewed.
Use the definition of probability.	$Probability of an event = \frac{number of favorable outcomes}{total number of outcomes}$
Substitute into the numerator and denominator.	$Probability = \frac{3}{8}$

ⓑ
Convert the fraction to a decimal.
Write the probability as a fraction.	$Probability = \frac{3}{8}$
Convert the fraction to a decimal.	$Probability equals 0.375$

Key Concepts

Calculate the mean of a set of numbers.
1. Write the formula for the mean $mean = \frac{sum of values in data set}{n}$
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 median of a set of numbers.
1. List the numbers from least to greatest.
2. Count how many numbers are in the set. Call this $n$ .
3. Is $n$ odd or even?
  If $n$ is an odd number, the median is the middle value.
  If $n$ is an even number, the median is the mean of the two middle values
Identify the mode of a set of numbers.
1. List the data values in numerical order.
2. Count the number of times each value appears.
3. The mode is the value with the highest frequency.

Practice Makes Perfect

Calculate the Mean of a Set of Numbers

In the following exercises, find the mean.

$3$ , $8$ , $2$ , $2$ , $5$

$6$ , $1$ , $9$ , $3$ , $4$ , $7$

$65$ , $13$ , $48$ , $32$ , $19$ , $33$

$34$ , $45$ , $29$ , $61$ , and $41$

$202$ , $241$ , $265$ , $274$

245.5

$525$ , $532$ , $558$ , $574$

$12.45$ , $12.99$ , $10.50$ , $11.25$ , $9.99$ , $12.72$

11.65

$28.8$ , $32.9$ , $32.5$ , $27.9$ , $30.4$ , $32.5$ , $31.6$ , $32.7$

Four girls leaving a mall were asked how much money they had just spent. The amounts were $$0$ , $$14.95$ , $$35.25$ , and $$25.16$ . Find the mean amount of money spent.

$18.84

Juan bought $5$ shirts to wear to his new job. The costs of the shirts were $$32.95$ , $$38.50$ , $$30.00$ , $$17.45$ , and $$24.25$ . Find the mean cost.

The number of minutes it took Jim to ride his bike to school for each of the past six days was $21$ , $18$ , $16$ , $19$ , $24$ , and $19$ . Find the mean number of minutes.

19.5 minutes

Norris bought six books for his classes this semester. The costs of the books were $$74.28$ , $$120.95$ , $$52.40$ , $$10.59$ , $$35.89$ , and $$59.24$ . Find the mean cost.

The top eight hitters in a softball league have batting averages of $.373$ , $.360$ , $.321$ , $.321$ , $.320$ , $.312$ , $.311$ , and $.311$ . Find the mean of the batting averages. Round your answer to the nearest thousandth.

0.329

The monthly snowfall at a ski resort over a six-month period was $60.3,$ $79.7,$ $50.9,$ $28.0,$ $47.4,$ and $46.1$ inches. Find the mean snowfall.

Find the Median of a Set of Numbers

In the following exercises, find the median.

$24$ , $19$ , $18$ , $29$ , $21$

$48$ , $51$ , $46$ , $42$ , $50$

$65$ , $56$ , $35$ , $34$ , $44$ , $39$ , $55$ , $52$ , $45$

$121$ , $115$ , $135$ , $109$ , $136$ , $147$ , $127$ , $119$ , $110$

$4$ , $8$ , $1$ , $5$ , $14$ , $3$ , $1$ , $12$

4.5

$3$ , $9$ , $2$ , $6$ , $20$ , $3$ , $3$ , $10$

$99.2$ , $101.9$ , $98.6$ , $99.5$ , $100.8$ , $99.8$

99.65

$28.8$ , $32.9$ , $32.5$ , $27.9$ , $30.4$ , $32.5$ , $31.6$ , $32.7$

Last week Ray recorded how much he spent for lunch each workday. He spent $$6.50$ , $$7.25$ , $$4.90$ , $$5.30$ , and $$12.00$ . Find the median.

$6.50

Michaela is in charge of 6 two-year olds at a daycare center. Their ages, in months, are $25$ , $24$ , $28$ , $32$ , $29$ , and $31$ . Find the median age.

Brian is teaching a swim class for $6$ three-year olds. Their ages, in months, are $38, 41, 45, 36, 40, and 42.$ Find the median age.

40.5 months

Sal recorded the amount he spent for gas each week for the past $8$ weeks. The amounts were $$38.65,$ $$32.18,$ $$40.23,$ $$51.50,$ $$43.68,$ $$30.96,$ $$41.37,$ and $$44.72.$ Find the median amount.

Identify the Mode of a Set of Numbers

In the following exercises, identify the mode.

$2$ , $5$ , $1$ , $5$ , $2$ , $1$ , $2$ , $3$ , $2$ , $3$ , $1$

$8$ , $5$ , $1$ , $3$ , $7$ , $1$ , $1$ , $7$ , $1$ , $8$ , $7$

$18$ , $22$ , $17$ , $20$ , $19$ , $20$ , $22$ , $19$ , $29$ , $18$ , $23$ , $25$ , $22$ , $24$ , $23$ , $22$ , $18$ , $20$ , $22$ , $20$

$42$ , $28$ , $32$ , $35$ , $24$ , $32$ , $48$ , $32$ , $32$ , $24$ , $35$ , $28$ , $30$ , $35$ , $45$ , $32$ , $28$ , $32$ , $42$ , $42$ , $30$

The number of children per house on one block: $1$ , $4$ , $2$ , $3$ , $3$ , $2$ , $6$ , $2$ , $4$ , $2$ , $0$ , $3$ , $0 .$

2 children

The number of movies watched each month last year: $2$ , $0$ , $3$ , $0$ , $0$ , $8$ , $6$ , $5$ , $0$ , $1$ , $2$ , $3 .$

The number of units being taken by students in one class: $12$ , $5$ , $11$ , $10$ , $10$ , $11$ , $5$ , $11$ , $11$ , $11$ , $10$ , $12$ .

11 units

The number of hours of sleep per night for the past two weeks: $8$ , $5$ , $7$ , $8$ , $8$ , $6$ , $6$ , $6$ , $6$ , $9$ , $7$ , $8$ , $8$ , $8$ .

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.)

Josue is in a book club with $20$ members. One member is chosen at random each month to select the next month’s book. Find the probability that Josue will be chosen next month.

$\frac{1}{20}, 0.05$

Jessica is one of eight kindergarten teachers at Mandela Elementary School. One of the kindergarten teachers will be selected at random to attend a summer workshop. Find the probability that Jessica will be selected.

There are $24$ people who work in Dane’s department. Next week, one person will be selected at random to bring in doughnuts. Find the probability that Dane will be selected. Round your answer to the nearest thousandth.

$\frac{1}{24}, 0.041 \bar{6} \approx 0.042$

Monica has two strawberry yogurts and six banana yogurts in her refrigerator. She will choose one yogurt at random to take to work. Find the probability Monica will choose a strawberry yogurt.

Michel has four rock CDs and six country CDs in his car. He will pick one CD to play on his way to work. Find the probability Michel will pick a rock CD.

$\frac{2}{5}, 0.4$

Noah is planning his summer camping trip. He can’t decide among six campgrounds at the beach and twelve campgrounds in the mountains, so he will choose one campground at random. Find the probability that Noah will choose a campground at the beach.

Donovan is considering transferring to a $4-year college.$ He is considering $10$ out-of state colleges and $4$ colleges in his state. He will choose one college at random to visit during spring break. Find the probability that Donovan will choose an out-of-state college.

$\frac{5}{7}, 0 . \overset{———}{714285} \approx 0.714$

There are $258,890,850$ number combinations possible in the Mega Millions lottery. One winning jackpot ticket will be chosen at random. Brent chooses his favorite number combination and buys one ticket. Find the probability Brent will win the jackpot. Round the decimal to the first digit that is not zero, then write the name of the decimal.

Everyday Math

Joaquin gets paid every Friday. His paychecks for the past $8$ Fridays were $$315,$ $$236.25,$ $$236.25,$ $$236.25, $315,$ $$315,$ $$236.25,$ $$393.75.$ Find the ⓐ mean, ⓑ median, and ⓒ mode.

ⓐ $285.47
ⓑ $275.63
ⓒ $236.25

The cash register receipts each day last week at a coffee shop were $$1,845,$ $$1,520,$ $$1,438,$ $$1,682,$ $$1,850,$ $$2,721,$ $$2,539.$ Find the ⓐ mean, ⓑ median, and ⓒ mode.

Writing Exercises

Explain in your own words the difference between the mean, median, and mode of a set of numbers.

Answers will vary.

Make an example of probability that relates to your life. Write your answer as a fraction and explain what the numerator and denominator represent.

Self Check

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

A self-assessment table for statistical skills, including calculating mean, median, mode, and using probability, with options to mark confidence levels: Confidently, With some help, or No-I don't get it!. — **Figure 5.14**

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