Averages and Probability

5.5 Averages and Probability

The topics covered in this section are:

  1. Calculate the mean of a set of numbers
  2. Find the median of a set of numbers
  3. Find the mode of a set of numbers
  4. Apply the basic definition of 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.

5.5.1 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 $3$.

$\large \frac{85+88+94}{3}$

$\large \frac{267}{3}$

$89$

His mean test score is $89$ points.

THE MEAN

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

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

HOT TO: Calculate the mean of a set of numbers.

  1. Write the formula for the mean
    • $\large \mathrm{mean} = \frac{\mathrm{sum\ of\ values\ in\ data\ set}}{n}$
  2. Find the sum of all the values in the set. Write the sum in the numerator.
  3. Count the number, 𝑛,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.

Example 1

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

Solution

Write the formula for the mean:$\mathrm{mean} = \frac{\mathrm{sum\ of\ values\ in\ data\ set}}{n}$
Write the sum of the numbers in the numerator.$\mathrm{mean} = \frac{8+12+15+9+6}{n}$
Count how many numbers are in the set. There are $5$ numbers in the set, so $n=5$.$\mathrm{mean} = \frac{8+12+15+9+6}{5}$
Add the numbers in the numerator.$\mathrm{mean} = \frac{50}{5}$
Then divide.$\mathrm{mean} = 10$
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.

Solution

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

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

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 2

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.

Solution

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

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$.

5.5.2 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 the table below.

AnnBiancaDoraEveFrancine
$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.

MEDIAN

The median of a set of data values is the middle value.

  • Half the data values are less than or equal to the median.
  • Half the data values are greater than or equal to 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.

HOW TO: Find the median of a set of numbers.

  1. Step 1. List the numbers from smallest to largest.
  2. Step 2. Count how many numbers are in the set. Call this $n$.
  3. Step 3. Is $n$ odd or even?
    • If $n$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.

Example 3

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

Solution

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 $n$.$n=7$
Is $n$ odd or even?odd
The median is the middle value..
The middle is the number in the $4$th position.So the median of the data is $13$.

Example 4

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.

Solution

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 $n$.$n=10$
Is $n$ odd or even?even
The median is the mean of the two middle values, the $5$th and the $6$th numbers..
Find the mean of $85$ and $86$.$\mathrm{mean} = \frac{85+86}{2}$
$\mathrm{mean} = 85.5$
Kirsten’s median score is $85.5$.

5.5.3 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.

MODE

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 the figure below.

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.

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.

HOW TO: 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.

Example 5

The ages of students in a college math class are listed below. Identify the mode. $18,18,18,18,19,19,19,20,20,20,20,20,20,20,21,21,22,22,22,22,22,23,24,24,25,29,30,40,44$.

Solution

The ages are already listed in order. We will make a table of frequencies to help identify the age with the highest frequency.

A table is shown with 2 rows. The first row is labeled 'Age' and lists the values: 18, 19, 20, 21, 22, 23, 24, 25, 29, 30, 40, and 44. The second row is labeled 'Frequency' and lists the values: 4, 3, 7, 2, 5, 1, 2, 1, 1, 1, 1, and 1.

Now look for the highest frequency. The highest frequency is $7$, which corresponds to the age $20$. So the mode of the ages in this class is $20$ years.

Example 6

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$
Solution

List each number with its frequency.

A table is shown with 2 rows. The first row is labeled “Number” and lists the values: 56, 60, 61, 62, 63, 64, 65, 66, 67, 70, and 74. The second row is labeled “Frequency” and lists the values: 1, 2, 2, 1, 5, 4, 3, 5, 6, 1, and 1.

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.

5.5.4 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.'

PROBABILTY

The probability of an event is the number of favorable outcomes divided by the total number of outcomes possible.

$\large \mathrm{Probability} = \frac{\mathrm{number\ of\ favorable\ outcomes}}{\mathrm{total\ number\ of\ outcomes}}$

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

$\large \mathrm{Probability\ of\ choosing\ a\ banana} = \frac{3}{5}$

$\large \mathrm{Probability\ of\ choosing\ a\ banana} = 0.6$

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 7

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.

  1. Find the probability she will win the prize.
  2. Convert the fraction to a decimal.
Solution

Part 1.
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.$\mathrm{Probability\ of\ an\ event} = \frac{\mathrm{number\ of\ favorable\ outcomes}}{\mathrm{total\ number\ of\ outcomes}}$
Substitute into the numerator and denominator.$\mathrm{Probability\ Cherie\ wins} = \frac{1}{100}$
Part 2.
Convert the fraction to a decimal.
Write the probability as a fraction.$\mathrm{Probability} = \frac{1}{100}$
Convert the fraction to a decimal.$\mathrm{Probability} = 0.01$

Example 8

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

  1. Find the probability the job is offered to a women.
  2. Convert the fraction to a decimal.
Solution

Part 1.
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.$\mathrm{Probability\ of\ an\ event} = \frac{\mathrm{number\ of\ favorable\ outcomes}}{\mathrm{total\ number\ of\ outcomes}}$
Substitute into the numerator and denominator.$\mathrm{Probability} = \frac{3}{8}$
Part 2.
Convert the fraction to a decimal.
Write the probability as a fraction.$\mathrm{Probability} = \frac{3}{8}$
Convert the fraction to a decimal.$\mathrm{Probability} = 0.375$
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