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# Mean

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

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Arithmetic Mean

#### Objective

In this lesson we will discuss the most common single measure of central tendency, the arithmetic mean, often called the average.

#### Concept

Suppose you just purchased your first car, and you are wondering what kind of gas mileage it gets. If you collect the data in the table below from your first 4 tanks of gas, how could you calculate your average gas mileage? Is the mean gas mileage different if it is calculated as the average of the mileages from each tank instead of calculated using the total miles and total gallons?

Credit: John Liu
Source: https://www.flickr.com/photos/8047705@N02/5560539738
License: CC BY-NC 3.0

 Gallons of Gas 18.7 17.8 16.3 19.1 Miles Driven 363 347 320 402

After the lesson, we’ll return to this question to review the answer.

#### Watch This

http://youtu.be/GzXlIfiZUqg statslectures - Arithmetic Mean for Samples and Populations

#### Guidance

Although there are actually three different relatively common types of mean, or average, the arithmetic mean is far and away the most common method of calculating the “middle” value. The other two, the geometric mean and harmonic mean, are the topics of other lessons.

The arithmetic mean is a very important calculation to master, because it is the first step in a multitude of more complex calculations. Fortunately, the mean is relatively easy to calculate:

1. Calculate the sum of all of the values in your set
2. Divide the sum by the number or count of values in the set
3. The quotient of the sum of the values divided by the number of values is the mean

In statistics, you will work with the arithmetic means of different groupings of units. Most importantly, you will need to know the difference between the population mean and the sample mean. The population mean is the mean of all of the members of an entire population, and the sample mean is the mean only of the members of a sample or subset of a population.

Example A

Calculate the arithmetic mean of the set.

\begin{align*}\begin{array}{c|c c c c c c c c c c c c} 26 & 5\\ 27&5&5\\ 28&0&0&1&1&1&2&4&5&5&8\end{array}\end{align*}

Solution: This set is organized in stem and leaf format, and contains 13 values between 265 and 288.

To find the mean, first calculate the sum of the values in the set:

\begin{align*}265+275+275+280+280+281+281+281+282+284+285+285+288=3,642\end{align*}

Since there are 13 values in the set, the mean is:

\begin{align*}\frac{3,642}{13}=280.154\end{align*}

Rounded to the nearest whole, the arithmetic mean is 280

Example B

Find the arithmetic means of the sets:

1. 261, 286, 257, 284, 258, 281, 258, 285, 267, 275, 258, 284, 260, 285, 258, 284
2. 43, 44, 45, 45, 46, 46, 47, 47, 47, 48, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 49, 50, 50, 50, 51
3. 14, 28, 42, 56, 23, 67, 12, 45, 93, 52

Solution: For each set, find the sum of the values in the set, and divide by the number of values:

a) The sum of the values is 4,341. There are 16 values.

\begin{align*}\frac{4,341}{16}=271.31\end{align*}

b) The sum of the values is 1,198. There are 25 values.

\begin{align*}\frac{1,198}{25}=47.92\end{align*}

c) The sum of the values is 432. There are 10 values.

\begin{align*}\frac{432}{10}=43.2\end{align*}

Example C

Using the data from the chart, find: a) the mean number of prank calls per five-year interval, and b) the mean number of reported prank calls per year.

Solution:

a) First, find the sum of the number of reported calls:

\begin{align*}188+149+161+172+154+172=996 \ calls\end{align*}

The chart indicates that there are six five-year intervals with recorded values. So, dividing the sum by the number of values, we get:

\begin{align*}\frac{996}{6}=166 \ calls \ per \ five \ year \ interval\end{align*}

Note that the value “166” is not the actual count of any of the recorded intervals. The arithmetic mean obviously does not need to be a member of the actual set.

b) The chart spans the years 1970 through 1999 (the end of 1999 is the beginning of 2000), so there are a total of 29 years recorded. Dividing the total number of calls by the number of years yields:

\begin{align*}\frac{996}{29} \approx 34 \ (rounded) \ calls \ per \ year\end{align*}

Concept Problem Revisited

a) If you collect the data in the table below from your first 4 tanks of gas, how could you calculate your average gas mileage?

b) Is the mean gas mileage different if calculated as the average of the mileages from each tank instead of the total miles and total gallons?

 Gallons of Gas 18.7 17.8 16.3 19.1 Miles Driven 363 347 320 402

a) Mileage is measured as miles travelled per gallon, so we can simply total the number of miles and divide by total gallons:

\begin{align*}\frac{1,432 \ miles}{71.9 \ gallons}=19.92 \ avg \ miles \ per \ gallon\end{align*}

b) If we calculate the mean mileage for each tank:

\begin{align*}\frac{363}{18.7}=19.41 \quad \frac{347}{17.8}=19.49 \quad \frac{320}{16.3}=19.63 \quad \frac{402}{19.1}=21.05\end{align*}

and find the mean of the mileages:

\begin{align*}\frac{19.41+19.49+19.63+21.05}{4}=\frac{79.58}{4}=19.90 \ miles \ per \ gallon\end{align*}

we see that the overall mileage appears slightly lower this way.

Can you figure out why this might be?

#### Vocabulary

The arithmetic mean is more commonly known as the average, and is calculated by dividing the sum of a set of values by the count of the number of values.

The geometric mean is another type of ‘middle value’, and is calculated by multiplying the values of each member of a set together and taking the \begin{align*}n^{th}\end{align*} root of the product, where \begin{align*}n\end{align*} is the number of values in the set.

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.

#### Guided Practice

1. Find the mean of the following data:

237, 258, 232, 232, 241, 238, 233, 245, 242, 243, 237, 232, 241, 242, 233

2. Find the mean of the following data:

22, 30, 24, 42, 24, 42, 18, 32, 24, 48, 24, 45, 28, 46, 22, 42

3. Find the mean of the following data:

\begin{align*}\begin{array}{c|c c c c c c c c c c c c c c c c c c c} 19 & 9\\ 20&0&5\\ 21&6&9\\ 22&0&1&1&2&2&4&7&7&8&9\end{array}\end{align*}

Solutions:

1. To find the mean, add all the numbers and divide by the count of numbers you started with.

\begin{align*}&237+258+232+232+241+238+233+245+242+243+237+232+241+242+233= 3586 \\ &=\frac{3586}{15}=239.07 \ (\text{rounded})\end{align*}

2. To find the mean, add all the numbers and divide by the quantity of numbers you started with.

\begin{align*}&22+30+24+42+24+42+18+32+24+48+24+45+28+46+22+42=513 \\ & = \frac{513}{16}=32.06 \ (\text{rounded})\end{align*}

3. To find the mean, add all the numbers and divide by the quantity of numbers you started with.

\begin{align*}&199+200+205+216+219+220+221+221+222+222+224+227+227+228+229=3280 \\ & =\frac{3280}{15}=218.67\end{align*}

#### Practice

Airon collected a sample of the five cheapest (out of a population of 10) different stores’ prices on the new laptop he wanted. His sample was $285,$290, $300,$305, and $315. The other five stores’ prices were$345, $380,$435, $480, and$500.

1. What was the arithmetic sample mean of laptop prices among the sample Airon collected?

2. What was the arithmetic mean of laptops that Airon did not add to his sample?

3. What was the arithmetic population mean of laptop prices?

Questions 4 – 6 refer to the numbers: 4, 7, 2, 5, 7, 8, 4, 9, 5, 6, 2, and 12

4. What is the arithmetic mean of the numbers?

5. What is the arithmetic mean of the odd numbers?

6. What is the arithmetic mean of the even numbers?

Chet’s class has 7 boys, ages 15, 16, 14, 15, 17, 14, and 17 years, and 6 girls, ages 15, 16, 14, 14, 15, and 16 years.

7. What is the arithmetic population mean of Chet’s class?

8. What is the arithmetic mean of girls in Chet’s class?

9. What is the arithmetic mean of boys in Chet’s class?

10. What is the arithmetic sample mean of a sample of students consisting of the youngest and oldest boys and youngest and oldest girls (a total of four students)?

Donna works in the customer service department at a large corporation. The employees in her department earn the following incomes: $24,560,$32,540, $29,540,$39,490, $42,100,$27,000, and $35,750. 11. What is the mean income of the department? 12. What is the mean income above$30,000?

13. What is the mean income below \$30,000?

### Answers for Explore More Problems

To view the Explore More answers, open this PDF file and look for section 5.1.

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### Vocabulary Language: English

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.

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.

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.