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11.1: Mean

Difficulty Level: At Grade Created by: CK-12
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This summer, Marco will wash cars five days a week to earn some money. He has worked one week already. Marco washed three cars on Wednesday, one car on Thursday, four cars on Friday, five cars on Saturday and two cars on Sunday. Marco gets paid $5 for every car he washes, and wants to know how much money he will have after eight weeks. Marco thinks he can wash the same average number of cars each week. How can Marco figure out how much money he will earn in total?

In this concept, you will learn how to calculate mean.

Calculating Mean

Data is a set of numerical or non-numerical information. Data can be analyzed in many different ways. In this concept we will analyze numerical data using the mean.

Mean is a numerical value that measures the spread of a data set. The mean is one way to determine the central or typical value of the set. Mean is also called the average.

To calculate the mean, first sum the data values in the set and then divide by the number of values in the set.

Let's look at an example.

Find the mean for the set of data: 47, 56, 51, 45, and 41.

First, add the data values to calculate the sum.

\begin{align*}47+56+51+45+41=240\end{align*}

Second, divide the sum by the number of values in the set. Since there are five values in the data set, you will divide by 5.

\begin{align*}240\div 5=48\end{align*}

The answer is the mean is 48.

Examples

Example 1

Earlier, you were given a problem about Marco and his dirty cars.

This summer, Marco is washing cars 5 days a week to earn money. Marco has kept track of the number of cars he washed in the first week. These values are his data. Marco organized his data in the following table.

Marco's Car Wash
Day Number of Cars Washed
Wednesday 3
Thursday 1
Friday 4
Saturday 5
Sunday 2

Marco wants to know how much money he will have after 8 weeks of washing cars. Marco gets paid $5 for each car he washes. If Marco washes the same average number of cars each week, how much money will he earn?

First, find the average number of cars washed in the first week. Then, use the average to calculate how much money Marco will earn.

To find the average number of cars Marco washed in the first week, add the number of cars washed.

\begin{align*}3+1+4+5+2=15\end{align*}
Next, divide by the number of days, 5.

\begin{align*}15\div 5=3\end{align*} 
The average number of cars Marco washed in a day is 3.

Next, calculate the average number of cars Marco washes in a week. Since Marco works 5 days a week, multiply 3 (the average number of cars washed per day) by 5 (days worked in a week).

\begin{align*}3\times 5=15\end{align*}

The average number of cars Marco washes in a week is 15.

Next, calculate the average number of cars Marco will wash in 8 weeks. To do this, multiply 15 (average number of cars washed per week) by 8 (number of weeks worked).  

\begin{align*}15\times 8=120\end{align*} 

The average number of cars Marco will wash in 8 weeks is 120.

Next, calculate how much money Marco will make in 8 weeks. To do this, multiply 120 (average number of cars washed in 8 weeks) by $5 (amount earned per car washed).

\begin{align*}1200\times 8=600\end{align*}
The answer is Marco will earn $600.

Example 2

West Elementary School has four classes of fifth grade students. Mr. Hernandez has 23 students, Mrs. Johnson has 26 students, Ms. Ruiz has 25 students, and Mr. Smith has 21. What is the average number of fifth grade students in each class?

Remember that to calculate the average is the same as calculating the mean. You must follow the same steps. First, find the sum of the fifth grade students. To do this, add the number of students in each class.

\begin{align*}24+26+25+21=96\end{align*}

Next, divide by the number of classes. In this case there are 4 classes.

\begin{align*}96\div 4=24\end{align*}

The answer is the average number of fifth grade students in each class is 24.

Example 3

Find the average of the following data set.

\begin{align*}11, 13, 13, 15, 16, 22, 24, 25, 30, 32\end{align*}

First, add the values.

\begin{align*}11+13+14+15+16+22+24+25+30+32=202\end{align*} 
Second, divide by the number of values in the set. In this set there are 10, so divide by 10. 

\begin{align*}202\div 10=20.2\end{align*}

This average in this set is 20.2. The average, or mean, is not always a whole number.
The answer is the average is 20.2.

Example 4

John wants to know his average quiz grade based on the following quiz scores.

\begin{align*}78, 90, 83, 88, 67, 90, 84, 69, 56\end{align*}

First, add the values.

\begin{align*}78+90+83+88+67+90+84+69+56=705\end{align*}

Second, divide the sum by the number of values in the set. In this set there are 9, so you divide by 9. 

\begin{align*}705\div 9=78.3... \approx 78.3\end{align*}

Notice the average, or mean, is not a whole number. In this problem, round the answer to the nearest tenth. 
The answer is John's average quiz grade is 78.3.

Example 5

The chart below shows the daily temperature in San Diego for the first seven days in August. Calculate the mean temperature.

Daily Temperatures for August
Date Temperature (\begin{align*}^o\end{align*}F)
Sunday 8/1 88
Monday 8/2 83
Tuesday 8/3 87
Wednesday 8/4 89
Thursday 8/5 82
Friday 8/6 79
Saturday 8/7 87

First, calculate the sum of the data values.

\begin{align*}88+83+87+89+82+79+87=595\end{align*}

Second, divide the sum by the number of data values in the set. Since there are seven values in this set, you will divide by 7.

\begin{align*}595\div 7=85\end{align*}

The answer is the mean temperature for the first week in August was \begin{align*}85^oF\end{align*}.

Review

Find the mean for each set of data. Round to the nearest hundredth when necessary.

  1. 4, 5, 4, 5, 3, 3, 6, 7, 8
  2. 6, 7, 8, 3, 2, 4, 9, 10, 11, 12
  3. 11, 10, 9, 13, 14, 16, 20, 22, 22
  4. 21, 23, 25, 22, 22, 27, 18, 20
  5. 27, 29, 29, 32, 30, 32, 31
  6. 34, 35, 34, 37, 38, 39, 39
  7. 43, 44, 43, 46, 39, 50
  8. 122, 100, 134, 156, 144, 110, 120, 123, 130
  9. 224, 222, 220, 222, 224, 224
  10. 540, 542, 544, 550, 548, 547
  11. 762, 890, 900, 789, 780, 645, 700
  12. 300, 400, 342, 345, 403, 302
  13. 200, 199, 203, 255, 245, 230, 211
  14. 1009, 1000, 1200, 1209, 1208, 1217
  15. 2300, 2456, 2341, 2400, 2541, 2321

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.1.

Resources

 

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Vocabulary

Average

The arithmetic mean is often called the average.

Data

Data is information that has been collected to represent real life situations, usually in number form.

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.

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Date Created:
Dec 21, 2012
Last Modified:
Sep 08, 2016
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