5.1: Mean
Your uncle is thinking of putting his house up for sale. He asks you to do some research to find out what prices home sold for in his neighborhood. You find some recent sold prices of $350,000; $475,000; $468,000; $550,000; $378,000 and $389,000. If his house is comparable to the others in the list, what might be a good listing price for his house? What would you recommend to him? What other factors might you consider before giving him advice?
Watch This
First watch this video to learn about the mean.
CK-12 Foundation: Chapter5MeanA
Then watch this video to see some examples.
CK-12 Foundation: Chapter5MeanB
Watch this video for more help.
Khan Academy Sample vs. Population Mean
Guidance
Now that you have had some fun discovering what you are finding when you are looking for the mean of a set of data, it is time to actually calculate the mean of your handfuls of blocks.
The term central tendency refers to the middle, or typical, value of a set of data, which is most commonly measured by using the 3 m’s\begin{align*}-\end{align*}mean, median and mode. The mean, median, and mode are known as the measures of central tendency. In this concept, we will explore the mean, and then we will move on to the median and the mode in the following concepts.
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. This is also referred to as the arithmetic mean. The mean is the balance point of a distribution.
To calculate the actual mean of your handfuls of blocks, you can use the numbers that were posted on your grid paper. These posted numbers represent the number of blocks that were picked by each student in your class. Therefore, you are calculating the mean of a population, which is a collection of all elements whose characteristics are being studied. You are not calculating the mean number of some of the blocks, but you are calculating the mean number of all of the blocks. We will use the example below for our calculations:
From the grid paper, you can see that there were 30 students who posted their numbers of blocks. The total number of blocks picked by all the students can be calculated as follows:
\begin{align*}& 1 \times 2 + 2 \times 3 + 3 \times 4 + 5 \times 5 + 3 \times 6 +4 \times 7 + 3 \times 8 + 2 \times 9 + 3 \times 10 + 3 \times 11 + 1 \times 12\\ & 2+6+12+25+18+28+24+18+30+33+12=208\end{align*}
The sum of all the blocks is 208, and the mean is the number you get when you divide the sum by the number of students who placed a post-it-note on the grid paper. The mean number of blocks is, therefore, \begin{align*}\frac{208}{30} \approx 6.93\end{align*}. This means that, on average, each student picked 7 blocks from the pail.
When calculations are done in mathematics, formulas are often used to represent the steps that are being applied. The symbol \begin{align*}\sum\end{align*} means “the sum of“ and is used to represent the addition of numbers. The numbers in every question are different, so the variable \begin{align*}x\end{align*} is used to represent the numbers. To make sure that all the numbers are included, a subscript is often used to name the numbers. Therefore, the first number in the example can be represented as \begin{align*}x_1\end{align*}. The number of data values for a population is written as \begin{align*}N\end{align*}. The mean of the population is denoted by the symbol \begin{align*}\mu\end{align*}, which is pronounced "mu." The following formula represents the steps that are involved in calculating the mean of a set of data:
\begin{align*}\text{Mean} = \frac{\text{sum of the values}}{\text{the number of values}}\end{align*}
This formula can also be written using symbols:
\begin{align*}\mu=\frac{\sum x_1+x_2+x_3+ \ldots + x_n}{N}\end{align*}
You can now use the formula to calculate the mean number of blocks per student:
\begin{align*}\mu &= \frac{\sum x_1+x_2+x_3+ \ldots + x_n}{N}\\ \mu &= \frac{2+6+12+25+18+28+24+18+30+33+12}{30}\\ \mu &= \frac{208}{30}\\ \mu &\approx 6.93\end{align*}
This means that, on average, each student picked 7 blocks from the pail.
Example A
Stephen has been working at Wendy’s for 15 months. The following numbers are the number of hours that Stephen worked at Wendy’s for each of the past 7 months:
\begin{align*}24, 24, 31, 50, 53, 66, 78\end{align*}
What is the mean number of hours that Stephen worked each month?
Step 1: Add the numbers to determine the total number of hours he worked.
\begin{align*}24+25+33+50+53+66+78=329\end{align*}
Step 2: Divide the total by the number of months.
\begin{align*}\frac{329}{7}=47\end{align*}
The mean number of hours that Stephen worked each month was 47.
Stephen has worked at Wendy’s for 15 months, but the numbers given above are for 7 months. Therefore, this set of data represents a sample, which is a portion of the population. The formula that was used to calculate the mean of the blocks must be changed slightly to represent a sample. The mean of a sample is denoted by \begin{align*}\bar{x}\end{align*}, which is called “\begin{align*}x\end{align*} bar.”
The number of data values for a sample is written as \begin{align*}n\end{align*}. The following formula represents the steps that are involved in calculating the mean of a sample:
\begin{align*}\text{Mean} = \frac{\text{sum of the values}}{\text{the number of values}}\end{align*}
This formula can now be written using symbols:
\begin{align*}\overline{x}=\frac{\sum x_1+x_2+x_3+ \ldots + x_n}{n}\end{align*}
You can now use the formula to calculate the mean number of hours that Stephen worked each month:
\begin{align*}\overline{x} &= \frac{\sum x_1+x_2+x_3+ \ldots + x_n}{n}\\ \overline{x} &= \frac{24+25+33+50+53+66+78}{7}\\ \overline{x} &= \frac{329}{7}\\ \overline{x} &= 47\end{align*}
The mean number of hours that Stephen worked each month was 47.
The formulas only differ in the symbol used for the mean and the case of the variable used for the number of data values (\begin{align*}N\end{align*} or \begin{align*}n\end{align*}). The calculations are done the same way for both a population and a sample. However, the mean of a population is constant, while the mean of a sample changes from sample to sample.
Example B
Mark operates a shuttle service that employs 8 people. Find the mean age of these workers if the ages of the 8 employees are as follows:
\begin{align*}55 \quad 63 \quad 34 \quad 59 \quad 29 \quad 46 \quad 51 \quad 41\end{align*}
If you were to take a sample of 3 employees from the group of 8 and calculate the mean age for these 3 workers, would the result change?
Since the data set includes the ages of all 8 employees, it represents a population. The mean age of the employees can be calculated as shown below:
\begin{align*}\mu &= \frac{\sum x_1+x_2+x_3+ \ldots + x_n}{N}\\ \mu &= \frac{55+63+34+59+29+46+51+41}{8}\\ \mu &= \frac{378}{8}\\ \mu &= 47.25\end{align*}
The mean age of all 8 employees is 47.25 years, or 47 years and 3 months.
Now let's take 2 samples of 3 employees from the group of 8 and calculate the mean age for these samples to see if the result changes. Let’s use the ages 55, 29, and 46 for one sample of 3, and the ages 34, 41, and 59 for another sample of 3:
\begin{align*}\overline{x} &= \frac{\sum x_1+x_2+x_3+ \ldots + x_n}{n} && \overline{x}=\frac{\sum x_1+x_2+x_3+ \ldots + x_n}{n}\\ \overline{x} &= \frac{55+29+46}{3} && \overline{x}=\frac{34+41+59}{3}\\ \overline{x} &= \frac{130}{3} && \overline{x}=\frac{134}{3}\\ \overline{x} &= 43.33 && \overline{x}=44.66\end{align*}
The mean age of the first group of 3 employees is 43.33 years.
The mean age of the second group of 3 employees is 44.66 years.
The mean age for a sample of a population depends upon what values of the population are included in the sample. From this example, you can see that the mean of a population and that of a sample from the population are not necessarily the same.
Example C
The selling prices of the last 10 houses sold in a small town are listed below:
\begin{align*}&\$125,000 \quad \$142,000 \quad \$129,500 \quad \$89,500 \quad \ \ \$105,000\\ &\$144,000 \quad \$168,300 \quad \$96,000 \quad \ \ \$182,300 \quad \$212,000\end{align*}
Calculate the mean selling price of the last 10 homes that were sold.
The prices are those of a sample, so the mean of the prices can be calculated as follows:
\begin{align*}\overline{x} &= \frac{\sum x_1+x_2+x_3+ \ldots + x_n}{n}\\ \overline{x} &= \frac{125,000+142,000+129,500+89,500+105,000+\$144,000+168,300+96,000+182,300+212,000}{10}\\ \overline{x} &= \frac{\$1,393,600}{10}\\ \overline{x} &= \$139,360\end{align*}
The mean selling price of the last 10 homes that were sold was $139,360.
The mean value is one of the 3 m’s and is a measure of central tendency. It is a summary statistic that gives you a description of the entire data set and is especially useful with large data sets, where you might not have the time to examine every single value. You can also use the mean to calculate further descriptive statistics, such as the variance and standard deviation. These topics will be explored in a future concept. The mean assists you in understanding and making sense of your data, since it uses all of the values in the data set in its calculation.
Guided Practice
During his final season with the Cadillac Selects, Joe Sure Shot played 14 regular season basketball games and had an average of 24.5 points per game. In the first 2 playoff games, Joe scored 18 and 26 points, respectively. Determine his new average for the season.
Answer:
Step 1: Multiply the given average by 14 to determine the total number of points he had scored before the playoff games.
\begin{align*}24.5 \times 14=343\end{align*}
Step 2: Add the points from the 2 playoff games to this total.
\begin{align*}343+18+26=387\end{align*}
Step 3: Divide this new total by 16 to determine the new average.
\begin{align*}\overline{x}=\frac{387}{16} \approx 24.19\end{align*}
Joe Sure Shot's new average for the season is about 24.19 points per game.
Interactive Practice
Practice
- What is the mean of the following numbers? \begin{align*}10, 39, 71, 42, 39, 76, 38, 25\end{align*}
- 42
- 39
- 42.5
- 35.5
- What symbol is used to denote the mean of a population?
- \begin{align*}\sum\end{align*}
- \begin{align*}\overline{x}\end{align*}
- \begin{align*}x_n\end{align*}
- \begin{align*}\mu\end{align*}
- What measure of central tendency is calculated by adding all the values and dividing the sum by the number of values?
- median
- mean
- mode
- typical value
- The mean of 4 numbers is 71.5. If 3 of the numbers are 58, 76, and 88, what is the value of the \begin{align*}4^{\text{th}}\end{align*} number?
- 64
- 60
- 76
- 82
- Determine the means of the following sets of numbers:
- 20, 14, 54, 16, 38, 64
- 22, 51, 64, 76, 29, 22, 48
- 40, 61, 95, 79, 9, 50, 80, 63, 109, 42
- The mean weight of 5 men is 167.2 pounds. The weights of 4 of the men are 158.4 pounds, 162.8 pounds, 165 pounds, and 178.2 pounds. What is the weight of the \begin{align*}5^{\text{th}}\end{align*} man?
- The mean height of 12 boys is 5.1 feet. The mean height of 8 girls is 4.8 feet.
- What is the total height of the boys?
- What is the total height of the girls?
- What is the mean height of the 20 boys and girls?
- The following data represents the number of advertisements received by 10 families during the past month. Calculate the mean number of advertisements received by each family during the month. \begin{align*}43 \qquad 37 \qquad 35 \qquad 30 \qquad 41 \qquad 23 \qquad 33 \qquad 31 \qquad 16 \qquad 21\end{align*}
- A group of grade 6 students each earned a mark on an in-class assignment. The marks for the boys were 90, 50, 70, 80, and 70. The marks for the girls were 60, 20, 30, 80, 90, and 20.
- Find the mean mark for the boys.
- Find the mean mark for the girls.
- Find the mean mark for all the students.
- The mean of 4 numbers is 31. (a) What is the sum of the 4 numbers? The mean of 6 other numbers is 28. (b) Calculate the mean of all 10 numbers.
- The following numbers represent the weights (in pounds) of 9 dogs: \begin{align*}22 \qquad 19 \qquad 26 \qquad 18 \qquad 29 \qquad 33 \qquad 20 \qquad 16 \qquad 30\end{align*}
- What is the mean weight of the dogs?
- If the heaviest and the lightest dogs are removed from the group, find the mean weight of the remaining dogs.
- To demonstrate her understanding of the concept of mean, Melanie recorded the daily temperature in degrees Celsius for her hometown at the same time each day for a period of 1 week. She then calculated the mean daily temperature.
Day | Sun | Mon | Tues | Wed | Thurs | Fri | Sat |
---|---|---|---|---|---|---|---|
Temperature \begin{align*}(^\circ \text{C})\end{align*} | \begin{align*}-7^\circ \text{C}\end{align*} | \begin{align*}0^\circ \text{C}\end{align*} | \begin{align*}-1^\circ \text{C}\end{align*} | \begin{align*}1^\circ \text{C}\end{align*} | \begin{align*}-4^\circ \text{C}\end{align*} | \begin{align*}-6^\circ \text{C}\end{align*} | \begin{align*}3^\circ \text{C}\end{align*} |
\begin{align*}\overline{x} &= \frac{-7+-1+1+-4+-6+3}{6}\\ \overline{x} &= \frac{-14}{6}\\ \overline{x} &= -2.3^\circ \text{C}\end{align*}
Melanie reported the mean daily temperature to be \begin{align*}-2.3^\circ \text{C}\end{align*}.
(a) Is Melanie correct? Justify your answer.
(b) If you do not agree with Melanie’s answer, can you tell Melanie what mistake she made?
- Below are the points scored by 2 basketball teams during the regular season for their first 12 games:
\begin{align*}&\text{Honest Hoopers} && 93 && 78 && 84 && 106 && 116 && 93 && 90 && 75 && 104 && 100 && 123 && 57\\ &\text{Bouncy Baskets} && 110 && 89 && 91 && 121 && 84 && 79 && 114 && 66 && 50 && 101 && 106 && 114\end{align*}
Which team had the higher mean score?
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Term | Definition |
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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. |
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. |
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. |