4.01 Calculating the mean

When we are given a big set of data, we would like to summarize it. The most common way to summarize a set of data is using a measure of center or a measure of central tendency.

There are three main measures of central tendency:

• Mean: the numerical average or balance point of a data set
• Median: the middle value of a data set ranked in order
• Mode: the data value that occurs most frequently

We will look at the mean for now.

The mean as a balance point

The mean can be thought of as the balance point for a data set. Below are two applets which will allow you to explore what this means.

Exploration

This applet has three data points that you can slide around. Consider the questions below.

1. What do you notice if you make all of the data points the same?
2. If you have two points close together and one far away, which is the mean closer to?
3. Can you make more than one data set with a mean of $4$4?
4. Will the mean ever be outside of the data set?
5. Does the mean do a good job of summarizing the data set? Explain.
6. Set up the points to $4$4, $6$6 and $11$11. How far is each value from the mean? Use negative values for below the mean and positive values for above the mean. What is the sum of these values? 

Credit: Steve Phelps

We should see that the mean is the balancing point, so the sum of the distances from the mean are 0.

For example, if we have the data set $4$4, $6$6 and $11$11, the mean is $7$7.

• $4$4 is $3$3 below $7$7
• $6$6 is $1$1 below $7$7
• $11$11 is $4$4 above $7$7
• $-3+\left(-1\right)+4=0$−3+(−1)+4=0

 

The applet below will allow us to explore how one value which is far away might change the mean. There is now only one point to slide, but you can get a new data set by clicking the button. Consider the questions below.

1. Start with the point in the middle of the data set and then slide it to either be higher or lower than all the other values. How does this change the mean?
2. How much does the sliding point change the mean? Explain.
3. If the sliding point is very far away, is the mean still a good summary of the data?
4. How can you change the mean the most with the sliding point?

 

Credit: Steve Phelps

The mean is the most appropriate for sets of data where there are no values which are far away from the rest of the data set. Later we will call these far away values outliers. 

 

Worked examples

Question 1

Calculate the mean temperature for the following temperature in degrees Fahrenheit $3,-5,6,8,-2$3,−5,6,8,−2.

Think: We need to add up all of the temperatures and divide by the number of values, $5$5. 

Do: 

$\frac{3+\left(-5\right)+6+8+\left(-2\right)}{5}$3+(−5)+6+8+(−2)5​	$=$=	$\frac{10}{5}$105​	Add up all of the values
	$=$=	$2$2	Divide by $5$5

 

The average, mean, temperature is $5$5$^\circ$°F.

 

Reflect: Does an average of $2$2 make sense based on the data set?

 

Question 2

Calculate the mean of the data set $1,1,1,1,2,2,2,3,3,3,3,3,3,9$1,1,1,1,2,2,2,3,3,3,3,3,3,9. Round to two decimal places.

Think: We need to add up all of the values and divide by the number of values, $14$14. Since we are adding the same numbers over and over, we can use multiplication to help us.

Do: 

	$\frac{1+1+1+1+2+2+2+3+3+3+3+3+3+9}{14}$1+1+1+1+2+2+2+3+3+3+3+3+3+914​	The long way to calculate the mean
$=$=	$\frac{4\times1+3\times2+6\times3+9}{14}$4×1+3×2+6×3+914​	Use multiplication to be more efficient
$=$=	$\frac{4+6+18+9}{14}$4+6+18+914​	Multiply
$=$=	$\frac{37}{14}$3714​	Add
$=$=	$2.64$2.64	Dividing and rounding

 

Reflect: How would the mean change if the $9$9 was removed from the data set?

 

Practice questions

Question 3

Question 4

Question 5