The mean, commonly referred to as the average, is a measure of centre that tells us something about the location of data.
We can use the interactive tool below to visualise the position of the mean for different data sets, and also how the mean changes as we move one of the scores around.
When we find the average of a sample of values, we use the symbol $\overline{x}$x (pronounced "x-bar").
The mean is calculated as the sum of values divided by the number of values. That is,
$\overline{x}=\frac{\text{sum of values}}{\text{number of values}}$x=sum of valuesnumber of values
We often see this written with summation notation as
$\overline{x}=\frac{\Sigma x}{n}$x=Σxn
When we are given data in a frequency table or graph, the number of values is the sum of all frequencies, so
$\overline{x}=\frac{\Sigma fx}{\Sigma f}$x=ΣfxΣf
For small data sets, the mean is readily calculated with a scientific calculator. For larger data sets the statistics capabilities of our CAS calculator will be faster and more accurate.
Tells us the location of data.
- mean - also called average, is the sum of values divided by the number of values.
- median - is the middle value when the values are sorted.
- mode - the value that occurs most often.
When we are working with grouped data, we can also refer to the modal class, which is the class interval with the highest frequency.
Example 1 - using CAS to calculate the mean
Calculate the mean for this data set: $5,75,75,80,80,80,80,92,107,107,107,107$5,75,75,80,80,80,80,92,107,107,107,107
Classpad
Using Statistics mode, enter class centres into "list1".
Use the Calc -> One-variable menu to calculate the mean (and other statistics), with "Freq" set to "1" because "list1" contains individual data values.
The mean is given as $\overline{x}=88.75$x=88.75.
Example 2 - using CAS to calculate the mean from a frequency table
Calculate the mean for the data represented in the frequency table.
| Value | Frequency |
|---|---|
| $75$75 | $3$3 |
| $80$80 | $4$4 |
| $92$92 | $1$1 |
| $107$107 | $4$4 |
Classpad
Using Statistics mode, enter values into "list1" and frequencies into "list2"
Use the Calc -> One-variable menu to calculate the mean (and other statistics), using the "Freq" setting to select frequencies from "list2"
This data set is equivalent to the previous examples so, once again, the mean is given as $\overline{x}=88.75$x=88.75.
Example 3 - using CAS to estimate the mean for grouped data
Estimate the mean for the data represented in the grouped frequency table:
| Class | Frequency |
|---|---|
| $30-<40$30−<40 | $12$12 |
| $40-<50$40−<50 | $16$16 |
| $50-<60$50−<60 | $25$25 |
| $60-<70$60−<70 | $4$4 |
Since we are given grouped data, we can only get an estimate of the mean. We first need to determine the class centres, which will be used to represent each class. For instance, the class centre for the first interval is $\frac{30+40}{2}=35$30+402=35.
| Class | Class Centre | Frequency |
|---|---|---|
| $30-<40$30−<40 | $35$35 | $12$12 |
| $40-<50$40−<50 | $45$45 | $16$16 |
| $50-<60$50−<60 | $55$55 | $25$25 |
| $60-<70$60−<70 | $65$65 | $4$4 |
Classpad
Using Statistics mode, enter class centres into "list1" and frequencies into "list2".
Use the Calc -> One-variable menu to calculate the mean (and other statistics), using the "Freq" setting to select frequencies from "list2".
For this data set, the mean is given as $\overline{x}\approx48.68$x≈48.68.
Example 4 - determine an unknown value
The mean of five values is $64$64. Four of the values are $84,77,72,70$84,77,72,70. What is the fifth value?
The sum of the four known values, plus the fifth value is divided by $5$5 to get the mean value of $64$64. This can be solved quickly with CAS.
ClassPad
The unknown score is $17$17.
Example 5 - determine the mean after adding a new value
Peter has an average of $34$34 runs after the first $7$7 cricket matches. If he scores $50$50 runs in his eighth match, what is his new average?
The mean of the first $7$7 matches is $34$34, so the sum of these marks must be $34\times7$34×7. The eighth match score is added on to this sum. Now we have $8$8 scores, so the result divided by $8$8 to get a new mean of $36$36.
The calculation for this example is shown below using CAS, but could also be done with a scientific calculator.
ClassPad
We can see that Peter's average will be $36$36 runs after the eighth match.
Example 6 - determine the new value needed to get a certain mean
James wants to get an average mark of $75$75 in his Mathematics course. If he has an average of $72$72 after the first $3$3 tests, what mark does he need in the final test?
The mean of the first $3$3 test marks is $72$72, so the sum of these marks must be $72\times3$72×3. The final test score is added on to this sum, and the result divided by $4$4 to get the target mean of $75$75.
ClassPad
We can see that James will need to score $84$84 in the final test.
Measures of centre for interval grouped data
For data grouped in intervals, such as continuous data, we cannot find the exact measures of centre as we do not have the individual scores. We can however find approximate measures by representing all scores in an interval by the class centre (midpoint) of the given interval.
Worked example
Example 4
Estimate the mean for the data represented in the grouped frequency table:
| Class | Frequency |
|---|---|
| $30-<40$30−<40 | $12$12 |
| $40-<50$40−<50 | $16$16 |
| $50-<60$50−<60 | $25$25 |
| $60-<70$60−<70 | $4$4 |
Think: To estimate the mean for the data we first need to determine the class centres, which will be used to represent all the scores in a class. For instance, the class centre for the first interval is $\frac{30+40}{2}=35$30+402=35.
We then use: $\text{Total sum of all scores}\approx\text{sum of}\ \left(\text{Class centre}\times\text{Frequency}\right)$Total sum of all scores≈sum of (Class centre×Frequency)
Do:
| Class | Class centre | Frequency |
|---|---|---|
| $30-<40$30−<40 | $35$35 | $12$12 |
| $40-<50$40−<50 | $45$45 | $16$16 |
| $50-<60$50−<60 | $55$55 | $25$25 |
| $60-<70$60−<70 | $65$65 | $7$7 |
| $\text{Mean}$Mean | $=$= | $\frac{\text{Total sum of all scores}}{\text{Number of scores}}$Total sum of all scoresNumber of scores |
| $\approx$≈ | $\frac{35\times12+45\times16+55\times25+65\times7}{60}$35×12+45×16+55×25+65×760 | |
| $=$= | $49.5$49.5 |
Thus, the mean for this data set is approximately $49.5$49.5.
Practice questions
Question 1
Consider the table below.
| Score | Frequency |
|---|---|
| $1$1 - $4$4 | $2$2 |
| $5$5 - $8$8 | $7$7 |
| $9$9 - $12$12 | $15$15 |
| $13$13 - $16$16 | $5$5 |
| $17$17 - $20$20 | $1$1 |
Use the midpoint of each class interval to determine an estimate for the mean of the following sample distribution. Round your answer to one decimal place.
Which is the modal group?
$1$1 - $4$4
A$17$17 - $20$20
B$13$13 - $16$16
C$5$5 - $8$8
D$9$9 - $12$12
E
Question 2
Consider the table below.
| Score (x) | Frequency |
|---|---|
| $0\le x<20$0≤x<20 | $4$4 |
| $20\le x<40$20≤x<40 | $15$15 |
| $40\le x<60$40≤x<60 | $23$23 |
| $60\le x<80$60≤x<80 | $73$73 |
| $80\le x<100$80≤x<100 | $45$45 |
Use the midpoint of each class interval to determine an estimate for the mean of the following sample distribution. Round your answer to one decimal place.
Which is the modal group?
$0\le x<20$0≤x<20
A$60\le x<80$60≤x<80
B$20\le x<40$20≤x<40
C$40\le x<60$40≤x<60
D$80\le x<100$80≤x<100
E
Measures of centre using technology
Throughout this chapter and in particular for moderate to large data sets, you should use appropriate technology such as a calculator with statistics program on your computer.
Tips:
- Familiarise yourself with the program and the types of calculations and graphs it is capable of creating.
- Ensure settings are correct for the data given, this is particularly important when changing between data that is in a simple list to data that is in a frequency table.
- Take note of the different symbols used for the different calculations we will encounter.