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.
Practice questions
Question 1
Answer the following based on the histogram given:
Find the total number of scores.
Use the histogram to approximate the sum of the scores.
Use the sum found in the previous part to approximate the mean of the scores, correct to two decimal places.
Question 2
What is the sum of a set of $29$29 scores with a mean of $37.7$37.7?
Question 3
The mean of a set of scores is $38.6$38.6 and the sum of the scores is $694.8$694.8. Calculate the number of scores.
Question 4
Han wants to try out as a batsman for a cricket team. In his last three matches, he scored $61$61, $75$75 and $66$66 runs. In his last match before trying out, he wants to lift his mean to $70.75$70.75. If $x$x is the number of runs he needs to score to achieve this, find $x$x.
Enter each line of working as an equation.
Question 5
A teacher calculated the mean of $25$25 students’ marks to be $64$64. A student who later completed the assessment got a mark of $55$55. What is the new mean of the class, to two decimal places?