The Mean and Mode
The mean and the mode are two ways to describe a typical value in a set of data. In NSW Year 7 maths they are among the first measures of centre that students calculate (NESA outcome MA4‑DAT‑C‑02).
You will find the mean by adding the values and dividing by how many there are, find the mode (including bimodal data and data with no mode), calculate the mean from a frequency table, and use \(\text{total} = \text{mean} \times \text{number of values}\) to find a missing score or a new mean. Cricket batting averages and typical cafe prices are everyday examples. This subtopic is part of Year 7 Analysing Data.
Theory
The mean and the mode are two measures of centre. The mean is the sum of the values divided by the number of values; the mode is the value that appears most often. This Year 7 maths page shows how to find both, from lists and frequency tables, and how to work backwards from a mean.
The mean is what each value would be if the total were shared out evenly: mean \(= \dfrac{\text{sum of the values}}{\text{number of values}}\). Its symbol is \(\overline{x}\). It is often called the average, and it can be a decimal that is not one of the values.
The mode is the value that appears most often (the value with the highest frequency). Put the values in order first, so repeats sit side by side. Data can have one mode, two modes (bimodal), more than two, or no mode.
Frequency table. Multiply each value by its frequency to get the total. Add the frequencies to get the number of values.
Working with totals. Total \(=\) mean \(\times\) number of values. Use it to find a missing value or a new mean.
Outliers. A value far from the rest pulls the mean towards it, but does not change the mode.
The rule, and examples of modes:
| Data | Mode |
|---|---|
| \(3, 5, 5, 5, 8\) | \(5\) (one mode) |
| \(2, 2, 6, 9, 9\) | \(2\) and \(9\) (bimodal) |
| \(1, 4, 6, 7\) | no mode |
How to find the mean
- Add all the values (in a table: value \(\times\) frequency, then add).
- Count the values (in a table: add the frequencies).
- Divide the sum by the count.
How to find a missing value from the mean
- Total \(=\) mean \(\times\) number of values.
- Subtract the known values from the total.
Sum: \(2 + 3 + 3 + 4 + 4\) \(+\, 4 + 5 + 5 + 6 + 7\), which is \(43\). There are \(10\) values.
| \(\overline{x}\) | \(=\) | \(\dfrac{43}{10}\) |
| \(\) | \(=\) | \(4.3\) |
Mean \(4.3\). The value \(4\) occurs \(3\) times, the highest frequency, so the mode is \(4\).
| \(6 + 11 + 8 + 11 + 9 + 6 + 12\) | \(=\) | \(63\) |
| \(63 \div 7\) | \(=\) | \(9\) |
Mean \(9\). In order: \(6, 6, 8, 9, 11, 11, 12\), so the modes are \(6\) and \(11\) (bimodal).
| Pets | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| Students | 4 | 7 | 6 | 3 |
Value \(\times\) frequency: \(0 \times 4 = 0\), \(1 \times 7 = 7\), \(2 \times 6 = 12\), \(3 \times 3 = 9\).
| \(0 + 7 + 12 + 9\) | \(=\) | \(28\) |
| \(4 + 7 + 6 + 3\) | \(=\) | \(20\) |
| \(\dfrac{28}{20}\) | \(=\) | \(1.4\) |
Mean \(1.4\) pets. Mode \(1\) pet (the highest frequency, \(7\)).
| \(16 \times 4\) | \(=\) | \(64\) |
| \(14 + 19 + 12\) | \(=\) | \(45\) |
| \(64 - 45\) | \(=\) | \(19\) |
\(19\). Check: \((14 + 19 + 12 + 19) \div 4 = 64 \div 4 = 16\).
(i) Find the mean and mode of the six.
(ii) Find the mean and mode of all seven.
(iii) Which is the better typical value now?
| \(24 \div 6\) | \(=\) | \(4\) |
| \((24 + 18) \div 7\) | \(=\) | \(6\) |
(i) Mean \(4\), mode \(4\)
(ii) Mean \(6\), mode \(4\)
(iii) The mode, \(4\): the \(18\) pulls the mean up, but six of the seven play \(5\) hours or less.
Common pitfalls
Frequently asked questions
How do you find the mean?
Add all the values and divide by how many there are. \(43 \div 10 = 4.3\).
How do you find the mode?
Find the value that appears most often.
Can there be more than one mode?
Yes. Two modes is called bimodal. There can also be no mode.
How do you find a missing value when you know the mean?
Multiply the mean by the number of values to get the total, then subtract the values you know.
What does an outlier do to the mean?
It pulls the mean towards it. The mode does not change.