Mean/median/mode | Year 11

01 / LEARN

Understand the idea.

The mean shares the total equally

The mean is the equal share of the total: add all observed values and divide by their number. Its unit is the same as the data, and it need not equal any individual observation.

In a frequency table, a value appearing $f$ times contributes $f x$ to the total. Divide the weighted total by $\sum f$ , not by the number of distinct values. Very large or small observations can pull the mean towards them.

\bar{x} = \frac{\sum x_{i}}{n} = \frac{\sum f x}{\sum f}

The median and mode answer different questions

The median locates the middle of the ordered observations. With odd $n$ , use position $(n + 1) / 2$ ; with even $n$ , average positions $n / 2$ and $n / 2 + 1$ . Ordering is essential, including when frequencies are supplied.

The mode is a most frequent value; there may be several tied modes or no unique mode. Median resists extreme observations better than mean. Mode can describe categories, whereas averaging category codes generally has no useful interpretation.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

An unusual value

Find mean, median and mode for 2,3,3,4,18. Which centre is more typical?

Worked solution

1
Find the mean from the total.
$\bar{x} = \frac{2 + 3 + 3 + 4 + 18}{5} = \frac{30}{5} = 6$
Every observation contributes to the arithmetic mean, including the unusually large 18.
2
Locate the middle ordered observation.
$median = x_{3} = 3$
The list is already ordered and contains five observations, so its third value is central.
3
Find the most frequent value.
$mode = 3$
Three occurs twice; every other value occurs once.
4
Choose a centre for the central cluster.
The value 18 pulls the mean above the cluster from 2 to 4; the median of 3 better describes that cluster.

Answer

Mean 6; median 3; mode 3. Median 3 better describes the central cluster.

EXAMPLE 02

Use frequencies

Scores 1,2,4 have frequencies 2,3,1 respectively. Find the mean and median.

Worked solution

1
Count observations using the frequencies.
$n = 2 + 3 + 1 = 6$
Each frequency counts observations, rather than just one distinct score.
2
Calculate the weighted total and mean.
$\begin{aligned} \sum f x & = 1 (2) + 2 (3) + 4 (1) = 12 \\ \bar{x} & = \frac{12}{6} = 2 \end{aligned}$
The values 1, 2 and 4 contribute twice, three times and once respectively.
3
Locate the middle pair in the ordered observations.
$1, 1, 2, 2, 2, 4$
Six observations place the median between the third and fourth values.
4
Average the middle pair.
$median = \frac{2 + 2}{2} = 2$
Both central observations are 2, so their average is 2.

Answer

Mean 2; median 2.

03 / YOUR TURN

Try it yourself.

Check your understanding

Find the median of 2,9,4,6.

Show answer & reasoning

Order the observations as $2, 4, 6, 9$ . There are four, so average the middle pair: $\frac{4 + 6}{2} = 5$ .

Keep practising

6 worksheet questions · 14 marks

Open worksheet Worked solutions

Read the questions online

Find mean, median and mode for 4,5,5,6,10.
2 marks
The mean of 6 values is 8. Find their total.
2 marks
Four scores are $3, 7, 8, x$ and their mean is 6. Find $x$ .
2 marks
Find the median and mode of 1,2,2,5,5,9.
2 marks
A set has mean 12. Adding a new observation of 12 changes what? Explain the mean.
3 marks
A small business has salaries 40,42,43,45 and 180 thousand dollars. Calculate mean and median and recommend one summary of a typical worker's pay.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Find mean, median and mode for 4,5,5,6,10.
2 marks
Sum 30 over 5 gives 6; the middle value is 5 and 5 is most frequent.
Mean 6; median 5; mode 5.
The mean of 6 values is 8. Find their total.
2 marks
$Total = mean \times number = 8 \times 6 = 48$ .
48.
Four scores are $3, 7, 8, x$ and their mean is 6. Find $x$ .
2 marks
Total must be 24; known total 18, so $x = 6$ .
6.
Find the median and mode of 1,2,2,5,5,9.
2 marks
The middle pair is 2,5.
Both 2 and 5 occur twice.
Median 3.5; modes 2 and 5.
A set has mean 12. Adding a new observation of 12 changes what? Explain the mean.
3 marks
The total increases by 12 at the same time the count increases by 1.
If old total $12 n$ , new mean $\frac{12 n + 12}{n + 1} = 12$ .
The mean stays 12.
A small business has salaries 40,42,43,45 and 180 thousand dollars. Calculate mean and median and recommend one summary of a typical worker's pay.
3 marks
Sum 350 divided by 5 is 70.
The middle value is 43; the 180 salary pulls the mean above most salaries.
Mean 70; median 43 thousand dollars. Median is more typical of this group.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 Mathematics: counting, binomial expansion, data, samples, summary measures and introductory normal distributions. These lessons provide the Counting and Statistics strand of the approved Methods preparation route.

Official curriculum source ↗

In this lesson

Understand the idea.

The mean shares the total equally

The median and mode answer different questions

See the method in action.

An unusual value

Find the mean from the total.

Locate the middle ordered observation.

Find the most frequent value.

Choose a centre for the central cluster.

Use frequencies

Count observations using the frequencies.

Calculate the weighted total and mean.

Locate the middle pair in the ordered observations.

Average the middle pair.

Try it yourself.

Check your understanding

Keep practising