Mean, median, mode and range | Year 7

01 / LEARN

Understand the idea.

Three different ideas of centre

The mean is the total shared equally among the observations. The median is the middle position after ordering; for an even-sized list, average the two middle values. The mode is the value occurring most often, rather than the number of times it occurs.

These measures describe different aspects of centre. A mean need not be an observed value. Two values can share the highest frequency and both be modes. When no value occurs more often than the others, the data may have no useful mode.

mean = \frac{sum of values}{number of values}

Spread and frequency

The range measures spread: subtract the lowest value from the highest. It uses only the extremes, so a single unusual value can change it greatly. Retain the data units for the mean, median and range.

In a frequency table, repeated observations still count. Multiply each value by its frequency to find its contribution to the total, then divide by total frequency. Averaging only the distinct values would wrongly give rare and common values equal weight.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

An odd-sized dataset

Find the mean, median, mode and range of 3, 5, 5, 7, 10.

Worked solution

1
Find the total and the mean.
$Mean = \frac{3 + 5 + 5 + 7 + 10}{5} = \frac{30}{5} = 6$
The total is shared across all five observations.
2
Locate the middle ordered value.
$Median = 5$
The list is already ordered, and the third of five values has two on each side.
3
Find the most frequent value.
$Mode = 5$
Five appears twice; every other value appears once.
4
Subtract the extremes.
$Range = 10 - 3 = 7$
Range measures the span from the smallest value to the largest.

Answer

Mean 6; median 5; mode 5; range 7.

EXAMPLE 02

Even number of observations and two modes

Summarise 12, 8, 4, 4, 6, 8.

Worked solution

1
Order the observations and calculate the mean.
$\begin{array}{c} 4, 4, 6, 8, 8, 12 \\ Mean = \frac{42}{6} = 7 \end{array}$
Ordering prepares the median calculation, while all six values contribute to the mean.
2
Average the two middle values.
$Median = \frac{6 + 8}{2} = 7$
An even-sized list has middle positions three and four.
3
Identify the joint highest frequency.
$f (4) = 2, f (8) = 2$
Four and eight both occur twice, so both are modes.
4
Calculate the range.
$Range = 12 - 4 = 8$
Use the extreme values, not the two middle ones.

Answer

Mean 7; median 7; modes 4 and 8; range 8.

03 / YOUR TURN

Try it yourself.

Check your understanding

Find the median of 2, 6, 8, 12.

Show answer & reasoning

There are four ordered values, so the middle positions contain $6$ and $8$ . Their average is $\frac{6 + 8}{2} = 7$ , the median.

Keep practising

6 worksheet questions · 15 marks

Open worksheet Worked solutions

Read the questions online

The data are 4, 6, 6, 8, 11.
5 marks
1. (a)
  Find the mean.
  2 marks
2. (b)
  Find the median.
  1 mark
3. (c)
  Find the mode.
  1 mark
4. (d)
  Find the range.
  1 mark
Find the median of 9, 3, 7, 5, 11, 13.
2 marks
What is the mode of 2, 4, 6, 8?
2 marks
Every score in a dataset increases by 3. What happens to the mean and range?
2 marks
Four scores have a mean of 9. Three scores are 6, 8 and 10. Find the fourth.
2 marks
Find the mean from this frequency table.
2 marks
Score frequencies
Score Frequency
2 1
4 2
6 1

Score frequencies
Score	Frequency
2	1
4	2
6	1

Read the worked solutions

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

The data are 4, 6, 6, 8, 11.
5 marks
a) The sum is 35; $35 \div 5 = 7$ .
b) The third of five ordered values is 6.
c) 6 occurs twice.
d) $11 - 4 = 7$ .
1. (a)
  Find the mean.
  2 marks
  The sum is 35; $35 \div 5 = 7$ .
  7.
2. (b)
  Find the median.
  1 mark
  The third of five ordered values is 6.
  6.
3. (c)
  Find the mode.
  1 mark
  6 occurs twice.
  6.
4. (d)
  Find the range.
  1 mark
  $11 - 4 = 7$ .
  7.
a) 7.; b) 6.; c) 6.; d) 7.
Find the median of 9, 3, 7, 5, 11, 13.
2 marks
Order: 3, 5, 7, 9, 11, 13.
The middle values are 7 and 9; $(7 + 9) \div 2 = 8$ .
8.
What is the mode of 2, 4, 6, 8?
2 marks
Every value occurs once, so no value is more frequent.
There is no mode.
Every score in a dataset increases by 3. What happens to the mean and range?
2 marks
Adding 3 to every value adds 3 to the equal share.
The new range is $(maximum + 3) - (minimum + 3)$ , so the added amounts cancel.
The mean increases by 3; the range stays the same.
Four scores have a mean of 9. Three scores are 6, 8 and 10. Find the fourth.
2 marks
$Required total = 4 \times 9 = 36$ .
$Known total = 6 + 8 + 10 = 24$ .
$Missing score = 36 - 24 = 12$ .
12.
Find the mean from this frequency table.
2 marks
Score frequencies
Score Frequency
2 1
4 2
6 1
Total frequency is $1 + 2 + 1 = 4$ .
Weighted sum is $2 \times 1 + 4 \times 2 + 6 \times 1 = 16$ .
$Mean = 16 \div 4 = 4$ .
4.

Score frequencies
Score	Frequency
2	1
4	2
6	1

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Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Mathematics Curriculum

Year 7 statistical variables, representations, summary statistics, outliers and an original small investigation; deeper sampling methods develop in Year 8.

Official curriculum source ↗

In this lesson

Understand the idea.

Three different ideas of centre

Spread and frequency

See the method in action.

An odd-sized dataset

Find the total and the mean.

Locate the middle ordered value.

Find the most frequent value.

Subtract the extremes.

Even number of observations and two modes

Order the observations and calculate the mean.

Average the two middle values.

Identify the joint highest frequency.

Calculate the range.

Try it yourself.

Check your understanding

Keep practising