Reviewing numerical displays and summary statistics | Year 8

01 / LEARN

Understand the idea.

Four different summaries

The mean shares the total equally across all observations. The median locates the middle after ordering; for an even number of values, average the two middle observations. The mode is the most frequent value, while range measures largest minus smallest.

A frequency table compresses repeated values. Multiply each value by its frequency to recover its contribution to the total, then divide by the total frequency. Averaging only the distinct listed values gives each value equal weight even when their frequencies differ.

A suitable picture

Choose a display that fits the variable: columns suit categories or discrete values, lines can show change through ordered time, and dot plots or stem-and-leaf displays retain individual values. Label quantities, units and scales clearly so readers can interpret the picture.

Compare more than one feature: centre, spread, shape and unusual values describe different aspects of a distribution. For column heights, a zero baseline avoids exaggerating differences. A single average can conceal the variation visible in the original observations.

Column graph of books read by ten students. The frequencies for zero, one, two, three and four books are one, two, four, two and one. The frequency axis starts at zero.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Worked example 1

Find mean, median, mode and range for 2,3,3,4,8.

Worked solution

1
Check the ordered data and calculate the mean.
$\begin{aligned} 2 + 3 + 3 + 4 + 8 & = 20 \\ \bar{x} & = \frac{20}{5} = 4 \end{aligned}$
There are five observations, including both occurrences of 3.
2
Locate the middle observation.
$Median = 3$
The third of five ordered observations is the middle value.
3
Identify the most frequent value.
$Mode = 3$
Three occurs twice, more often than any other value.
4
Measure the spread between the extremes.
$Range = 8 - 2 = 6$
Range uses largest minus smallest, rather than the middle observations.

Answer

Mean 4; median 3; mode 3; range 6.

EXAMPLE 02

Worked example 2

Use the supplied table to find mean books read.

Example data: books read in one week
Books	Frequency
0	1
1	2
2	4
3	2
4	1

Worked solution

1
Count all observations in the frequency table.
$n = 1 + 2 + 4 + 2 + 1 = 10$
The frequencies count the people, not just the five distinct book totals.
2
Find the weighted book total.
$\begin{aligned} Total & = 0 \times 1 + 1 \times 2 + 2 \times 4 \\ + 3 \times 2 + 4 \times 1 \\ = 20 books \end{aligned}$
Each book count contributes once for every person at that value.
3
Divide the total by the observation count.
$Mean = \frac{20}{10} = 2 books$
Twenty books shared across ten observations gives the mean count.

Answer

2 books.

03 / YOUR TURN

Try it yourself.

Check your understanding

Find the median of 2,4,7,9.

Show answer & reasoning

5.5.

With four ordered observations, the middle two are 4 and 7. Their average is $\frac{4 + 7}{2} = 5.5$ , so the median can lie between observed values.

Keep practising

6 worksheet questions · 16 marks

Open worksheet Worked solutions

Read the questions online

Find the mean, median, mode and range of 5,6,6,7,11.
4 marks
How many students read at least 3 books in the supplied table?
2 marks
Example data: books read in one week
Books Frequency
0 1
1 2
2 4
3 2
4 1
Find the median books read from the supplied table.
2 marks
Example data: books read in one week
Books Frequency
0 1
1 2
2 4
3 2
4 1
A stem-and-leaf display has rows $1 ∣ 2 4 6$ and $2 ∣ 0 3$ , with key $1 ∣ 2 = 12$ . List the data and range.
3 marks
Choose a display for rainfall measured daily across a fortnight. Explain.
2 marks
Columns for counts 45 and 50 start their vertical axis at 40. Explain the distortion.
3 marks

Read the worked solutions

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

Find the mean, median, mode and range of 5,6,6,7,11.
4 marks
Total 35; $35 \div 5 = 7$ .
Middle value 6; 6 occurs twice; $11 - 5 = 6$ .
Mean 7; median 6; mode 6; range 6.
How many students read at least 3 books in the supplied table?
2 marks
Example data: books read in one week
Books Frequency
0 1
1 2
2 4
3 2
4 1
Add frequencies for 3 and 4 books: $2 + 1 = 3$ .
3 students.
Find the median books read from the supplied table.
2 marks
Example data: books read in one week
Books Frequency
0 1
1 2
2 4
3 2
4 1
The fifth and sixth observations are both 2.
2 books.
A stem-and-leaf display has rows $1 ∣ 2 4 6$ and $2 ∣ 0 3$ , with key $1 ∣ 2 = 12$ . List the data and range.
3 marks
Read place values from the key; $23 - 12 = 11$ .
12,14,16,20,23; range 11.
Choose a display for rainfall measured daily across a fortnight. Explain.
2 marks
Ordered time allows day-to-day change to be seen.
A line graph with day horizontally and rainfall in millimetres vertically.
Columns for counts 45 and 50 start their vertical axis at 40. Explain the distortion.
3 marks
Use a zero baseline for fair column-height comparisons.
Displayed heights 5 and 10 make 50 look twice 45, although actual counts differ by only 5.

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Curriculum reference & lesson scope

Australian Curriculum v9 and SA Curriculum

Population, census and sample; observational versus experimental data; random and non-random sampling; primary/secondary data; reviewing numerical displays and summary statistics; comparing samples and sample sizes; bias, variation and uncertainty; planning, conducting and reporting a fair investigation.

Official curriculum source ↗

In this lesson

Understand the idea.

Four different summaries

A suitable picture

See the method in action.

Worked example 1

Check the ordered data and calculate the mean.

Locate the middle observation.

Identify the most frequent value.

Measure the spread between the extremes.

Worked example 2

Count all observations in the frequency table.

Find the weighted book total.

Divide the total by the observation count.

Try it yourself.

Check your understanding

Keep practising