Statistics

Averages and range

Worked example (data written for this app): the long axes of seven pebbles at a beach site, in cm: 3, 5, 5, 6, 8, 9, 12.

add them up and divide by how many: 48 ÷ 7 ≈ 6.9 cm.
the middle value in order: the 4th of 7, 6 cm.
the most common value: 5 cm.
highest minus lowest: 12 − 3 = 9 cm.

For grouped data, such as pedestrian counts in classes (0–9: 2 sites; 10–19: 5 sites; 20–29: 3 sites), the modal class is the class with the most: 10–19.

The median isn't pulled up or down by one extreme value, as the mean is.

Key terms

Quartiles and interquartile range

The (IQR) shows the spread of the middle half of the data, ignoring extremes.

With the same seven pebbles, in order from the largest: 12, 9, 8, 6, 5, 5, 3.

Count (n + 1) ÷ 4 = 2 values in from each end.
Upper quartile: the 2nd from the top: 9 cm.
Lower quartile: the 2nd from the bottom: 5 cm.
IQR = 9 − 5 = 4 cm.

A small IQR means the values are bunched together; a large one means they are spread out.

adds up the values as you go, and a cumulative frequency graph can be used to read off percentiles, such as the 90th.

Key terms

Percentage change

= (new − old) ÷ old × 100.

Increase: a town grew from 12,000 to 15,000 people.

Change: 15,000 − 12,000 = 3,000.
3,000 ÷ 12,000 × 100 = 25% increase.

Decrease: traffic at a site fell from 80 to 60 vehicles an hour.

Change: 60 − 80 = −20.
−20 ÷ 80 × 100 = −25%, a 25% decrease.

Common mistake

Always divide by the old value, not the new one.

Key terms

Trends and misleading statistics

A trend line (line of best fit) shows the overall pattern on a scatter graph.
is estimating a value between known points, from the trend line: for example, the river velocity 3.5 km downstream.
is estimating beyond the data, by extending the line. It is less reliable, because the pattern may not continue.

Misleading statistics: watch out for

a vertical axis that doesn't start at zero, which exaggerates differences;
very small samples;
averages that hide extremes;
data from different years or places being compared as if alike.

Key terms

Exam practice

Plan your own answer first, then open the model answers to see what each level looks like.

4 marks · about 4 minutes

Explain why a student should use both an average and a measure of spread to compare two sets of fieldwork data.

Explain means: Give the reasons why something happens, developing each point with "because" or "so".

Level 1 (Basic)· 2/4 marks (band 1–2)

The average shows the middle value and the spread shows how spread out the data is.

Why this level: Correct definitions, but it doesn't explain why both are needed. To reach Level 2: show how two sets could have the same average but different spreads.

Level 2 (Clear)· 4/4 marks (band 3–4)

An average, such as the median, shows the typical value in each set, so you can say which is generally higher. But two sets with the same median can be very different: one bunched together, the other spread out. A measure of spread, such as the interquartile range, shows this, and because it ignores the extremes, one unusual value doesn't distort it.

Why this level: A clear, developed explanation of what each measure adds. Full marks.

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