GCSE · Geography · AQA · Spec 8035

Numerical skills in geography

Twelve desert sites, one set of July rainfall data, three averages: 6.25 mm, 3.5 mm and 0 mm. None of them is lying. None tells the whole story.

Spread · Quartiles by position

42 values in a line. Where do the quartiles sit?

42 life-expectancy values are written in order, from the lowest (position 1) to the highest (position 42). The median marks the middle. Q1 marks the top of the lowest 25% of the values, and Q3 marks the bottom of the highest 25%. Before any maths, drag each marker to the POSITION where you think it falls, then check.

Q1 — Lower quartile (Q1)Place MedianQ3 — Upper quartile (Q3)

The middle · Mean, median, mode

Three averages, one desert, three different answers

Average July precipitation was measured at 12 hot desert sites. That's univariate data: values for one variable only. The 12 values add up to 75 mm. The mean is 75 ÷ 12 = 6.25 mm. With the values in order, the middle two are 3 and 4, so the median is (3 + 4) ÷ 2 = 3.5 mm. The most common value, the mode, is 0 mm.

Three averages, three different numbers. Which idea is closest to what you think right now?

The mean is the one to trust. It uses every single value.A mode of 0 mm means it never rained at any of these sites.The median is always the best, because it's the value right in the middle.Each average does a different job. Which one to use depends on the data, especially how big the data set is.

How sure are you?

I'm sureI think soI'm guessing

The mean is the one to trust. It uses every single value.

This is a common misconception

Using every value sounds fair, but watch what it does here. A mean of 6.25 mm makes every site look a little bit wet. It hides the fact that some sites had no precipitation at all. Using every value doesn't make an average more reliable. It just means it hides different things.

A mode of 0 mm means it never rained at any of these sites.

This is a common misconception

The mode is only the most common value. 0 mm turning up more often than any other value tells you nothing about the rest. The sites add up to 75 mm, so some of them clearly got rain. Here the mode gives a misleading picture of a desert with no rain at all.

The median is always the best, because it's the value right in the middle.

This is a common misconception

The median really is the central value. With 12 values it lands between the 6th and 7th, so you take their mean: 3.5 mm. But the median is meaningful in a large data set. In a very small one like this, it isn't representative of the data.

Each average does a different job. Which one to use depends on the data, especially how big the data set is.

This matches the evidence

Exactly. Mean, median and mode each summarise the data with a different purpose, and none of them is thought of as more reliable than the others. Think about the size of the data set first, then choose. Whichever you pick, know what it leaves out.

Every average compresses the data and hides something. Here it's 6.25 mm, 3.5 mm and 0 mm from the same 12 sites. The skill isn't finding THE answer. It's choosing the summary that fits the job and knowing what it leaves out.

Change over time · Percentage change

Spot the line that loses the mark

A town's yearly energy consumption rose from 800 units to 1,000 units (made-up numbers for practice). Calculate the percentage change.

A student's answer — which line goes wrong?

Original (earlier) value = 800 units. New value = 1,000 units.
Difference = 1,000 − 800 = 200 units
200 ÷ 1,000 = 0.2
0.2 × 100 = 20
Answer: a 20% increase

Predict, then check

Two made-up countries. No numbers yet, just your instinct.

Country A's total carbon dioxide emissions are far higher than Country B's. Which country emits more carbon dioxide per person?

Reading graphs · Structure, then GRaDE

A scatter graph, read in the right order

View: 1 · First look. Showing 2 layers: Axes, title and key, Plotted points

View

Layers

Explore

Start with structure and don't explain anything yet. What type of graph is it? What does the title say it shows? What does the key tell you? Which variable is on each axis? What's the scale, how far do the values extend, and in what units?

tap each part of the graph

Choosing graphs · What fits the data?

Match each data set to the graph that shows it

Pick a data set, then pick the graph you'd draw for it.

Still to sort

Climate graph (0)

Months along the bottom, two y-axes.

Population pyramid (0)

Age groups up the middle, bars either side.

Triangular graph (0)

Three axes forming a triangle.

Radial graph (0)

An axis for each variable, radiating from the centre.

Box and whisker plot (0)

One data set's spread on one set of axes.

Scatter graph (0)

Two variables plotted against each other.

7 of 7 still to sort.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

Mean = total ÷ number of values. Median = the middle value in order (the mean of the middle two when there's an even number of values). Mode = the most common value.
No average is more reliable than the others. Each has its own purpose, so think about the size of the data set before you choose.
Percentage of a total = part ÷ total × 100. Value from a percentage = percentage ÷ 100 × total.
Percentage change = difference ÷ original (earlier) value × 100, stated as an increase or a decrease.
Per capita (per person) data makes huge values manageable and lets you compare places fairly.
Quartiles come from positions in the ordered data. Find the median position first. Then count the n whole positions on one side of it, and go (n + 1) ÷ 2 positions in from that end. IQR = Q3 − Q1.
Read a graph's structure first (type, title, key, axes, scale, units), then examine it with GRaDE: General trend, Range, Data quotation, Exceptions.

The big picture

Geographers squeeze big data sets into a few numbers: averages for the middle, percentages for parts of a whole and for change over time, per capita figures for fair comparisons, and quartiles and the IQR for spread. Every one of those numbers leaves something out. Graphs are read in a set order: structure first, then GRaDE, then prediction with a trendline.

Key points

1Every numerical summary compresses the data and hides something. The mean can hide sites with no rain, and the mode can suggest there was no rain at all.

2A quartile is 25% of the data VALUES, found by counting positions in ordered data. It isn't 25% of the categories.

3A larger IQR means more spread-out data. On a box and whisker plot, the box is the IQR.

4Percentage change is always measured against the original, earlier value.

5A country with a far higher total can still have a lower figure per person.

6On a scatter graph, the trendline shows the correlation. An anomaly sits away from it. Interpolation estimates within the data range, and extrapolation estimates beyond it with a dotted line.

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Worked example

Problem

31% of a country's total GDP of 9,053 million US dollars comes from one part of its economy. Find the value of that share, then check your answer by turning it back into a percentage.

⚠ Watch out

Finding Q1 by chopping the scale, or the categories, into four equal pieces. Quartiles come from counting positions in the ordered values. With 42 values, Q1 is the 11th value, not whatever sits a quarter of the way along the axis.

🧠

Memory hook

Line them up and let position do the work. The middle position is the median, the middle of each half is a quartile, and the gap between the quartiles is the IQR.

✓

Check yourself

30 values are in order. At which position is the median, and at which positions are Q1 and Q3? (Median: 15.5th. Q1: 8th. Q3: 23rd.)

Flashcards

(16)

What is univariate data?

Values against one variable only, like average July precipitation at 12 hot desert sites.

What does frequency data show?

The actual count of values that fall into each category.

Two ways to put data into categories?

Equal-size categories (the count in each varies), or equal-frequency categories (the width of each one is adjusted).

Why be careful grouping a SMALL data set?

Grouping helps with big data sets, but with a small one it encourages generalisations the data can't support.

How do you find the mean, and what can it hide?

Add all the values and divide by how many there are. It can hide things like sites with no precipitation at all.

Median of an even number of values?

Put them in order and take the mean of the middle two. For example, 3 and 4 give 3.5.

Why can the mode mislead?

It's only the most common value. A mode of 0 mm suggests no rain at all, even when some sites had rain.

Which average is the most reliable?

None of them. Each has its own purpose. Consider the size of the data set before choosing.

Percentage of a total, both directions?

Part ÷ total × 100 gives the percentage. Percentage ÷ 100 × total gives the real value.

Formula for percentage change?

(Difference ÷ original, earlier value) × 100, stated as an increase or a decrease.

Why use per capita data?

It makes very large values manageable and makes places fairly comparable.

Decile vs quartile?

A decile is 10% of the data values and a quartile is 25% of the data values. They're never 10% or 25% of the categories.

What does the IQR tell you?

It's Q3 − Q1, a measure of spread. A larger IQR means more spread-out data.

First look at any graph: what do you check?

The type, title, key, the axes and their variables, and the scale, extent of values and units. No explaining yet.

What is an anomaly on a scatter graph?

A value that doesn't follow the trend and sits away from the trendline. It's more likely due to an unmeasured variable than to human error.

Interpolation vs extrapolation?

Interpolation estimates a value within the data's range. Extrapolation estimates beyond it, shown as a dotted extension of the trendline.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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42 values in a line. Where do the quartiles sit?

Three averages, one desert, three different answers

Spot the line that loses the mark

Match each data set to the graph that shows it

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