Numerical and statistical skills

What you'll learn

How to use number, area, scale and units accurately in geography.
How to calculate percentages, ratios, averages, spread and percentiles.
How to collect reliable fieldwork data and spot weaknesses in data presentation.
How to describe relationships, draw lines of best fit and justify conclusions from evidence.

Why numerical skills matter in Geography

Geography is full of data: rainfall in mm, wind speeds in km/h, GNI per capita in US$, river velocity, population density, carbon emissions and deprivation scores. In OCR GCSE Geography A, these skills can appear in any theme and are especially important in the Geographical Skills component.

You might use numbers at a global scale to compare countries, at a national scale to compare UK regions, or at a local scale in your own fieldwork.

Number, units, area and scale

A unit tells you what a number measures, such as metres, km, mm, people per km², or US$ per person. Always keep the unit with the number.

Definition

Scale

In numerical geography, scale means the relationship between distance on a map or graph and distance in the real world. A map scale of 1:50,000 means 1 cm on the map represents 50,000 cm on the ground.

Area is the amount of surface covered by a place or feature. For regular shapes, use length multiplied by width. For irregular shapes on a map, estimate by counting grid squares.

Example

Converting scale and estimating area

On a 1:50,000 OS map, a footpath measures 7.2 cm. The scale means 1 cm represents 50,000 cm, so $7.2 \times 50{,}000 = 360{,}000$ cm.
Convert to kilometres: 100,000 cm = 1 km, so 360,000 cm = 3.6 km.
If a flood outline covers six full 1 km² grid squares and about four half-squares, estimate the area as 6 + 2 = 8 km².

Tip

Unit sanity check

If your answer to a local fieldwork distance is 360,000 km, the units have gone wrong. Convert cm to m or km before deciding whether the answer is realistic.

Proportion, ratio, magnitude and frequency

A proportion is a part of a whole. A ratio compares two quantities, such as 3:1. Magnitude means the size or strength of something, such as flood discharge or earthquake magnitude. Frequency means how often something happens, such as flood events per year.

These are useful when comparing places fairly. For example, comparing total numbers of migrants can be misleading because countries have different population sizes. A proportion or percentage is often fairer.

Key Idea

Fair comparison

Use rates, proportions and ratios when places have different sizes or populations. Raw totals often favour larger places.

Example

Calculating a proportion and ratio

A class fieldwork survey records 18 pedestrians and 12 cyclists passing a point in 10 minutes. The total count is 30.
The proportion that are cyclists is $12 \div 30 = 0.4$ , which is 40%.
The pedestrian-to-cyclist ratio is 18:12. Divide both sides by 6 to simplify it to 3:2.

Averages and spread

A measure of central tendency is a way of describing a typical value. The main ones are:

Mean: add all values and divide by the number of values.
Median: the middle value when the data is in order.
Mode: the most common value.
Modal class: the most common group in grouped data, such as 10–19 mm.

A measure of spread shows how varied the data is:

Range: maximum value minus minimum value.
Quartiles: values that split ordered data into four equal parts.
Interquartile range, or IQR: upper quartile minus lower quartile; it shows the spread of the middle 50% of values.

The diagram below shows how quartiles and the IQR sit within an ordered data set.

Diagram showing minimum, quartiles, median, range and interquartile range on an ordered data set

Example

Summarising pebble size data

A coastal fieldwork sample from a beach such as Chesil Beach gives pebble sizes in mm: 4, 6, 8, 11, 13, 15, 18, 20, 24, 28. The data is already ordered.
The mean is $147 \div 10 = 14.7$ mm, because the total is 147 and there are 10 values.
The median is halfway between the 5th and 6th values: $13 + 15 = 28, then $28 \div 2 = 14$ mm.
The range is $28 - 4 = 24$ mm.
The lower quartile is 8 mm and the upper quartile is 20 mm, so the IQR is $20 - 8 = 12$ mm.

Common Mistake

Using the mean without thinking

The mean can be distorted by an extreme value. If one street has a very high house price, the median may better represent a typical value for an urban study.

Percentages, percentage change and percentiles

A percentage means “out of 100”. Percentages help you compare changes and proportions across different places.

For percentage increase or decrease, use:

value×100\text{percentage change} = \frac{\text{new value} - \text{old value}}{\text{old value}} \times 100

A percentile tells you the value below which a certain percentage of data falls. For example, the 90th percentile is higher than 90% of the values. Percentiles are often found from ordered data or a cumulative frequency graph. Cumulative frequency means a running total of frequencies.

Example

Calculating percentage increase and a percentile

If a city’s population rises from 8 million to 12 million, the increase is 4 million.
Divide the increase by the original value: $\div 8 = 0.5$ .
Convert to a percentage: $0.5 \times 100 = 50$ , so the population increased by 50%.
For 20 ordered river-depth readings, the 75th percentile is around the value at position $0.75 \times 20 = 15$ , so look near the 15th value in the ordered list.

Fieldwork data: accuracy, sample size and reliability

A data collection sheet is a planned table used to record observations or measurements consistently. In fieldwork, good data depends on:

Accuracy: how close a measurement is to the true value.
Sample size: how many measurements or questionnaires you collect.
Procedure: the exact method followed, so data is collected consistently.
Control group or control site: a comparison where the factor being tested is absent or lower.
Reliability: whether repeating the method would give similar results.

For example, in river fieldwork, you might measure width, depth and velocity at several sites downstream. In urban fieldwork, you might compare environmental quality scores in a regenerated area with a similar non-regenerated street.

Example

Designing a fieldwork data sheet

Decide the enquiry question, such as “Does river width increase downstream on the River Tillingbourne in Surrey?”
Choose columns that match the question: site number, grid reference, distance downstream, width, depth readings, velocity, weather notes and anomalies.
Improve reliability by using the same measuring procedure at each site and taking repeat readings.
Improve the sample by using enough sites along the river, rather than only one convenient location.
Include a control or comparison where useful, such as an upstream site before a tributary joins.

Interpreting tables of data

A table organises data into rows and columns. Read the title, headings and units first. Then look for:

highest and lowest values
overall patterns or trends
anomalies, which are values that do not fit the pattern
differences between places, groups or times

Example

Interpreting a river data table

If a table shows river width increasing from 2.1 m upstream to 8.4 m downstream, identify the overall downstream increase.
If one middle site is narrower than the previous site, treat it as an anomaly and consider a reason, such as bank reinforcement or a measurement taken near a bridge.
Use data in your conclusion: “Width generally increased downstream, from 2.1 m to 8.4 m, although Site 3 was an exception.”

Bivariate data, scatter plots and lines of best fit

Bivariate data means two variables are measured for each case, such as distance downstream and river width. A scatter plot places one variable on each axis and plots each pair of values as a point.

A relationship describes how the variables are linked:

Positive relationship: as one variable increases, the other generally increases.
Negative relationship: as one increases, the other generally decreases.
No clear relationship: the points are scattered with no pattern.

A trend line or estimated line of best fit shows the overall pattern. It should pass through the middle of the points, with a roughly balanced number above and below. It does not need to touch every point.

Scatter plot showing positive correlation, interpolation, extrapolation and a line of best fit

Example

Using a line of best fit to make predictions

On the scatter plot, the data shows a positive relationship because river width generally increases as distance downstream increases.
To interpolate, read from the line within the data range. At about 10 km downstream, the estimated width is about 50 m.
To extrapolate, extend beyond the measured data range. At about 18 km, the estimate is about 82 m, but this is less reliable because no data was collected that far downstream.

Common Mistake

Joining the dots

A line of best fit is not a dot-to-dot line. It summarises the overall trend, so it can pass through empty space between points.

Spotting weaknesses and drawing conclusions

A statistical presentation is any way of showing data, such as a table, graph, choropleth map, pie chart or scatter plot. Be ready to identify weaknesses, including:

missing source, date, units or sample size
uneven or misleading class intervals
axes that exaggerate patterns
a mean used when there are strong outliers
too few data points for a secure conclusion
extrapolation beyond the data range
using totals when rates or percentages would be fairer

A strong geographical conclusion should make a judgement, use evidence, and recognise limitations. For example, for a Lagos, Nigeria case study, exact population estimates vary by source, so you should name the data source if given and avoid pretending figures are perfectly certain.

Key Idea

Evidence-led conclusions

Do not just say “there is a relationship”. Say what the relationship is, support it with values, name any exceptions, and comment on reliability.

Exam technique

In the exam

Keep units attached to every value, especially when converting map scale, rainfall, distance, population density or money.
For calculations, show enough working to make your method clear, even if your final answer is rounded.
When describing data, use the pattern plus evidence: trend, highest or lowest value, anomaly, and a possible geographical reason.

Self review

Check yourself

What is the difference between range and interquartile range?
Why is interpolation usually more reliable than extrapolation?
How could you improve the reliability of a local fieldwork data set?

What you'll learn

How to use number, area, scale and units accurately in geography.
How to calculate percentages, ratios, averages, spread and percentiles.
How to collect reliable fieldwork data and spot weaknesses in data presentation.
How to describe relationships, draw lines of best fit and justify conclusions from evidence.

Why numerical skills matter in Geography

You might use numbers at a global scale to compare countries, at a national scale to compare UK regions, or at a local scale in your own fieldwork.

Number, units, area and scale

A unit tells you what a number measures, such as metres, km, mm, people per km², or US$ per person. Always keep the unit with the number.

Definition

Scale

Area is the amount of surface covered by a place or feature. For regular shapes, use length multiplied by width. For irregular shapes on a map, estimate by counting grid squares.

Example

Converting scale and estimating area

On a 1:50,000 OS map, a footpath measures 7.2 cm. The scale means 1 cm represents 50,000 cm, so $7.2 \times 50{,}000 = 360{,}000$ cm.
Convert to kilometres: 100,000 cm = 1 km, so 360,000 cm = 3.6 km.
If a flood outline covers six full 1 km² grid squares and about four half-squares, estimate the area as 6 + 2 = 8 km².

Tip

Unit sanity check

If your answer to a local fieldwork distance is 360,000 km, the units have gone wrong. Convert cm to m or km before deciding whether the answer is realistic.

Proportion, ratio, magnitude and frequency

Key Idea

Fair comparison

Use rates, proportions and ratios when places have different sizes or populations. Raw totals often favour larger places.

Example

Calculating a proportion and ratio

A class fieldwork survey records 18 pedestrians and 12 cyclists passing a point in 10 minutes. The total count is 30.
The proportion that are cyclists is $12 \div 30 = 0.4$ , which is 40%.
The pedestrian-to-cyclist ratio is 18:12. Divide both sides by 6 to simplify it to 3:2.

Averages and spread

A measure of central tendency is a way of describing a typical value. The main ones are:

Mean: add all values and divide by the number of values.
Median: the middle value when the data is in order.
Mode: the most common value.
Modal class: the most common group in grouped data, such as 10–19 mm.

A measure of spread shows how varied the data is:

Range: maximum value minus minimum value.
Quartiles: values that split ordered data into four equal parts.
Interquartile range, or IQR: upper quartile minus lower quartile; it shows the spread of the middle 50% of values.

The diagram below shows how quartiles and the IQR sit within an ordered data set.

Diagram showing minimum, quartiles, median, range and interquartile range on an ordered data set

Example

Summarising pebble size data

A coastal fieldwork sample from a beach such as Chesil Beach gives pebble sizes in mm: 4, 6, 8, 11, 13, 15, 18, 20, 24, 28. The data is already ordered.
The mean is $147 \div 10 = 14.7$ mm, because the total is 147 and there are 10 values.
The median is halfway between the 5th and 6th values: $13 + 15 = 28, then $28 \div 2 = 14$ mm.
The range is $28 - 4 = 24$ mm.
The lower quartile is 8 mm and the upper quartile is 20 mm, so the IQR is $20 - 8 = 12$ mm.

Common Mistake

Using the mean without thinking

The mean can be distorted by an extreme value. If one street has a very high house price, the median may better represent a typical value for an urban study.

Percentages, percentage change and percentiles

A percentage means “out of 100”. Percentages help you compare changes and proportions across different places.

For percentage increase or decrease, use:

value×100\text{percentage change} = \frac{\text{new value} - \text{old value}}{\text{old value}} \times 100

Example

Calculating percentage increase and a percentile

If a city’s population rises from 8 million to 12 million, the increase is 4 million.
Divide the increase by the original value: $\div 8 = 0.5$ .
Convert to a percentage: $0.5 \times 100 = 50$ , so the population increased by 50%.
For 20 ordered river-depth readings, the 75th percentile is around the value at position $0.75 \times 20 = 15$ , so look near the 15th value in the ordered list.

Fieldwork data: accuracy, sample size and reliability

A data collection sheet is a planned table used to record observations or measurements consistently. In fieldwork, good data depends on:

Accuracy: how close a measurement is to the true value.
Sample size: how many measurements or questionnaires you collect.
Procedure: the exact method followed, so data is collected consistently.
Control group or control site: a comparison where the factor being tested is absent or lower.
Reliability: whether repeating the method would give similar results.

Example

Designing a fieldwork data sheet

Decide the enquiry question, such as “Does river width increase downstream on the River Tillingbourne in Surrey?”
Choose columns that match the question: site number, grid reference, distance downstream, width, depth readings, velocity, weather notes and anomalies.
Improve reliability by using the same measuring procedure at each site and taking repeat readings.
Improve the sample by using enough sites along the river, rather than only one convenient location.
Include a control or comparison where useful, such as an upstream site before a tributary joins.

Interpreting tables of data

A table organises data into rows and columns. Read the title, headings and units first. Then look for:

highest and lowest values
overall patterns or trends
anomalies, which are values that do not fit the pattern
differences between places, groups or times

Example

Interpreting a river data table

If a table shows river width increasing from 2.1 m upstream to 8.4 m downstream, identify the overall downstream increase.
If one middle site is narrower than the previous site, treat it as an anomaly and consider a reason, such as bank reinforcement or a measurement taken near a bridge.
Use data in your conclusion: “Width generally increased downstream, from 2.1 m to 8.4 m, although Site 3 was an exception.”

Bivariate data, scatter plots and lines of best fit

A relationship describes how the variables are linked:

Positive relationship: as one variable increases, the other generally increases.
Negative relationship: as one increases, the other generally decreases.
No clear relationship: the points are scattered with no pattern.

Scatter plot showing positive correlation, interpolation, extrapolation and a line of best fit

Example

Using a line of best fit to make predictions

On the scatter plot, the data shows a positive relationship because river width generally increases as distance downstream increases.
To interpolate, read from the line within the data range. At about 10 km downstream, the estimated width is about 50 m.
To extrapolate, extend beyond the measured data range. At about 18 km, the estimate is about 82 m, but this is less reliable because no data was collected that far downstream.

Common Mistake

Joining the dots

A line of best fit is not a dot-to-dot line. It summarises the overall trend, so it can pass through empty space between points.

Spotting weaknesses and drawing conclusions

A statistical presentation is any way of showing data, such as a table, graph, choropleth map, pie chart or scatter plot. Be ready to identify weaknesses, including:

missing source, date, units or sample size
uneven or misleading class intervals
axes that exaggerate patterns
a mean used when there are strong outliers
too few data points for a secure conclusion
extrapolation beyond the data range
using totals when rates or percentages would be fairer

Key Idea

Evidence-led conclusions

Do not just say “there is a relationship”. Say what the relationship is, support it with values, name any exceptions, and comment on reliability.

Exam technique

In the exam

Keep units attached to every value, especially when converting map scale, rainfall, distance, population density or money.
For calculations, show enough working to make your method clear, even if your final answer is rounded.
When describing data, use the pattern plus evidence: trend, highest or lowest value, anomaly, and a possible geographical reason.

Self review

Check yourself

What is the difference between range and interquartile range?
Why is interpolation usually more reliable than extrapolation?
How could you improve the reliability of a local fieldwork data set?

What you'll learn

Why numerical skills matter in Geography

Number, units, area and scale

Proportion, ratio, magnitude and frequency

Averages and spread

Percentages, percentage change and percentiles

Fieldwork data: accuracy, sample size and reliability

Interpreting tables of data

Bivariate data, scatter plots and lines of best fit

Spotting weaknesses and drawing conclusions

Recap questions

Review Numerical and statistical skills by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Numerical and statistical skills

What you'll learn

Why numerical skills matter in Geography

Number, units, area and scale

Proportion, ratio, magnitude and frequency

Averages and spread

Percentages, percentage change and percentiles

Fieldwork data: accuracy, sample size and reliability

Interpreting tables of data

Bivariate data, scatter plots and lines of best fit

Spotting weaknesses and drawing conclusions

Recap questions

Review Numerical and statistical skills by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Numerical and statistical skills

What you'll learn

Why numerical skills matter in Geography

Number, units, area and scale

Proportion, ratio, magnitude and frequency

Averages and spread

Percentages, percentage change and percentiles

Fieldwork data: accuracy, sample size and reliability

Interpreting tables of data

Bivariate data, scatter plots and lines of best fit

Spotting weaknesses and drawing conclusions

Recap questions

Review Numerical and statistical skills by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Numerical and statistical skills Revision Guide

Numerical and statistical skills

What you'll learn

Why numerical skills matter in Geography

Number, units, area and scale

Proportion, ratio, magnitude and frequency

Averages and spread

Percentages, percentage change and percentiles

Fieldwork data: accuracy, sample size and reliability

Interpreting tables of data

Bivariate data, scatter plots and lines of best fit

Spotting weaknesses and drawing conclusions

Recap questions

Review Numerical and statistical skills by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Numerical and statistical skills Revision Guide