Mathematical Skills (WJEC Eduqas GCSE Geography B): Revision Note

Specification link

Mathematics and statistics in WJEC Eduqas GCSE Geography B

This page covers part of the techniques required by Appendix A in the WJEC Eduqas specification (opens in a new tab).

The weighting for the assessment of maths and statistics skills will be at least 10%

• 1 Numerical skills

  • 1.1 Demonstrate an understanding of numbers, areas, and scales, as well as the quantitative and qualitative relationships between units

  • 1.3 Understand and correctly use proportions and ratios, magnitudes, and frequencies

• 2 Statistical skills

  • 2.1 Use appropriate measures for central tendency, spread, and cumulative frequency

  • 2.2 Calculate percentage increase or decrease and understand the use of percentiles

  • 2.3 Describe relationships in bivariate data

  • 2.4 Identify weaknesses in selective statistical presentation of data

Key terminology

• Bivariate data

  • Bivariate data is data which is collected on two variables, and it is used to look at how one of the variables affects the other

• Continuous data

  • Numerical data that can take any value within a given range, e.g. heights and weights

• Discrete data

  • Numerical data that can only take certain values, e.g. shoe size

• Quantitative data

  •  Results that can be expressed using numerical values

• Qualitative data

  • Results that can’t be expressed as numbers, e.g. opinions

Percentages

• Percentages are frequently used in geography

  • 'Percent' simply means 'out of one hundred'

  • For example, 25 of 360 homes in a town were burgled. What is the percentage (to the nearest whole number)?

    • 

• Percentages can be used in many ways, for example, literacy rates or the numbers of people in different age groups in a country

• Percentages can be converted into fractions and back again

  • If 20% of the population cannot read or write, this is 1/5th of the population

Rounding

• In the example above, the answer to 7 divided by 45 is 0.1555

• In geography, significant figures can be used to make the numbers easier to work with

• In the example, 0.15555 was rounded up to 0.16 to make the calculations easier

• To round up:

  • Identify the digit in the required place value

  • Circle the number to the right of the required place value

    • If the circled number is 5 or more, then you round to the bigger number

    • If the circled number is less than 5, then you round to the smaller number

    • Put a zero in any following place values before the decimal point

Significant figures

• To find the first significant figure when reading from left to right, find the biggest place value that has a non-zero digit

  • The first significant figure of 3097 is 3

  • The first significant figure of 0.0062070 is 6

    • The zeros before the 6 are not significant

    • The zero after the 2 but before the 7 is significant

    • The zero after the 7 is not significant

• Count along to the right from the first significant figure to identify the position of the required significant figure 

  • Do count zeros that are between other non-zero digits

    • E.g. 0 is the second significant figure of 3097

    • 9 is the third significant figure of 3097

  • Use the normal rules for rounding

  • For large numbers, complete places up to the decimal point with zeros

    • E.g. 34 568 to 2 significant figures is 35 000

  • For decimals, complete places between the decimal point and the first significant figure with zeros

    • E.g. 0.003 435 to 3 significant figures is 0.003 44

Proportion

• Direct proportion

  • As one quantity increases/decreases by a certain rate (factor)

  • The other quantity will increase/decrease by the same rate 

• The ratio of the two quantities is constant

  • A map has a scale of 1:25,000. This means that 1 unit on the map (e.g. 1 cm) represents 25,000 units (25,000 cm or 0.25km) in real life

• An inverse proportion means that as one variable increases, the other decreases by a proportionate amount

  • As urban populations increase, rural populations decrease

Ratio

• A ratio is a way of comparing one part of a whole to another

  • Ratios are used to compare one part to another part

What do ratios look like?

• Ratios involve two or three different numbers separated using a colon

  • E.g. 2 : 5,  3 : 1,  4 : 2 : 3 

• Dependency ratios compare the number of dependents (individuals aged 0-14 and over 65) to the working-age population (aged 15-64)

  • In the ratio 2:1, when referring to the dependency ratio, this means that for every 2 working-age people, there is one dependent person

Magnitude

• In geography, the term 'magnitude' has two meanings

• In mathematical skills, it is the relative size or scale of a quantity when comparing different geographical data

  • For example, if Country A has a population of 40 million and Country B has a population of 10 million, we can say that Country A's population is four times greater than Country B's

    • This means Country A's population is greater by a magnitude of 4.

• Magnitude can also refer to the amount of energy released in an earthquake

Frequency

• Frequency refers to how often a particular value or category appears within a set of data

• In a traffic survey, the number of times each type of vehicle (car, bus, bicycle) passes by is recorded

  • The count of each vehicle type represents its frequency.

• To organise and interpret this data effectively, geographers use frequency tables

• These tables list each category alongside its corresponding frequency, making it easier to identify patterns and trends

  • For example, a frequency table can help to identify the most common mode of transport used in a particular area

Statistics

• This is the study and handling of data, which includes ways of gathering, reviewing, analysing, and drawing conclusions from data

Mean, median and mode

• These are measures of central tendency

  • Mean = average value

    • The mean is calculated by adding up all of the values in the data set and then dividing by the total number of values in the data set

  • The median is the middle value of a set of data

    • Arrange the numbers in rank order, and then select the middle value

  • If there are an even number of data sets

    • First, order the numbers from lowest to highest

    • Find the two middle numbers and calculate their average by adding them together and dividing by two

    • E.g. the following are sample sizes; find the median: 90 64 98 142 159 95 184 64

    • After reordering by size 64 64 90 [95 98] 142 159 184, (95 and 98) are the middle values

    • is the median figure

  • The mode is the value which occurs most frequently in a set of data

  • When a dataset has two or more numbers that appear with the same highest frequency, those numbers are the mode

    • To find the mode, first count how many times each number appears; then identify the two numbers with the highest counts

    • For example, if the set was [64 64] 90 [95 95] 142 159 184, the modes are (64) and (95) because each appears twice, which is more than any other number in the set 

Modal class

• In a grouped frequency distribution, the modal class is the class/group interval that shows up most often

• It shows the range or group where most of the data points are found

• To find it, just look for the class/group with the most observations or occurrences 

How to find the modal class 

1. Look for the highest frequency:

   • Examine the frequency column of a grouped frequency table

2. Identify the corresponding class interval:

   • The class/group interval that lines up with the highest frequency is the modal class

Range 

• A measure of dispersion: the spread of data around the average

  • Range is the distance between the highest and lowest value

• The interquartile range is the part of the range that covers the middle 50% of the data

Percentage change

• A percentage change shows by how much something has either increased or decreased

• A percentage change shows by how much something has either increased or decreased

• 

  • In 2020, 25 out of 360 homes in a town were burgled. In 2021, 21 houses were burgled. What is the percentage change?

  • 

  • There has been a decrease of 16% in the rate of burglaries in the town

• Do remember that a positive figure shows an increase, but a negative is a decrease

Relationship in bivariate data

• Another term used to show the relationship between bivariate data is correlation

• A scatter diagram is a way of graphing bivariate data

  • For example, in a river study, the relationship between the width and depth of the river channel is plotted, with a line of best fit drawn to show if there is any correlation

  • In this instance, there is a positive correlation

• Strengths

  • Clearly shows data correlation

  • Shows the spread of data

  • Makes it easy to identify anomalies and outliers

• Limitations

  • Data points cannot be labelled

  • Too many data points can make it difficult to read

  • Can only show the relationship between two sets of data

  • Correlation cannot be measured without the use of a statistical test (Spearman’s Coefficient of Correlation)

Types of correlation

• Positive correlation

  • As one variable increases, so too does the other

  • The line of best fit goes from the bottom left to the top right of the graph

• Negative correlation 

  • As one variable increases, the other decreases

  • The line of best fit goes from the top left to the bottom right of the graph

• No correlation

  • Data points will have a scattered distribution

  • There is no relationship between the variables

Understand the use of percentiles

• Percentiles divide a data set into 100 equal parts

  • n% of the data values will be less than the n^th percentile

    • e.g. 10% of data values will be less than the 10^th percentile (and 90% will be greater than it)

    • 99% of data values will be less than the 99^th percentile (and 1% will be greater than it)

• Quartiles divide a set of data into 4 equal parts

• Percentiles do not need to be whole numbers

  • The 2.5^th percentile is where 2.5% of the data will be less than that (and 97.5% will be greater than it)

• Percentiles can be useful for discussing the distribution of data in a data set

  • For example, a student wanted to compare incomes in the UK

    • They decided to compare the highest 1% of earners (the ones above the 99^th percentile)

    • with the median income (the 50^th percentile)

    • and then with the lowest 10% of earners (those below the 10^th percentile)

Identifying weaknesses in the statistical presentation of data

• Some problems with presenting statistical data are:

  • Using misleading graphs

  • Picking data that isn't representative (cherry-picking)

  • Making comparisons that aren't balanced

  • Making broad conclusions from small samples

• Other problems include:

  • Misrepresenting the statistical significance

  • Not saying what the study's limitations or funding sources are

  • Using vague or misleading language to present data

• To find these flaws:

  • Look for graphs that don't make sense

  • Graphs that are missing context

  • Graphs/charts with conflicts of interest or claims that are too broad

• Always think about where the data came from and how it was collected