📏 Units to know
1 km = 1,000 m. 1 m = 100 cm. 1 cm = 10 mm. 1 hectare = 10,000 m2.
Numerical and Statistical Skills » Averages, Range and Quartiles
What you'll learn this session
Study time: 30 minutes
AQA spec: 3.4.3, 3.4.4
- Use number, area, units, proportion and ratio correctly
- Design a fieldwork data collection sheet and collect accurate data
- Find the mean, median, mode, modal class and range
- Use quartiles, the inter-quartile range and cumulative frequency
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Unlock This CourseWorking with numbers, area and units
Mean, median, mode and range help you sum up data
Geography is full of numbers. Distances, areas, rainfall totals and populations all need to be read and handled with care. The most common slip is mixing up units, so always check them first.
Key terms:
- Area: the amount of flat surface something covers, such as a field or a city.
- Proportion: a part compared with the whole, written as a fraction or percentage.
- Ratio: a comparison of two amounts, written with a colon, such as 3:1.
- Magnitude: how large a value is.
Units and area
▦ Finding area
For a rectangle, area = length × width. The answer is in square units, such as m2 or km2.
Worked example
A field is 200 m long and 150 m wide. Area = 200 × 150 = 30,000 m2. As 1 hectare is 10,000 m2, the field is 3 hectares.
On a map at 1:50 000, one grid square is 2 cm wide on the page, which is 1 km on the ground. So each square covers 1 km × 1 km = 1 km2. You can estimate an area by counting squares.
Proportion, ratio, magnitude and frequency
A proportion shows how much of the whole something makes up. If 30 of 120 people surveyed walk to the shops, the proportion is 30/120 = 1/4, or 25%.
A ratio compares two amounts. If 30 walk and 90 do not, the ratio is 30:90, which simplifies to 1:3. Divide both sides by the same number to simplify.
Magnitude is about size. A city of 5 million people is a much greater magnitude than a town of 5 thousand. Frequency is how many times something happens, so use it when you count.
Collecting accurate data
A data collection sheet is a table you design before fieldwork to record results neatly. A good sheet has:
- a clear title, the date and the location
- headed columns with the units, such as width (cm)
- enough rows for every site or person you will sample
- a space for notes, such as weather or an odd result
Accuracy means your measurements are close to the true value. Use the same equipment each time, read it carefully and measure to the same unit. Remember a bigger sample size gives more trustworthy results, as you saw in Collecting Fieldwork Data.
A control group is a comparison site or group where the factor you are testing is missing. If you test whether a path has eroded because of walkers, a similar stretch of grass with no walkers is your control. Comparing the two shows whether the walkers made the difference.
Results are reliable if repeating the fieldwork gives similar answers. Repeating measurements and using a bigger sample both help.
Mean, median, mode and range
These are measures of central tendency (the middle of the data) and spread (how far the data are scattered).
- Mean: add all the values, then divide by how many there are.
- Median: the middle value when the data are in order.
- Mode: the value that appears most often.
- Range: the highest value minus the lowest value.
Worked example
Cars counted in nine ten-minute periods: 3, 5, 5, 6, 8, 9, 10, 12, 23.
Mean: total = 81, and 81 ÷ 9 = 9 cars.
Median: the 5th value in order = 8 cars.
Mode: 5 cars.
Range: 23 − 3 = 20 cars.
Notice the mean (9) is pulled upwards by the one very high value, 23. The median is not affected as much, so it is often a fairer middle when there is an extreme value.
Quartiles and the inter-quartile range
Quartiles split ordered data into four equal parts
Quartiles split ordered data into four equal parts. The lower quartile (Q1) is a quarter of the way through and the upper quartile (Q3) is three quarters of the way through. The inter-quartile range (IQR) is Q3 − Q1. It shows the spread of the middle half of the data, so extreme values do not distort it.
Worked example
Using the same data: 3, 5, 5, 6, 8, 9, 10, 12, 23.
The median is 8. Q1 is the median of the lower half (3, 5, 5, 6): (5 + 5) ÷ 2 = 5.
Q3 is the median of the upper half (9, 10, 12, 23): (10 + 12) ÷ 2 = 11.
IQR = 11 − 5 = 6 cars.
The range was 20, but the IQR is only 6. That tells you most of the counts were close together and 23 was an unusual result.
Modal class
When data are grouped into class intervals, you cannot pick out one mode. Instead you give the modal class, which is the class with the highest frequency.
Worked example
Distance people travelled to a shop:
0 to under 1 km: 6
1 to under 2 km: 14
2 to under 3 km: 9
3 to under 4 km: 3
The modal class is 1 to under 2 km, because 14 is the highest frequency.
Cumulative frequency
Cumulative frequency is a running total of the frequencies. It lets you find the median and quartiles from grouped data.
- Add up the frequencies as you go down the table.
- Plot each total against the upper end of its class.
- Join the points with a smooth curve. It is usually an S shape.
- Read across from n ÷ 2 for the median, n ÷ 4 for Q1 and 3n ÷ 4 for Q3.
Worked example
From the shop data, the running totals are 6, 20, 29 and 32, so n = 32.
Median: read across at 16, which gives about 1.7 km.
Q1: read across at 8, which gives about 1.1 km.
Q3: read across at 24, which gives about 2.4 km.
IQR = 2.4 − 1.1 = about 1.3 km.
Drawing conclusions from numbers
A good conclusion uses the numbers. Say what the data show, quote figures, and link back to your question. For example: most people live close to the shop. Half travel less than about 1.7 km, and the middle half travel between 1.1 km and 2.4 km. Always mention any odd values and say how much they affect the mean.
Common mistakes
Forgetting to put the data in order before finding the median or quartiles. Writing the modal class as a frequency (14) instead of a class (1 to under 2 km). Mixing units, such as metres and kilometres, in one calculation. Plotting cumulative frequency at the start of each class instead of the upper end.
Exam-style question
A student counted the number of pebbles in ten quadrats on a beach: 4, 6, 6, 7, 9, 10, 11, 13, 15, 19.
(a) Calculate the median. [1 mark]
(b) Calculate the range. [1 mark]
(c) Calculate the inter-quartile range. [3 marks]
Model answer
(a) Middle two values are 9 and 10, so median = (9 + 10) ÷ 2 = 9.5 (1).
(b) 19 − 4 = 15 (1).
(c) Lower half is 4, 6, 6, 7, 9, so Q1 = 6 (1). Upper half is 10, 11, 13, 15, 19, so Q3 = 13 (1). IQR = 13 − 6 = 7 (1).
Exam tip
With ten values there is no single middle one, so split the data into two halves of five and find each quartile as the middle value of its half. Show each step so you pick up marks even if you slip.