GCSE · Geography · AQA · Spec 8035

Statistical skills - central tendency and spread

Eight of these stones are 10 cm or smaller, and one is a 32 cm whopper. So what's the 'average' stone? That depends on which average you pick.

Guess first, then check

Where does the 'average' stone really sit?

Illustrative data: a student measures the longest side of nine stones from one river bed (cm): 2, 3, 3, 3, 5, 6, 8, 10, 32. Quick reminder: the mode is the most common value, the median is the middle one when they're in order, and the mean is the total shared out equally. Don't calculate yet. Drag each marker to where you think it lands, then check.

Place ModePlace MedianPlace MeanLow — Smallest stone (bottom of the range)High — Largest stone (top of the range)Mean* — The mean if the 32 cm stone is left out

Grouped data

What's the mode when you can't see the values?

Illustrative survey: 40 visitors in a town centre were asked how far they had travelled, and the answers were grouped. Distance x (km) and number of visitors: 0 < x ≤ 10: 4 · 10 < x ≤ 20: 12 · 20 < x ≤ 30: 10 · 30 < x ≤ 40: 8 · 40 < x ≤ 50: 4 · 50 < x ≤ 60: 2.

Someone asks for the mode of these distances. Which answer is closest to what you'd say right now?

The mode is 12, because that's the biggest number in the table.The mode is 15 km, the middle of the class with the most visitors.The modal class is 20 < x ≤ 30, because that's where the middle visitor is.The modal class is 10 < x ≤ 20 km, because more visitors are in that class than any other.

How sure are you?

I'm sureI think soI'm guessing

The mode is 12, because that's the biggest number in the table.

This is a common misconception

12 is a frequency: it counts visitors, not kilometres. It tells you HOW MANY people are in the busiest class, not how far they travelled. The answer to 'what's the mode distance?' has to be a distance. So look for which class that 12 belongs to.

The mode is 15 km, the middle of the class with the most visitors.

This is a common misconception

You found the right class, which is the hard part. But 15 km is a guess. The 12 visitors in that class could have travelled 11 km, 19 km or anything in between, and grouping has hidden the exact values. You can't name a single most common distance, so you name the class.

The modal class is 20 < x ≤ 30, because that's where the middle visitor is.

This is a common misconception

Good instinct to count through: the 20th visitor does land in 20 < x ≤ 30. But that's the class holding the MEDIAN, the middle value. The mode is about what's most common, not what's in the middle. Which class has the most visitors in it?

The modal class is 10 < x ≤ 20 km, because more visitors are in that class than any other.

This matches the evidence

Exactly. Once data is grouped, the separate values are hidden, so there's no single mode to find. You give the modal class instead: the class with the highest frequency. Here that's 10 < x ≤ 20 km, with 12 visitors.

Grouped data hides the individual values, so you can't give an exact mode. Give the modal class: the class with the highest frequency, written as a class, not as the frequency.

Same 40 visitors

Finding the middle of data you can't see

1. Complete the cumulative frequency column.
Distance travelled (km)	Frequency	Cumulative frequency
0 < x ≤ 10	4
10 < x ≤ 20	12
20 < x ≤ 30	10
30 < x ≤ 40	8
40 < x ≤ 50	4
50 < x ≤ 60	2

A cumulative frequency is a running total: how many values so far. Each total is plotted at the END of its class, because that's the point by which all those values have been counted. For 40 visitors, the median is the 20th, so you read across at 20; the quartiles sit a quarter and three-quarters of the way through, at 10 and 30. The inter-quartile range (upper quartile − lower quartile) is the spread of the middle half. The range here, 58 − 2 = 56 km, depends only on the nearest and farthest visitors. The IQR ignores the top and bottom quarters, so one unusually long journey can't stretch it.

Your turn: new data

Calculate it, then choose and justify

Illustrative data: a student counts the pedestrians passing seven points along one street in five minutes: 14, 9, 16, 11, 93, 12, 13. (a) Calculate the mean. (b) Find the median. (c) Which average better represents a typical point on this street? Which measure of spread would you use, the range or the inter-quartile range? Explain both choices. [5 marks]

Your answer

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The mark scheme

(a) Mean = 168 ÷ 7 = 24 pedestrians. (1 mark)
(b) Median = 13: the 4th of the seven counts once they're in order (9, 11, 12, 13, 14, 16, 93). (1 mark)
(c) Chooses the median, because the 93 pulls the mean up so that it's higher than six of the seven counts. (1 mark)
(c) Says the median (13) sits among the typical counts, because it isn't affected by the one extreme count. (1 mark)
(c) Chooses the inter-quartile range, because the range (93 − 9 = 84) is set by the one extreme count, while the IQR uses only the middle half of the data. (1 mark)

Model answer

(a) 14 + 9 + 16 + 11 + 93 + 12 + 13 = 168, and 168 ÷ 7 = 24 pedestrians. (b) In order: 9, 11, 12, 13, 14, 16, 93. The middle (4th) value is 13 pedestrians. (c) The median is better. The count of 93 is far above the rest and drags the mean up to 24, which is higher than six of the seven points, so the mean doesn't describe a typical point. The median, 13, sits right among the typical counts. For spread I'd use the inter-quartile range, not the range. The range, 93 − 9 = 84, is almost all down to that one extreme point, but the IQR only looks at the middle half of the counts, so the extreme value can't stretch it.

Before you choose, scan the data for a value far from the rest. If there is one, the median and the IQR usually describe the data better, because the mean and the range both get pulled by it.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Mean, median and mode; range and inter-quartile range. Each describes the same data differently, and the skill is picking the one that tells the truth about it.

What you need to know

Calculate the mean, median and mode of a set of values, and its range
Give the modal class for grouped data, and say why it isn't a single mode
Build a cumulative frequency graph and read the median and the lower and upper quartiles from it
Work out the inter-quartile range from the quartiles
Choose and justify the average and the measure of spread that best represent a data set

The big picture

Mean, median and mode are three different ways of describing a typical value, and range and inter-quartile range are two ways of describing spread. Grouped data gives a modal class, and a cumulative frequency graph gives the median and quartiles. The key skill is noticing when an extreme value makes the mean and range misleading, and choosing the median and inter-quartile range instead.

Key points

1Mean = total of the values ÷ number of values. It uses every value, so one extreme value can drag it a long way.

2Median = the middle value once the data is in order (halfway between the middle two if there's an even number of values). An extreme value barely moves it.

3Mode = the most common value. Grouped data hides the individual values, so you give the modal class: the class with the highest frequency.

4Range = largest value − smallest value. It depends only on the two end values, so one extreme value stretches it.

5Cumulative frequency is a running total, plotted at each class's upper bound. For n values, read across at n/4, n/2 and 3n/4 for the lower quartile, median and upper quartile.

6Inter-quartile range = upper quartile − lower quartile. It measures the spread of the middle half, so extreme values at either end can't stretch it.

7When data has an extreme value, the median and IQR usually represent it better than the mean and range.

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Worked example

Problem

Illustrative fieldwork: a student measures the depth of two streams at 11 points each. Stream A depths (cm): 12, 4, 21, 18, 9, 25, 15, 7, 30, 16, 22. Stream B has a median of 17 cm and an inter-quartile range of 5 cm. Find the median and inter-quartile range of Stream A, then decide which stream has the more consistent depth.

⚠ Watch out

Finding the 'middle' of the list as it was written, without putting the values in order first. The median is the middle of the ORDERED data. The middle of an unsorted list is just whichever value happened to be written there.

🧠

Memory hook

The mean is a people-pleaser. It tries to be fair to every value, so one giant drags it along. The median just stands in the middle of the queue and doesn't care how tall the last person is.

✓

Check yourself

Cover the page. Find the mean, median, mode and range of 6, 2, 9, 4, 4. Then predict which one changes most if the 9 becomes 49, and check.

Flashcards

(7)

How do you calculate the mean?

Add up all the values, then divide by how many values there are.

How do you find the median?

Put the values in order and take the middle one. With an even number of values, it's halfway between the middle two.

Mode or modal class: which do you give for grouped data, and why?

The modal class: the class with the highest frequency. Grouping hides the individual values, so there's no exact mode to find.

How do you find the range, and what is its weakness?

Largest value − smallest value. It depends only on the two end values, so one extreme value can stretch it a lot.

What does the inter-quartile range tell you that the range doesn't?

How spread out the middle half of the data is. Because it leaves out the top and bottom quarters, an extreme value can't stretch it.

On a cumulative frequency graph for n values, where do you read across for the lower quartile, median and upper quartile?

At n/4, n/2 and 3n/4 on the cumulative frequency axis, then straight down to read the value.

Where do you plot each point on a cumulative frequency graph?

At the upper bound of its class, because that's the point by which all the values in the running total have been counted.

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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Where does the 'average' stone really sit?

What's the mode when you can't see the values?

Finding the middle of data you can't see

Calculate it, then choose and justify

More AQA GCSE Geography topics