Comparing Two Sets of Data

Compare two sets of data with one comment on the average and one on the spread, and say which group is higher or more consistent, with a reason.

In the exam for Cambridge IGCSE Maths 0580, Edexcel IGCSE Maths 4MA1, Cambridge O Level Maths 4024 · 17 real past-paper questions below

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Two Things to Compare

To compare two groups, look at two things: where the data sits, and how spread out it is.

Measure	What it tells you
Average: mean or median	which group is higher on average
Spread: range or IQR	which group is more consistent

Rule

A smaller spread means the values are closer together, so the group is more consistent.

Key idea

IQR means interquartile range. You find it in the lesson Finding the Median and Quartiles.

Key idea

One extreme value, far from the rest, pulls the mean and the range a long way. The median and the IQR hardly change. So if a question asks why the mean is not a good average, say: there is an extreme value that makes the mean too high (or too low).

Two Comparisons, in Context

The 2-mark question: "Write down two comparisons." 20 students sat a Science test and a Maths test, each out of 50.

Test	Median, IQR
Science	31, 9
Maths	26, 14

Worked example

1
$31 > 26$ — Compare the medians first. The median is the middle mark, so it shows a typical student. Science's is higher.
2
$Science marks were higher on average$ — Comparison 1, in context: say what the bigger median means for these two tests, not just “31 is bigger”.
3
$9 < 14$ — Now compare the IQRs. The IQR is how spread out the middle half of the marks is. Science's is smaller.
4
$Science marks were more consistent$ — Comparison 2: a smaller IQR means the marks sit closer together, so they are more consistent.

Rule

One sentence about the average, one about the spread. Name the groups, say what it means for them, and use the numbers: “On average the Science marks were higher, 31 against 26.”

Common mistake

Writing "Science 31, Maths 26". That only copies the numbers. Say which is higher and what it means.

State Which, With a Reason

The 1-mark question: "State, giving a reason, who scored more on average" or "who was more consistent". Two basketball players scored points in 10 games.

Player	Median, IQR
Rana	14, 6
Omar	11, 3

Worked example

1
$14 > 11$ — “Who scored more on average?” is about the average, so look at the medians. Rana's is higher.
2
$Rana, as her median is higher$ — Give the name AND the reason. The name on its own gets no mark.
3
$3 < 6$ — “Who was more consistent?” is about spread, so look at the IQRs. Omar's is smaller: his points stay closer together from game to game.
4
$Omar, as his IQR is smaller$ — Name plus reason again. Use the IQR here, never the median.

Common mistake

Writing only the name, or only the numbers (“Rana 14, Omar 11”). Both get no mark: say who, and that their median is higher. Do not add the other measure either: “Rana, as she has a higher median and a bigger IQR” can lose the mark. Also, the bigger IQR is the less consistent one.

Key idea

"Is Lina correct?" works the same way: say yes or no, then give the two numbers. With a mean and a range, use the same rules.

Mean and Range: Same Rules

Many questions give a mean and a range instead. The rules are the same: the higher mean is higher on average, and the smaller range is more consistent.

Café	Mean, range (customers a day)
Café A	64, 30
Café B	71, 48

Worked example

1
$71 > 64$ — Compare the means first. Café B's is higher.
2
$On average, Caf \overset{e}{ˊ} B had more customers$ — Comparison 1, in context: say what the higher mean means for the cafés, more customers a day.
3
$30 < 48$ — Now compare the ranges. Café A's is smaller, so its daily numbers stay closer together.
4
$Caf \overset{e}{ˊ} A’s customer numbers were more consistent$ — Comparison 2. That is one comment on the average and one on the spread, so both marks.

Common mistake

“Which café had greater variation?” asks the opposite way round. Greater variation, or more spread out, means the bigger range: Café B, as its range is 48 against 30.

Key idea

Each comparison is usually 1 mark. If you worked out the mean or range yourself in an earlier part, the marks follow your own numbers, so check them first. Any number you quote must be correct.

Which Method? Read the Question

Decide first: is it asking about the average, the spread, or both?

You see	You do
Write two comparisons	one on average, one on spread, in context
More on average	higher mean or median
More consistent	smaller range or IQR
Greater variation, more spread out	bigger range or IQR
Two bar charts or pie charts	compare the modes, or how often a value comes up
Why is the mean not suitable?	an extreme value drags it; use the median
Is he correct? Give a reason	yes or no, then the numbers
Find the median and IQR first	Finding the Median and Quartiles
Find the mean or range of a list	Mean, Median, Mode and Range
Two lists back to back	Stem-and-Leaf Diagrams

Key idea

A comparison describes these two groups only. It does not explain why one group did better.

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