IMAT Logical Reasoning · Data Interpretation

Averages, Percentages and Change from Data

Find the mean, median and mode from a frequency table, and describe changes in data with absolute differences, percentage changes and shares of a total.

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Why this matters

The 2024 paper asked for a percentage worked out from a frequency table. The Cambridge papers asked which category contributed most to a rise, which ratio was closest to a given one, and a fuel consumption from a logbook.

Concept 1 of 2: Mean, median and mode from a frequency table

A frequency table is a compressed list: "3 appears 6 times" stands for six 3s. So the total of all values is each value times its frequency, added up, and the number of values is the sum of the frequencies. For the median, count along the frequencies until you reach the middle position.

Definition

For values $x$ with frequencies $f$ :

Number of values $n = \sum f$ .
Mean $= \dfrac{\sum f x}{\sum f}$ .
Median: the value in position $\dfrac{n+1}{2}$ when listed in order. Add the frequencies up (cumulative frequency) until you pass that position. For even $n$ , take the mean of the two middle values.
Mode: the value with the highest frequency (the value, not the frequency itself).
Range: largest value minus smallest value that actually occurs.
Proportion above a mark: add the frequencies above it and divide by $n$ . Read "higher than 5" and "5 or more" carefully.
For grouped data (10 to 20, 20 to 30), use each group's midpoint as $x$ ; the mean is then an estimate.

Mean of a frequency table

\bar{x} = \frac{\sum f x}{\sum f}

$x$ each value (or group midpoint)
$f$ how many times that value occurs

Worked example

A class records how many siblings each student has:

Siblings	0	1	2	3	4
Number of students	5	9	7	3	1

Find the mean, median and mode.

Practice this conceptself-check · 3 quick reps

Divide by the number of values, not the number of columns

A table with 5 columns can describe 40 people. The mean divides

\sum f x

\sum f

(40 here), never by the number of columns.

The mode and median are values

If the value 7 occurs 12 times, the mode is 7, not 12. The same goes for the median: it is the value in the middle position, not the frequency of that value.

Concept 2 of 2: Percentage change, share and contribution from data

There are three different ways to say which category "grew most": by the largest absolute rise, by the largest percentage rise, or by the largest contribution to the total rise. They can give three different answers, so decide which one the question means before calculating. A share is a percentage of the total at one moment; it can fall even while the amount itself rises.

Definition

Absolute change = new $-$ old (in the table's units).
Percentage change = $\dfrac{\text{new} - \text{old}}{\text{old}} \times 100\%$ .
Share of a category = category ÷ total for the same year.
Contribution to a total rise = the category's absolute change ÷ the total's absolute change.
Comparing ratios (per person, per kg, per 100 km): compute the same ratio for every row, with the same units, before comparing.

Percentage change and share

\%\ \text{change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\% \qquad \text{share} = \frac{\text{part}}{\text{total}}

$\text{old}, \text{new}$ the values at the start and end of the period
$\text{part}, \text{total}$ a category and the sum of all categories in the same year

Worked example

A company's sales (units) in three regions:

Region	2020	2024
North	1,200	1,500
South	800	1,080
East	2,000	2,400
Total	4,000	4,980

Which region had the largest absolute rise, which the largest percentage rise, and what happened to East's share of total sales?

Practice this conceptself-check · 3 quick reps

Largest rise and largest percentage rise are different questions

A big category can gain the most units while growing by the smallest percentage. Check whether the question asks for the change in units, the percentage change, or the share.

A falling share is not a falling amount

If the total grows faster than one category, that category's share falls even though it grew. Options saying "East's sales fell" from a smaller share are wrong.

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Mean, median and mode from a frequency table
Mean of a frequency table
$\bar{x} = \frac{\sum f x}{\sum f}$
Percentage change, share and contribution from data
Percentage change and share
$\%\ \text{change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\% \qquad \text{share} = \frac{\text{part}}{\text{total}}$

Watch out for (4)

Divide by the number of values, not the number of columns
→ Mean, median and mode from a frequency table
The mode and median are values
→ Mean, median and mode from a frequency table
Largest rise and largest percentage rise are different questions
→ Percentage change, share and contribution from data
A falling share is not a falling amount
→ Percentage change, share and contribution from data

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