Statistics Toolkit (Edexcel IGCSE Maths): Revision notes

Section 1

What are mean, median, mode and range, and when do we use each?

For a list of data, you should be able to calculate all four measures quickly.

Measure	Definition	Formula/Method
Mean	The average value	$\bar{x} = \dfrac{\sum x}{n}$ (sum of values ÷ number of values)
Median	The middle value when data is ordered	Order the data; middle value, or mean of two middle values if $n$ is even
Mode	The most frequently occurring value	Value(s) with the highest frequency
Range	A measure of spread	Largest value − smallest value

Worked example: Find the mean, median, mode and range of: 4, 7, 4, 9, 6, 4, 8

Ordered: 4, 4, 4, 6, 7, 8, 9
Mean: $\dfrac{4+7+4+9+6+4+8}{7} = \dfrac{42}{7} = 6$
Median: middle (4th) value = 6
Mode: 4 (appears three times)
Range: $9 - 4 = 5$

When $n$ is even, the median is the mean of the two middle values, e.g. for 8 data points, average the 4th and 5th values once ordered.

Key termsmeanmedianmoderange

Exam tip

Always order the data first before finding the median — it's easy to pick the wrong middle value from an unordered list.

Common mistake

Range is a single number (largest − smallest), not an interval like "4 to 9". Writing "4 to 9" instead of 5 loses marks.

Section 2

How do we find the mean, median and mode from a frequency table?

When data is given as discrete values with frequencies, use a running total.

$x$ (goals scored)	Frequency ( $f$ )	$x \times f$
0	3	0
1	8	8
2	6	12
3	3	9
Total	20	29

Mean: $\bar{x} = \dfrac{\sum fx}{\sum f} = \dfrac{29}{20} = 1.45$

Mode: the $x$ value with the highest frequency — here it is 1 (frequency 8).

Median: with $n = 20$ , the median is the mean of the 10th and 11th values. Use cumulative frequency to locate them: 0 (positions 1–3), 1 (positions 4–11), 2 (positions 12–17). Both the 10th and 11th values fall in the "1 goal" group, so the median is 1.

Key termsfrequency tablecumulative frequency

Example

Use a running (cumulative) total of frequency down the table to work out which position the median falls in, then read across to the matching $x$ value.

Common mistake

Do not divide $\sum fx$ by the number of rows in the table — always divide by $\sum f$ , the total frequency (total number of data items).

Section 3

How do we estimate the mean of grouped data?

When data is grouped into class intervals, we cannot find the exact mean because individual values are unknown — we can only estimate it using midpoints.

Height, $h$ (cm)	Frequency ( $f$ )	Midpoint ( $x$ )	$x \times f$
$150 \le h < 160$	4	155	620
$160 \le h < 170$	10	165	1650
$170 \le h < 180$	6	175	1050
Total	20		3320

Method:

Find the midpoint of each class interval: $\text{midpoint} = \dfrac{\text{lower bound} + \text{upper bound}}{2}$
Multiply each midpoint by its frequency.
Divide the total of $x \times f$ by the total frequency: $\text{estimated mean} = \dfrac{\sum fx}{\sum f} = \dfrac{3320}{20} = 166 \text{ cm}$

This is always called an estimate because we assume every value in a class is at the midpoint, which is not exactly true.

Key termsgrouped dataclass intervalmidpointestimated mean

Exam tip

Always write "estimated mean" (or "estimate of the mean") in your answer for grouped data — examiners look for this exact wording.

Common mistake

Do not use the class boundary or width as $x$ — always use the midpoint of each interval.

Section 4

What are the modal class and median class, and how do we find them?

For grouped data, we cannot state a single mode or median value — instead we identify a class.

Modal class: the class interval with the highest frequency (no calculation needed — just read off the table).
Median class: the class interval containing the middle value(s). Use cumulative frequency to find the position of the median ( $\frac{n}{2}$ or between the $\frac{n}{2}$ th and $(\frac{n}{2}+1)$ th values), then identify which class that position falls into.

Worked example (using the height table above, $n = 20$ ):

Modal class: $160 \le h < 170$ (frequency 10, the highest)
Median position: between the 10th and 11th values. Cumulative frequencies: 4 (up to 160), 14 (up to 170). Both the 10th and 11th values fall in $160 \le h < 170$ , so the median class is $160 \le h < 170$ .

Key termsmodal classmedian class

Think of it like this

Think of the modal class as the "most popular neighbourhood" (highest frequency) and the median class as the "neighbourhood where the middle person lives" (found by counting along cumulative frequency).

Common mistake

Never give a single number for the modal class or median class of grouped data — always state the whole class interval, e.g. " $160 \le h < 170$ ", not "165".

Section 5

How do we compare two distributions?

Exam questions often give two data sets (e.g. two classes' test scores) and ask you to compare them. You must comment on both average and spread, using the context.

Feature	What to compare	Example comment
Average	Mean, median or mode	"Class A has a higher mean score than Class B, so on average Class A performed better."
Spread	Range (or interquartile range)	"Class B has a smaller range, so Class B's scores were more consistent."

Key rule: a comparison needs two parts — one sentence about average, one sentence about spread — both written in context (referring to what the data represents, e.g. "scores", "heights"), not just "Set A is bigger than Set B".

Key termsdistributionconsistency

Exam tip

Structure every comparison answer as: (1) compare an average, (2) compare a spread measure, both in context. This is what secures both comparison marks.

Common mistake

Comparing only averages ("Group A did better") without mentioning spread/consistency will lose at least one mark on comparison questions.

Must Know

Mean = $\dfrac{\sum x}{n}$ (or $\dfrac{\sum fx}{\sum f}$ for frequency tables); median = middle value when ordered; mode = most frequent value; range = largest − smallest.
For grouped data, always say estimated mean, calculated from $\dfrac{\sum fx}{\sum f}$ using class midpoints.
Modal class = interval with highest frequency; median class = interval containing the middle value(s), found via cumulative frequency.
Always state modal class and median class as a full interval, never a single number.
When comparing distributions, always give one comment on average AND one comment on spread, both in context.
Order data before finding the median, and remember the range is a single value, not a description of an interval.

That's the notes covered.

Carry on to the next subtopic.