Statistics Toolkit Notes

Q: What are the modal class and median class?

For grouped data you name a class interval, not a single value. The modal class is the interval with the highest frequency, read straight from the table.

Edexcel IGCSE Maths: Revision notes

Key facts

Mean $=\dfrac{\sum x}{n}$ (or $\dfrac{\sum fx}{\sum f}$ ); median is the middle value of ordered data; mode is the most frequent; range is largest $-$ smallest.
For grouped data say estimated mean, using class midpoints.
Modal class has the highest frequency; median class contains the middle value, found with cumulative frequency.
State modal and median classes as full intervals.
Compare distributions with one comment on average and one on spread, both in context.

Averages and range

Four quick measures: the average, the middle, the most common, and the spread.

When $n$ is even the median is the mean of the two middle values; for 8 values, average the 4th and 5th once ordered. Always order the data first. The mode can be more than one value, and the range is a single number.

\bar{x}=\dfrac{\sum x}{n}

	Definition	Method
Mean	The average value	$\bar{x}=\dfrac{\sum x}{n}$
Median	The middle value when ordered	Middle value, or mean of the two middle values
Mode	The most frequent value	Value(s) with the highest frequency
Range	Largest value − smallest value	A measure of spread

Definition

Mean:: The average value
Median:: The middle value when ordered
Mode:: The most frequent value
Range:: Largest value − smallest value

Method

Mean:: $\bar{x}=\dfrac{\sum x}{n}$
Median:: Middle value, or mean of the two middle values
Mode:: Value(s) with the highest frequency
Range:: A measure of spread

Worked example

Find the mean, median, mode and range of 4, 7, 4, 9, 6, 4, 8.

1
Ordered: 4, 4, 4, 6, 7, 8, 9.
2
Mean: $\dfrac{42}{7}=6$ . Median: the 4th value, 6. Mode: 4. Range: $9-4=5$ .

Find the median of 3, 9, 4, 8, 6, 5.

Frequency tables

Multiply each value by its frequency for the mean; use a running total to locate the median.

With goals $x=0,1,2,3$ and frequencies $3,8,6,3$ : $\sum f=20$ and $\sum fx=0+8+12+9=29$ .

$\bar{x}=\frac{\sum fx}{\sum f}=\frac{29}{20}=1.45$

The mode is the value with the highest frequency (1). For the median with $n=20$ , average the 10th and 11th values: positions 4 to 11 are "1 goal", so the median is 1.

Goals scored: frequency of each value

A table shows x = 1, 2, 3 with frequencies 2, 5, 3. What is the mean?

Grouped data

Use each class midpoint as if every value sat there; the result is an estimate.

With classes we cannot know individual values, so we calculate an estimated mean using midpoints, assuming every value is at the midpoint. For heights $150\le h<160$ , $160\le h<170$ , $170\le h<180$ with frequencies 4, 10, 6, the midpoints are 155, 165, 175 and $\sum fx=620+1650+1050=3320$ .

1
Find each midpoint
(lower + upper) ÷ 2
2
Multiply
each midpoint by its frequency
3
Add up
the x × f column
4
Divide
by the total frequency

Estimating the mean of grouped data

Worked example

Use the height table to estimate the mean height.

1
$\sum fx=620+1650+1050=3320$ and $\sum f=20$ .
2
Estimated mean $=\dfrac{3320}{20}$ .

Why is the mean of grouped data only an estimate?

Modal and median class

For grouped data you name a class interval, not a single value.

The modal class is the interval with the highest frequency, read straight from the table. The median class is the interval containing the middle value: find its position with cumulative frequency ( $\frac{n}{2}$ , or between the $\frac{n}{2}$ th and next value) and see which class it falls in.

Frequency of each class: the modal class is 160 ≤ h < 170

150\le h<160

	Frequency	Cumulative frequency	Midpoint x
$150\le h<160$	4	4	155
$160\le h<170$	10	14	165
$170\le h<180$	6	20	175

Frequency

$150\le h<160$ :: 4
$160\le h<170$ :: 10
$170\le h<180$ :: 6

Cumulative frequency

$150\le h<160$ :: 4
$160\le h<170$ :: 14
$170\le h<180$ :: 20

Midpoint x

$150\le h<160$ :: 155
$160\le h<170$ :: 165
$170\le h<180$ :: 175

Worked example

Find the modal class and the median class.

1
Modal class: $160\le h<170$ has the highest frequency, 10.
2
The median is between the 10th and 11th values. Cumulative frequency is 4 then 14, so both fall in $160\le h<170$ .

A grouped table has 40 values. Which values give the median position?

Comparing distributions

Make two comments: one on average and one on spread, both in context.

Exam questions give two data sets and ask for a comparison. Use an average (mean, median or mode) and a spread (range or interquartile range), and refer to what the data represents, such as "scores" or "heights", not just "Set A is bigger".

1
Compare averages
mean, median or mode, with the higher or lower named
2
Compare spread
range or interquartile range: smaller means more consistent
3
Say it in context
refer to the scores, heights or times, not Set A and Set B

Comparing two distributions

Average Spread What to compare Mean, median or mode Range or interquartile range Example comment "Class A has a higher mean score, so on average Class A did better." "Class B has a smaller range, so its scores were more consistent."

	Average	Spread
What to compare	Mean, median or mode	Range or interquartile range
Example comment	"Class A has a higher mean score, so on average Class A did better."	"Class B has a smaller range, so its scores were more consistent."

Average

What to compare:: Mean, median or mode
Example comment:: "Class A has a higher mean score, so on average Class A did better."

Spread

What to compare:: Range or interquartile range
Example comment:: "Class B has a smaller range, so its scores were more consistent."

Class A has mean 62 and range 40. Class B has mean 58 and range 15. Which comparison earns full marks?

Try an exam question

The table shows the times, $t$ minutes, of $20$ runners: $10\le t<20$ (5 runners), $20\le t<30$ (8 runners), $30\le t<40$ (7 runners). (a) Estimate the mean time. (b) Write down the modal class.

[4 marks]

That's the notes covered.

Carry on to the next subtopic.