Statistics — Histograms, Cumulative Frequency & Averages

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Corbett MathsHistograms — frequency density

Corbett MathsCumulative frequency and box plots

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GCSE Maths TutorBox Plots & Cumulative Frequency Graphs | Grade 6+ Series | GCSE Maths Tutor

Key facts & methods

Histograms

Frequency density = frequency ÷ class width. This is what you plot on the y-axis.
The area of each bar = frequency (not the height).
To find frequency from histogram: frequency = frequency density × class width.
Unequal class widths: must use frequency density. Equal class widths: can plot frequency directly.
Reading histograms: area = frequency. To find how many in a group, calculate frequency density × class width.

Cumulative frequency and box plots

Cumulative frequency: running total of frequencies. Plot at the upper class boundary.
S-shaped (ogive) curve when drawn correctly.
Median = value at n/2 on the cumulative frequency axis.
Lower quartile (Q1) = value at n/4. Upper quartile (Q3) = value at 3n/4.
Interquartile range (IQR) = Q3 − Q1. Measure of spread unaffected by outliers.
Box plot: shows minimum, Q1, median, Q3, maximum on a number line.

Exam questions — 5 questions · 12 marks · Edexcel 4MA1 style

3 marksHistogram — find frequency4MA1 style 🔢 Calculator

A histogram shows frequency density for the class 20 < x ≤ 30 as 3.5 and for 30 < x ≤ 50 as 1.8. How many values are in each class? (3 marks)

Hint: Frequency = frequency density × class width. Note the class widths are different (10 and 20).

+30 XP

4 marksCumulative frequency4MA1 style 🔢 Calculator

From a cumulative frequency diagram, 80 values have a maximum of 100. The median is read as 42, Q1 as 28, and Q3 as 58. (a) State the interquartile range. (b) A box plot is drawn. An outlier is any value more than 1.5 × IQR above Q3 or below Q1. What values are outliers? (4 marks)

Hint: IQR = Q3 − Q1. Outlier boundaries: Q3 + 1.5×IQR and Q1 − 1.5×IQR. Check if any values fall outside these boundaries.

+40 XP

3 marksMean from grouped data4MA1 style 🔢 Calculator

The table shows grouped data: 0(3 marks)

Hint: For each class, use the midpoint. Multiply each midpoint by its frequency. Sum all (midpoint × frequency). Divide by total frequency.

+30 XP

1 markFrequency density4MA1 style

In a histogram, the bar for the class 10 < x ≤ 20 has frequency density 4.5 and the class 20 < x ≤ 30 has frequency density 3. The class 30 < x ≤ 40 has frequency 18. What is the frequency density for this class?

A 18

B 180

C 1.8

D 0.18

1 markBox plot interpretation4MA1 style

Two box plots show the exam scores for class A and class B. Class A has median 65, IQR 15. Class B has median 58, IQR 28. Which statement is correct?

A Class A performed better on average and their scores were more consistent

B Class B performed better on average

C Class A has greater spread of results

D Both classes performed equally

Statistics — Histograms, Cumulative Frequency & Averages

Histograms

Cumulative frequency and box plots

Module complete! 🎉