Problem solving with cumulative frequency and histograms

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Lesson details

Learning outcome

I can use my knowledge of cumulative frequency and histograms to solve problems.

Key learning points

Different representations of the same data set give a fuller picture
It is important to identify what features are used to analyse a data set
It is important to identify what features are used to compare two data sets

Keywords

Box plot - A box plot is a diagram that clearly shows the minimum and maximum value of a data set long with the three quartiles.
Lower quartile - The lower quartile is the value under which 25% of data points are found when they are arranged in increasing order. Also known as the first quartile (Q1).
Upper quartile - The upper quartile is the value under which 75% of data points are found when they are arranged in increasing order. Also known as the third quartile (Q3).
Histogram - A histogram is a diagram consisting of rectangles whose area is proportional to the frequency in each class and whose width is equal to the class interval.

Common misconception

Histograms with unequal bar widths are plotted against the frequency and/or pupils read the frequency density as the frequency of the class width.

Show what a histogram with unequal bar widths looks like when plotted against frequency. This visual aid will show pupils the distribution of data is difficult to see.

Teacher tip

Discuss pros and cons to the assumption that data is assumed to be equally distributed in each bar. Pros - easier to calculate, and accurately represents the frequency within the bar. Cons - Not accurate or true reflection of how the data is distributed in the bar, hence why we estimate.

Licence

This content is © Oak National Academy Limited (2025), licensed on Open Government Licence version 3.0

except where otherwise stated. See Oak's terms & conditions

(Collection 2).

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Prior knowledge starter quiz

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6 Questions

Q1.
Which of the following is true for a histogram with unequal bar widths?

The highest bar shows the modal class.

Correct answer: The bar with the largest area shows the modal class.

The frequency is the frequency density $$\div$$ class width.

The lowest bar shows the class width with the lowest frequency.

The highest bar shows the class width with the highest frequency.

Q2.
Here are the times taken to type the alphabet. Match the class width with its frequency.

Correct Answer:$$0 < t \leq 10$$,Frequency = 4

Frequency = 4

Correct Answer:$$10 < t \leq 15$$,Frequency = 8

Frequency = 8

Correct Answer:$$15 < t \leq 20$$,Frequency = 20

Frequency = 20

Correct Answer:$$20 < t \leq 30$$,Frequency = 16

Frequency = 16

Correct Answer:$$30 < t\leq 50$$,Frequency = 5

Frequency = 5

Q3.
Here is a histogram showing the time it takes to solve a puzzle. Match the class width with its frequency.

Correct Answer:$$0 < t\leq 2$$,10

Correct Answer:$$2 < t \leq 4$$,12

Correct Answer:$$4 < t \leq 5$$,15

Correct Answer:$$5 < t\leq 10$$,14

Correct Answer:$$10 < t \leq 12$$,5

Q4.
Which class' scores were more consistent?

A - median is lower

B - median is higher

Correct answer: A - similar IQR to B but smaller range

B - similar IQR to A but smaller range

Q5.
Calculate the interquartile range from this cumulative frequency graph.

Correct Answer: 8

Q6.
This table show information about the times taken by some children and adults to complete a puzzle. Compare their times. Which is the correct response to this question?

On average the children did better and their times are more consistent.

On average the children did better and their times are less consistent.

The children's median was lower and their interquartile range was higher.

Correct answer: On average the adults did better but their times are less consistent.

The adults median was better and their interquartile range was lower.

Assessment exit quiz

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6 Questions

Q1.
Match the statement with the definition.

Correct Answer:A boxplot,a diagram that clearly shows the min and max values and the quartiles

a diagram that clearly shows the min and max values and the quartiles

Correct Answer:A histogram,a diagram with rectangles whose area is proportional to the frequency

a diagram with rectangles whose area is proportional to the frequency

Correct Answer:Lower quartile,the value under which 25% of data points are found

the value under which 25% of data points are found

Correct Answer:Upper quartile,the value under which 75% of data points are foun

the value under which 75% of data points are foun

Q2.
Here are some test scores for Class A and Class B. Select the statement which is true.

Correct answer: 50% of Class A scores were more consistent with the median than Class B.

50% of Class B scores were more consistent with the median than Class A.

Both classes were equally consistent.

Q3.
Here are two box plots showing the scores of two classes. Select the statements which are true.

The medians are the same.

Correct answer: The ranges are the same.

Correct answer: The IQRs are the same.

Correct answer: The medians are different.

The ranges are different.

Q4.
Here are three different cumulative frequency graphs showing different classes' test percentages. The scale is unlabelled but is the same for all graphs. Which class has the lowest average score

Correct answer: C

Q5.
Here are three box plots showing different classes' test percentages. The scale is unlabelled but is the same for all graphs.

Correct Answer:Class C,has the most consistent scores as the IQR and range is the smallest

has the most consistent scores as the IQR and range is the smallest

Correct Answer:Class B,is on average the lowest performing class

is on average the lowest performing class

Correct Answer:Class A and B,have the same IQR and range

have the same IQR and range

Correct Answer:Class A and C,have on average the higher performing students

have on average the higher performing students

Q6.
Here are some masses of sweets in a bag. 50% of the sweets are ≤ 120 g but greater than 80 g. What percentage are greater than 120 g but less than or equal to 180 g?

Correct Answer: 25%, 25

To help you plan your 11 maths lesson on: Problem solving with cumulative frequency and histograms, download all teaching resources for free and adapt to suit your pupils' needs...

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Problem solving with cumulative frequency and histograms

Lesson slides

Lesson details

Learning outcome

Key learning points

Keywords

Common misconception

Teacher tip

Licence

Lesson video

Worksheet

Prior knowledge starter quiz

6 Questions

Q1.Which of the following is true for a histogram with unequal bar widths?

Q2.Here are the times taken to type the alphabet. Match the class width with its frequency.

Q3.Here is a histogram showing the time it takes to solve a puzzle. Match the class width with its frequency.

Q4.Which class' scores were more consistent?

Q5.Calculate the interquartile range from this cumulative frequency graph.

Q6.This table show information about the times taken by some children and adults to complete a puzzle. Compare their times. Which is the correct response to this question?

Assessment exit quiz

6 Questions

Q1.Match the statement with the definition.

Q2.Here are some test scores for Class A and Class B. Select the statement which is true.

Q3.Here are two box plots showing the scores of two classes. Select the statements which are true.

Q4.Here are three different cumulative frequency graphs showing different classes' test percentages. The scale is unlabelled but is the same for all graphs. Which class has the lowest average score

Q5.Here are three box plots showing different classes' test percentages. The scale is unlabelled but is the same for all graphs.

Q6.Here are some masses of sweets in a bag. 50% of the sweets are ≤ 120 g but greater than 80 g. What percentage are greater than 120 g but less than or equal to 180 g?

How to plan a lesson using our resources

Q1.
Which of the following is true for a histogram with unequal bar widths?

Q2.
Here are the times taken to type the alphabet. Match the class width with its frequency.

Q3.
Here is a histogram showing the time it takes to solve a puzzle. Match the class width with its frequency.

Q4.
Which class' scores were more consistent?

Q5.
Calculate the interquartile range from this cumulative frequency graph.

Q6.
This table show information about the times taken by some children and adults to complete a puzzle. Compare their times. Which is the correct response to this question?

Q1.
Match the statement with the definition.

Q2.
Here are some test scores for Class A and Class B. Select the statement which is true.

Q3.
Here are two box plots showing the scores of two classes. Select the statements which are true.

Q4.
Here are three different cumulative frequency graphs showing different classes' test percentages. The scale is unlabelled but is the same for all graphs. Which class has the lowest average score

Q5.
Here are three box plots showing different classes' test percentages. The scale is unlabelled but is the same for all graphs.

Q6.
Here are some masses of sweets in a bag. 50% of the sweets are ≤ 120 g but greater than 80 g. What percentage are greater than 120 g but less than or equal to 180 g?