Problem solving with cumulative frequency and histograms | Maths

Starter quiz

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

The highest bar shows the modal class.

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

$0 < t \leq 10$

⇔

Frequency = 4 ✓

$10 < t \leq 15$

⇔

Frequency = 8 ✓

$15 < t \leq 20$

⇔

Frequency = 20 ✓

$20 < t \leq 30$

⇔

Frequency = 16 ✓

$30 < t\leq 50$

⇔

Frequency = 5 ✓
Here is a histogram showing the time it takes to solve a puzzle. Match the class width with its frequency.

$0 < t\leq 2$

⇔

10 ✓

$2 < t \leq 4$

⇔

12 ✓

$4 < t \leq 5$

⇔

15 ✓

$5 < t\leq 10$

⇔

14 ✓

$10 < t \leq 12$

⇔

5 ✓
Which class' scores were more consistent?

A - median is lower

B - median is higher

A - similar IQR to B but smaller range ✓

B - similar IQR to A but smaller range
Calculate the interquartile range from this cumulative frequency graph.

'8' ✓
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.

On average the adults did better but their times are less consistent. ✓

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

Exit quiz

Match the statement with the definition.

A boxplot

⇔

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

A histogram

⇔

a diagram with rectangles whose area is proportional to the frequency ✓

Lower quartile

⇔

the value under which 25% of data points are found ✓

Upper quartile

⇔

the value under which 75% of data points are foun ✓
Here are some test scores for Class A and Class B. Select the statement which is true.

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

The medians are the same.

The ranges are the same. ✓

The IQRs are the same. ✓

The medians are different. ✓

The ranges are different.
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

A

B

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

Class C

⇔

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

Class B

⇔

is on average the lowest performing class ✓

Class A and B

⇔

have the same IQR and range ✓

Class A and C

⇔

have on average the higher performing students ✓
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?

'25%' ✓

Worksheet

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Lesson Details

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

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.

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.