Charts, averages and cumulative frequency – Year 11 GCSE maths revision notes to download as PDF

Worked example 1: averages

Find the mean, median, mode and range of \(4,7,7,9,12,15,16\).

Mean \(=\dfrac{70}{7}=10\), median \(=9\), mode \(=7\), range \(=16-4=12\).

Worked example 2: a frequency table

Scores 1 to 5 have frequencies 3, 4, 6, 5, 2. Find the mean, mode and median.

Total \(=20\); \(\sum fx=3+8+18+20+10=59\); mean \(=2.95\). The mode is 3 and the median (10th and 11th values) is 3.

Worked example 3: estimated mean

Frequencies 4, 6, 10, 5 in classes of width 10 starting at 0. Estimate the mean and give the modal class.

Midpoints are 5, 15, 25, 35. \(\sum fx=20+90+250+175=535\) and \(\sum f=25\), so the estimated mean is \(21.4\). The modal class is \(20\lt h\le30\).

Worked example 4 (Higher): frequency density

Class \(10\lt t\le30\) has frequency 50. A bar \(30\lt t\le50\) has height 1.5. Find the frequency density of the first class and the frequency of the second.

The class width is 20, so the frequency density is \(50\div20=2.5\). The second bar has frequency \(1.5\times20=30\).

Worked example 5 (Higher): reading quartiles

Ages have class frequencies 5, 15, 30, 35, 15 in classes of width 10 from 0. Find the median and the interquartile range.

The cumulative frequencies are 5, 20, 50, 85, 100. The median is at the 50th value, 30. \(Q_1\) at the 25th value is \(20+\dfrac{5}{30}\times10\approx21.7\) and \(Q_3\) at the 75th value is \(30+\dfrac{25}{35}\times10\approx37.1\). So \(IQR\approx15.5\).

Worked example 6: comparing two teams

Team A has mean 52 and range 10; Team B has mean 50 and range 30. Compare them. Then compare two box plots.

Team A scores higher on average and is more consistent. For box plots, for example: "Group B has a higher median, so they are generally faster, and a smaller IQR, so their times are more consistent."

Worked example 7: Choosing the best average

The times (in seconds) of eight pupils: 12, 15, 15, 18, 21, 24, 24, 35. Find the mean, median and modes, and explain which average best represents the group.

Total \(=12+15+15+18+21+24+24+35=164\), so the mean is \(164\div8=20.5\). The median is the mean of the 4th and 5th values: \((18+21)\div2=19.5\). There are two modes, 15 and 24. The value 35 is much larger than the others, which pulls the mean up, so the median (19.5) is the better description of a typical time.

Worked example 8: Comparing two data sets

Team A has mean 14 and range 6. Team B has mean 12 and range 15. Compare the two teams.

Compare an average and a spread. Team A has the higher mean (14 compared with 12), so they score more on average. Team A also has the smaller range (6 compared with 15), so they are more consistent. Always write the comparison in the context of the question and use the word "consistent" for spread.

Worked example 9: Frequency density without a table

A histogram bar for \(20\lt x\le50\) has height 1.6 on the frequency density axis. How many items does it represent?

The class width is \(50-20=30\). Frequency \(=\) frequency density \(\times\) class width \(=1.6\times30=48\). So the bar represents 48 items. The area of a bar in a histogram is the frequency, which is why the bars can have different widths.

Worked example 10: Using a cumulative frequency graph (Higher)

From a cumulative frequency graph for 200 pupils, the reading at 60 marks is 150. How many pupils scored more than 60? Give the percentage.

150 pupils scored 60 or fewer, so \(200-150=50\) pupils scored more than 60. As a percentage: \(50\div200\times100=25\%\). If the question asks for the median, read across from 100, which is half of 200. For the lower quartile read across from 50 and for the upper quartile from 150.

Worked example 11: Drawing and reading a box plot (Higher)

A data set has minimum 4, lower quartile 9, median 13, upper quartile 20 and maximum 31. Work out the range and the interquartile range, and say what the box contains.

Range \(=31-4=27\). Interquartile range \(=20-9=11\). The box runs from 9 to 20 and so contains the middle half of the data. The median line is closer to the lower quartile (\(13-9=4\) against \(20-13=7\)), which tells us the data is positively skewed: the upper half is more spread out.