### Overview

In order to understand and use cumulative frequency, you need knowledge covered in Data Collection, Interpreting and Representing Data and Range and Averages. None of the concepts are too difficult. You need to be able to complete cumulative frequency tables, draw box-plots and be able to calculate range, median, lower and upper quartiles. You then need to pull these skills together to compare data sets. I’ve tried to pull this altogether in one question.

### Cumulative Frequency Question

Mr. Wenger wanted to choose his penalty taker scientifically. He decided to record the speed of shots recorded by his squad. It soon became clear that Robert and Theo had the fastest shot. Robert’s and Theo’s last 100 shot speeds were:

a) Complete the cumulative frequency table.

b) Graph the cumulative frequencies for Robert’s and Theo’s shot speeds and estimate the median, lower quartile, upper quartile and inter quartile range for both players’ shot speeds.

c) Draw box-plots for Robert’s and Theo’s shot speeds.

d) Advise Mr. Wenger which player he should select for penalty taking duties (assuming that shot speed is the most important factor). Explain why you would give this advice.

### Cumulative Frequency Question Approach

a) In completing the cumulative frequency table, make sure that the cumulative frequency is equal to the sum of individual frequencies, in this case one hundred.

b) You can plot at least 2 cumulative frequencies on the same graph. This help you to compare sets of data. You can estimate the median value by drawing lines on your graph. In range and averages we saw that median = (n+1)/2 th value. However when we have a large number of data items (which is the usually the case with cumulative frequency questions) we simplify the formula to median = (n/2)th value.

The median value = (n/2) :- where n = number of values, so in this question the meidan = 100/2 = 50^{th} value

The lower quartile value = (n/4) :- where n= number of values, so in this question the lower quartile value = 100/4 = 25^{th} value

The upper quartile = (3n/4), so in this question the upper quartile value = (3×100)÷4 = 75^{th} value.

So to get the median, lower quartile and upper quartiles draw lines from 50,25 and 75 respectively from the Cumulative Frequency axis on the graph and read off the values on the Shot Speed axis. If this is not clear, just look at the graph in the answer below and you will see what I mean.

The inter-quartile range = Upper quartile — Lower quartile.

c) A box-plot is drawn on a scale in this format:-

The only way to learn these box-plots (sometimes known as box-and-whisker diagrams is to actually do a few.

d) Usually the data in the questions will be prepared so that the conclusions to be drawn are clear. So don’t try and be clever, just use common sense to give your answer but most importantly in your explanation demonstrate your knowledge and understanding of this subject.

### Cumulative Frequency Answer

a)

b)

Reading from the graph above:

Shot Speeds (kph):-

Robert Theo

Median 87 81

Lower Quartile 81 76

Upper Quartile 95 92

Inter-quartile range 14 16

c) Box-plots for Robert and Theo shot speed (kph):-

d) Mr Wenger should choose Robert as his penalty-taker. The box plots graphically illustrate that Robert is more consistent than Theo. The median speed of his shots is higher than Theo’s. To emphasise this point, Robert’s lower quartile shot speed equals Theo’s median shot speed.

Although Robert and Theo’s range of shot speeds is similar and Theo seems to be able to hit the occasional very fast shot, Robert is able to hit speedy shots more consistently.