Cumulative Frequency


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:

Cumulative Frequency Question












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 = 50th value

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

The upper quar­tile = (3n/4), so in this ques­tion the upper quar­tile value = (3×100)÷4 = 75th 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:-

Box-Plot (Box-and-Whisker Diagram)

Box-Plot (Box-and-Whisker Diagram)


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


Cumulative Frequency Table Answer

Cumu­la­tive Fre­quency Table Answer



Cumulative Frequency Graph

Cumu­la­tive Fre­quency Graph













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):-

Box Plot Answer

Box Plot Answer


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.


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