# Cumulative Frequency question

Discussion in 'Mathematics' started by SMC21, Feb 25, 2012.

1. Any views on part (d) of the following question would be very much appreciated. Thank you "The weight of 80 bags of rice are measured. The table summarises the results:
Minimum 480g
Lower Quartile 500g
Median 540g
Upper Quartile 620g
Maximum 720g
a) Draw a box plot to show this information
b) Write down the interquartile range for these data
c) How many bags weigh (i) less than 480g (ii) less than 500g
d) Draw a cumulative frequency diagram to show the information."
The axes and grid are given for plotting. The x axis is labelled Weight (g) and graduated such that there is indication of missing section for lower values of weight and then 500 550 ..up to .. 700
The Mark Scheme says "5 correct plots" are required.
The Examiners' Report says, "Some candidates plotted four of the points but omitted the point corresponding to a cumulative frequency of 0 at 480g"

2. Any views on part (d) of the following question would be very much appreciated. Thank you "The weight of 80 bags of rice are measured. The table summarises the results:
Minimum 480g
Lower Quartile 500g
Median 540g
Upper Quartile 620g
Maximum 720g
a) Draw a box plot to show this information
b) Write down the interquartile range for these data
c) How many bags weigh (i) less than 480g (ii) less than 500g
d) Draw a cumulative frequency diagram to show the information."
The axes and grid are given for plotting. The x axis is labelled Weight (g) and graduated such that there is indication of missing section for lower values of weight and then 500 550 ..up to .. 700
The Mark Scheme says "5 correct plots" are required.
The Examiners' Report says, "Some candidates plotted four of the points but omitted the point corresponding to a cumulative frequency of 0 at 480g"

3. Hi SMC
I'm not sure if I'm misunderstanding something, but if the minimum value is 480, then it would be reasonable to plot 0 at 479.99 (or as close to 480 as you wish).
However I think I can see that a student might worry that there is a value at 480 and so they can't plot 0 at that point...(on the other hand, they might also deduce that they can't put any other value at 480 because there could be up to 1/4 of the data points at 480)
Perhaps students can be encouraged to realise that the data must be based on grouped data, (otherwise it wouldn't be necessary to use a cumulative frequency diagram). The minimum value must be at the lower end of the group containing the smallest data item(s), but the cumulative frequency is always plotted at the top of the relevant group range.
Does that help?
Liz

4. I think this is a bit ambigous as it depends how the data is grouped. If my first group was -- More than 470 up to and including 480 then the value of 480 would be incuded in this group and the cumulative frequecy at this point would not be 0.

5. How so? I can group the data howeevr I like. The minimum value could be inm the middle of the first group - or even on the upper limit of the first group as my example above.

6. Thanks for that. Unfortunately I don't have access to Exampro.

7. I can see your point Mike. The examiner presumably thought that the part c answers (what proportion of the answers were less than 480, and what proportion were less than the LQ), would be sufficient to guide students to plot the points they were looking for...

8. It's an old AQA module 1 question - March 02 - higher. The question has been given fully in the first post.

9. This is, of course, a highly artificial question. However, as we know the lowest weight, it seems best to me to plot the cumulative frequency at (OK, just below) that value, as we know that it would be accurate to do so. Like Casy, I remember the question - it came up on a paper I was looking at with Year 10 in my GTP year. My recollection is that most of the students got the idea.

10. I recollect the same! They got it! In fact it led to some good discussions.

11. Thank you all for your comments on this question. It is one which has been bugging me - and it is very hard to find similar questions (with convincing explanations) in textbooks - so I appreciate you sharing your thoughts on this topic.
Since no table is given showing how the data was or should be grouped, nor the class boundaries used, do you think that:
1) A point plotted at (475,0) could be considered a correct answer - and if not, why not?
2) Does 480 seem a natural choice of class boundary given the graduations on the x axis?
3) The class boundary at 480 could be considered as a lower bound of the type 480 <= rather than just < in which case the point can be plotted exactly at 480 and not just infinitely close to it? I.e. if our first group was 480<=x<500 then the minimum value would fall into this class and we would plot (480,0) for the lower bound of this class.
4) Without the information being prescribed, is it not reasonable for a pupil to envisage a number of equally valid alternatives scenarios for how the raw data could be organised and the resulting CF diagram?

12. I think we are going deeper than the question intended. I doubt if there was any intention to make people think along the lines of what groups were involved in the first place. If that was the idea, we would also be arguing about the upper end. It was simply for the students to recognise which points on the cumulative frequency curve could be noted with certainty. After all, the quartiles are quoted as if they are exact rather than estimated as is usually the case with Cumulative Frequency graphs.
When my class tried it (selective school bottom set 10s), nobody started at a lower value; a few missed the x-intercept out. But then, some do when drawing any Cumulative Frequency curve!
As I said before, a rather artificial problem, and it has no bearing on how Statistics are used in real life.

13. The question says nothing about grouping,. The minimum value was given and would probably be unique in reality. It seems perfectly reasonable to plot the first point at (480,0)

14. Hopefully, the question said 'mass' rather than 'weight'.
Then the 'g' would refer to grams.

15. The question said 'weight'.

16. Since the Mark Scheme only stipulated '5 correct points plotted' - and noting the variety of views in this thread - does this mean that not all pupils would have been marked to the same criteria?
The mention of '0 at 480g' only appears in the Examiners' Report.

17. Hi Piranha, thanks for your post.
As you can see, this question is still bugging me Regarding going deeper than the question intended:
Does that make other answers incorrect?
Does that mean the pupil has more or less understanding?
Do you think that a point plotted at (475,0) could be considered a correct answer - and if not, why not?
Is there a definitive method for dealing with this kind of question?

I believe that every mark matters and the loss of a single mark will, in some cases, have life-changing consequences for a pupil in terms of future opportunities and choices.
There should no room for ambiguity in questions set on exam papers.

18. The problem here is more with the markscheme than the original question.
To say that '5 correct plots are required' wouldn't get past me.
Furthermore, the 'Examiners' Report' is unhelpful about the fifth point issue. I would have said more.
I'm assuming teachers know that published markschemes do not correspond exactly with those used for marking. The working markschemes contain notes about sins that will be condoned by the markers. These are then deleted.

19. ...but I'm not a GCSE setter.
My impression is that the lower the level the more autocratic the examiner!

20. But....
If a pupil had a point plotted at (475,0), would you tell them it was a correct answer - and if it isn't, why isn't it?
Is there a definitive method for dealing with this kind of question?

I know I must sound pedantic but I would like to feel clear about what exactly is acceptable as a correct answer and why 