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One Cumulative Frequency Curve To Rule Them All

Tuition given in the topic of E-Maths Tuition Questions from the desk of Miss Loi at 1:10 pm (Singapore time)

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Prepare for MODERATION!!! Rejoice! For our scouts bring forth great tidings from the frontline once more!

An awesome sight to behold, as entire divisions of our enemy were annihilated by the great phalanxes of our brave warriors, with their remnants fleeing even as we speak.

Long-feared nightmares in the likes of Relative Velocity and Kinematics finally emerged from the enemy’s ranks but were swiftly slayed, and their spirits banished to the infernal lair from where they came. Woe to the schools that deprive their charges the chance to partake in the slaying of these mighty beasts!

So in the end, there were no Level 99 fireballs. As our alchemists and mages began preparing their chemical spells for their battle later today, amidst the self-congratulations and widespread burning of A-Maths notes, some, though, are beginning to question the ease of this victory.

With the exact mechanics of allocating the spoils of war veiled in secrecy by the shady War Council, and with a major chunk of the top grades almost certain to be taken up by the foreign cyborg mercenaries, an uneasiness exists within our ranks.

Yes Miss Loi’s old adage

85% for A-Maths and 90% for E-Maths to get an A1!

looks ominously true now.

For a hypothetical look at the inner workings of the War Council, and to let you practise something that is almost certain to appear in your upcoming E-Maths Paper 1, take a sample of our students from yesterday’s battles:

120 candidates took a Mathematics examination which consists of two papers. Each paper was marked out of 100. The diagram shows,on the same axes, the cumulative frequency curves for Paper 1 and Paper 2.

Cumulative Frequency Graph

Use the graph for Paper 2 to estimate

  1. the median mark.
  2. the inter-quartile range.
  3. the number of candidates who scored more than 70 marks.
  4. the pass mark such that 60% of the pupils will pass the examination.
  5. the minimum mark required to gain a distinction if the top 5% of the pupils are awarded a distinction.

State, with reason, which was the more difficult paper.

Yes even the War Council rely on that little bit of E-Maths to seal your fate.

Cumulative Frequency Summary The chart on the right summarizes all you need for this question. But in the heat of battle it’s sometimes easy to forget that quartiles are determined by the percentages on the cumulative frequency axis, and that their values are actually taken from the horizontal axis.

Parts 4 and 5 of the question once again involve that additional topic not found in your textbook called common sense, which we usually assume that members of the War Council possess. Ask yourself these: Will you pass your exam if you scored zero? On the contrary, will you fail your exam if you scored 100 marks?

Now, say Hi and wave to yourself on the vertical axis!


Revision Exercise

To show that you have understood what Miss Loi just taught you from the centre, you must:

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Comments & Reactions


  1. shihui's Avatar
    shihui commented in tuition class


    huh! then top how many percent will get distinctions! what if everyone get 90 plus then the A1 will be 90 plus! oh dear!

  2. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    Shihui, no point worrying about things beyond your control. But do focus on things that are within your control i.e. go study your Geography!

  3. Karen's Avatar
    Karen commented in tuition class


    I was told that according to Cambridge report, the A1 cut-off for A Maths would never exceed 80, because they still need to take into consideration the "less able". Ah well I'm just worried about losing my A1 because I was so terribly careless. 🙁

  4. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    Welcome to Jφss Sticks Karen!

    As mentioned earlier, no point worrying about things beyond your control okay?

    Rumours are now flying all over the place as to what's the A1 cut-off.

    But the perennial question remains: who assigns your grades? Is it Cambridge or SEAB aka the War Council? If it's Cambridge then the pool of people is more spread out but if it's SEAB then you're probably competing with a pool of people at the higher-end, namely Singaporeans and our foreign 'talents'.

    Miss Loi has heard from somewhere that gradings were being determined through some 'collaborative meetings' between Cambridge and SEAB reps. And if that's true, then it's highly probable we're looking at the latter case due to SEAB's involvement.

    In any case, Miss Loi isn't 100% sure on this and is still waiting for anyone in the know to confirm this.

    But then again it's always good to aim high 🙂

  5. 123's Avatar
    123 commented in tuition class


    Lol, Haha, Foreign cyborg mercenaries....i agree

  6. Prodigy's Avatar
    Prodigy commented in tuition class


    [1] 60

    That's right. The median is the value of the middle term of the set of data arranged in increasing order. In this case, the middle term is the 60th out of the 120 students, and it's corresponding marks for Paper 2 is 60.

    [2] 72-45
    = 27

    Yes if you refer to the little diagram Miss Loi has drawn up. The interquartile rage is Q3-Q1

    = Marks at 75% of 120
    = Marks at 90
    ≅ 72

    = Marks at 25% of 120
    = Marks at 30
    ≈ 45

    So Q3-Q1 ≈ 72-45 ≈ 27

    Again Miss Loi would like to stress that we're looking for the values i.e. those on the horizontal axis corresponding to the data on the vertical axis. Many students make the careless of mistake of reading directly off the vertical axis.

    [3] 120-85
    = 35

    Righto! The keyword here is more than 70 marks, which includes all who scored between 70-100. From the Paper 2 graph you get the lower limit of 85 students (who scored 70) and the upper limit of 120 students (who scored 100). So the number of students in this range is 120-85 = 35 students.

    [4] From the graph,
    pass mark would be 65.

    You got pwned here! A good way to check is if you set 65 as the passing mark, and if you look at the graph again, 0-72 students scored between 0-65 marks. Will you PASS your exam if you score zero? No way! So you actually FAILED 60% of the students with your cutoff of 65!

    Will you PASS if you score 100? Of course. So you should instead look at the top 60% of the frequency axis in the graph, i.e. 60% of 120 starting from 120 (instead of starting from 0). Therefore you should be getting the passing mark from 40% of 120 = 48 ⇒ passing mark is around 55.

    BE VERY CAREFUL IN THESE KIND OF QUESTIONS! Many students got caught out here!

    [5] From the graph,
    mark= 83 {approx}

    Let's check again ... if you set 83 as your distinction mark, you'll get 114 (95%) students scoring 0-83, which means they failed to attain the distinction grade, which means only 5% get it. This time you correctly used the top 5% so you're right!

    Actually part 5 requires the same method to solve as part 4, and your 'instincts' served you well when it's the top 5%. But things don't look so obvious when it's the top 60% 😉

    Paper 1 is more difficult, as median mark of Paper 1 is lower than the median mark of Paper 2.

    Absolutely! Those who're still not sure let's chant the mantra together!

    Higher median = easier paper.
    Lower median = more difficult paper.

    Repeat this 20x before you go to bed and always remember that for medians and quartiles we're interested in the corresponding values on the horizontal axis. Don't make this careless mistake in your exam!


  7. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    123, well it's the sign of the times. Gone are the carefree days when you only have to compete with your cousin, or your mother's friend's daughter, or your uncle's friend's friend's son etc. etc.

  8. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    Prodigy, Miss Loi has marked your answers. Please pay particular attention to Parts 4 and 5.

    A little scary that the values in this question seem a tad too 'realistic' eh?

  9. BP stabbed's Avatar
    BP stabbed commented in tuition class


    Seems like the "war" has begun. best wishes to all those participating.

One Reaction


    [...] Loi managed an ironic smile. At least there won’t be too much moderation this time round. I only lose marks at the similarity one. Cause I anyhow give reason. But maybe [...]

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