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E Math (IGCSE)

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H2 Math

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IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The height of students at a school is shown as a cumulative frequency graph below.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769331137120-oubtwaa4dlf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzExMzcxMjAtb3VidHdhYTRkbGYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.pRDBA5nIkBqXplK1RymtUm1hUxfGD56sAI2Z3egzZKQ" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Cumulative frequency graph showing Number of students vs Height (cm)" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to estimate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the median,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the inter-quartile range.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">If&nbsp;$20$% of the students are taller than $h$ cm, find the value of $h$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Copy and complete the grouped frequency table of students' heights.</span></div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Heights (cm)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of students</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$120 \le x &lt; 130$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$130 \le x &lt; 140$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$140 \le x &lt; 150$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$150 \le x &lt; 160$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$160 \le x &lt; 170$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$170 \le x &lt; 180$</td><td style="height: 23px; min-height: 23px; text-align: left; --cell-justify: flex-start"></td></tr></tbody></table></div></div>
<ol class="abc-list" start="4"><li><div class="list-item-paragraph">Using your grouped frequency table, calculate an estimate of the standard deviation.</div></li><li><div class="list-item-paragraph">The students at the school took measurements of their weights using a faulty weighing machine. The box and whisker graph below shows the students' weights (in kg) measured by the faulty weight machine.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769331223446-mab2gukchu8.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEyMjM0NDYtbWFiMmd1a2NodTgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.cj4dLWwFD_MXT75262DXq_55JD0coWkEmnqvEIJ7NjU" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div><ol class="i-list"><li><div class="list-item-paragraph">If every student's true weight was actually $5$ kg lower, describe briefly how the box and whisker graph will differ from the given graph.</div></li><li><div class="list-item-paragraph">State the correct value of the median weight.</div></li></ol></li></ol>

The height of students at a school is shown as a cumulative frequency graph below.

If $20$% of the students are taller than $h$ cm, find the value of $h$.

Copy and complete the grouped frequency table of students' heights.

Using your grouped frequency table, calculate an estimate of the standard deviation.

The students at the school took measurements of their weights using a faulty weighing machine. The box and whisker graph below shows the students' weights (in kg) measured by the faulty weight machine.

Percentiles and Quartiles from Cumulative Frequency Curves

Box-and-Whisker Plots

Effects of Changing Data on Statistical Measures

Cumulative Frequency Table and Curve

Mean and Standard Deviation for Grouped Data

Statistics

<p>The weights of $200$ secondary two pupils at Ace Secondary School are represented by the cumulative frequency curve shown below.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769330982039-i6xy3zqu22q.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA5ODIwMzktaTZ4eTN6cXUyMnEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.CxiEL3lHRWVOX0grsquH695HuypsYdCjTP3KbZ0a13g" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Cumulative Frequency Curve for Weights of Pupils" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to find</span></div><ol class="i-list"><li><div class="list-item-paragraph">the median weight of the pupils,</div></li><li><div class="list-item-paragraph">the interquartile range,</div></li><li><div class="list-item-paragraph">the probability that at least one of two pupils chosen at random weighs at least $55$ kg.</div></li></ol></li><li><div class="list-item-paragraph">The weights of the pupils are represented in the box-and-whiskers plot shown below. Write down the values of $a$, $b$, $c$, $d$ and $e$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769331007631-b9z4edd5xs5.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEwMDc2MzEtYjl6NGVkZDV4czUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.RlrYdHv_tPp2szkm6nWJTyTO58bx2xONdD34s0rwDDM" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div></li><li><div class="list-item-paragraph">The weights of another $200$ secondary two pupils at Best Secondary School are represented in another box-and-whiskers plot shown below. Compare briefly the weights of the secondary two pupils from the two schools.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769331018689-izkd9lsqqf.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEwMTg2ODktaXprZDlsc3FxZi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODU3LCJleHAiOjE3OTI2OTYxNTd9.QeXfx17Z92gCmbZ1ti7-8PM5vUuWluGUws8Oqom2mp0" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div></li></ol>

The weights of $200$ secondary two pupils at Ace Secondary School are represented by the cumulative frequency curve shown below.

The weights of the pupils are represented in the box-and-whiskers plot shown below. Write down the values of $a$, $b$, $c$, $d$ and $e$.

Probabilities of Successive Events

Comparing Distributions Using Centre and Spread

Probability

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769330783936-6uq3u26qw5f.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA3ODM5MzYtNnVxM3UyNnF3NWYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.bmyUP4GJJmpHY6K-M-UNK3UOsvGG7K1BEz7pDD1RQ6U" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The cumulative frequency curve above represents daily wages of $60$ male adults in factory $X$. Use the graph to estimate, as accurately as possible,</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the median daily wage,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the interquartile range,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the $30^{\mathrm{th}}$ percentile,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the number of male adults whose daily wage is less than or equal to $\$45$,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the fraction of the $60$ male adults whose daily wage is more than $\$25$.</span></div></li></ol>

The cumulative frequency curve above represents daily wages of $60$ male adults in factory $X$. Use the graph to estimate, as accurately as possible,

the number of male adults whose daily wage is less than or equal to $\$45$,

the fraction of the $60$ male adults whose daily wage is more than $\$25$.

<p>The cumulative frequency curve below illustrates the marks obtained by $120$ students in a Science test for Class $A$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769330518184-k7awj2ogt0h.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA1MTgxODQtazdhd2oyb2d0MGgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.U8ovfftNcEA6588mtvCmI9qe3F-VgUSFqLP1wJJydWc" style="max-width: min(400px, 100%); height: auto; display: block;" alt="Cumulative frequency curve for Science test marks" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to find,</span></div><ol class="i-list"><li><div class="list-item-paragraph">the $80^{th}$ percentile.</div></li><li><div class="list-item-paragraph">the interquartile range.</div><div class="list-item-paragraph">In this test, only $15$% of the students scored a distinction.</div></li><li><div class="list-item-paragraph">Find the least marks for a student to be considered for distinction.</div><div class="list-item-paragraph">Two student are chosen at random.</div></li><li><div class="list-item-paragraph">Find the probability that both the students scored above $80$ marks.</div></li></ol></li><li><div class="list-item-paragraph">The cumulative frequency curve can be represented using the following box-and-whisker diagram.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769330604036-b08esn5p68u.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA2MDQwMzYtYjA4ZXNuNXA2OHUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.CMvtXL-tklfMHH3Z3nqXocbAbBlR-9hG9QpAW4F9Z9M" alt="List item image" class="aligncenter" style="max-width: 100%; height: auto;" /></div><ol class="i-list"><li><div class="list-item-paragraph">State the value of $x_{3}$.</div></li><li><div class="list-item-paragraph">State the value of $x_{5}-x_{1}$. What does $x_{5}-x_{1}$ represent?</div></li></ol></li><li><div class="list-item-paragraph">The box-and-whisker diagram below illustrates the marks obtained by $120$ students in a Science test for Class $B$.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769499239663-n5u3oytoth.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk0OTkyMzk2NjMtbjV1M295dG90aC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODU3LCJleHAiOjE3OTI2OTYxNTd9.uGQt_jYeJJRhBCzN4oy_ntwOUNAxLee1ih7MF5a0jqw" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div><div class="list-item-paragraph">Compare the test results for the two classes in two different ways.</div></li><li><div class="list-item-paragraph">The results of the science test of Class $C$ of $120$ students have the same median but a smaller standard deviation. Describe how the cumulative frequency curve will differ from the given curve.</div></li></ol>

The cumulative frequency curve below illustrates the marks obtained by $120$ students in a Science test for Class $A$.

<p>A teacher recorded the individual time taken by $200$ students to complete a cross-country race. The results are presented in the form of a cumulative frequency curve.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769330452944-u09rnsjaqz.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA0NTI5NDQtdTA5cm5zamFxei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODU3LCJleHAiOjE3OTI2OTYxNTd9.nARSYRDT2sK83DL-6V17Dy-BzvSYqyTfy7vU79Ho3kg" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the curve to estimate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the median time taken to run the race,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the interquartile range of the distribution,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the percentage of students who took longer than $66$ min to finish the race,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the probability of choosing a student who took $45$ min or less time to finish the race.</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">The frequency table relates to the same cross-country race results.</span></div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr style="height: 23px;"><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Frequency</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Time ($t$ min)</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$13$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$30 &lt; t \le 40$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$r$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$40 &lt; t \le 50$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$90$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$50 &lt; t \le 60$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$s$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$60 &lt; t \le 70$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$9$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$70 &lt; t \le 80$</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$4$</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">$80 &lt; t \le 90$</td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of $r$ and the value of $s$.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Showing your method clearly, estimate the mean time required to run the race.</span><span class="inline-marks">[3]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Estimate the standard deviation of the distribution.</span><span class="inline-marks">[2]</span></div></li></ol>

A teacher recorded the individual time taken by $200$ students to complete a cross-country race. The results are presented in the form of a cumulative frequency curve.

The frequency table relates to the same cross-country race results.

Find the value of $r$ and the value of $s$.

Showing your method clearly, estimate the mean time required to run the race.

Estimate the standard deviation of the distribution.

Probability of a Single Event

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1769330191844-v3siqj5rqg.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzAxOTE4NDQtdjNzaXFqNXJxZy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODU3LCJleHAiOjE3OTI2OTYxNTd9.wXUI4g_qc9AiC4vqPTqh4x2Ir7gKsra0cmzQtzi4jLk" style="max-width: min(400px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The diagram above is the cumulative frequency curve for the daily time spent surfing Facebook of $250$ students.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to find</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the median time spent surfing Facebook,</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the number of students who spend more than $20$ minutes a day surfing Facebook,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the $90^{th}$ percentile,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the interquartile range.</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">A second group of students have a higher median and a larger standard deviation. Describe how the cumulative frequency graph will differ from this given curve.</span><span class="inline-marks">[2]</span></div></li></ol>

The diagram above is the cumulative frequency curve for the daily time spent surfing Facebook of $250$ students.

A second group of students have a higher median and a larger standard deviation. Describe how the cumulative frequency graph will differ from this given curve.

<p>The cumulative frequency curve below shows the amount of weight loss, measured to the nearest kg, of 200 men in Glow Fitness Centre.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767947420004-9gmcbn2jul.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDc0MjAwMDQtOWdtY2JuMmp1bC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODU3LCJleHAiOjE3OTI2OTYxNTd9.Fsxjoa-8idY0jm90_9VIrS4opjkE50udbE-bF02IonE" style="max-width: min(500px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>The corresponding cumulative frequency table for this distribution is as shown below.</p>
<div class="content-block content-block--table"><div class="table-container"><table><colgroup><col style="width: 150px;"><col style="width: 150px;"></colgroup><thead><tr style="height: 43px;"><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Weight loss ($x$ kg)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Cumulative Frequency</th></tr></thead><tbody><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$\le 10$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">22</td></tr><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$\le 15$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$a$</td></tr><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$\le 20$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">100</td></tr><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$\le 25$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">168</td></tr><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$\le 30$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">190</td></tr><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">$\le 35$</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">200</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Determine the value of $a$.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Use your graph to find the</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">median weight loss,</span></div></li><li><div class="list-item-content"><span class="list-item-text">interquartile range,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$44^{\mathrm{th}}$ percentile weight loss.</span></div></li></ol></li><li><div class="list-item-paragraph">A second fitness centre, Saturn Fitness Centre also measured the amount of weight loss by $200$ of their men. The box-and-whisker diagram below illustrates their weight loss achieved.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767947675184-72xpkvh4e0y.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDc2NzUxODQtNzJ4cGt2aDRlMHkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.Ar59fqwCjVKL_U3KhCr7Mq0h31R4gTL5VH_q1gFlEMk" alt="List item image" class="aligncenter custom-width" style="max-width: min(400px, 100%); height: auto;" /></div><ol class="i-list"><li><div class="list-item-paragraph">Find the interquartile range.</div></li><li><div class="list-item-paragraph">A physical instructor claimed that the weight loss program at Glow Fitness Centre is more effective than Saturn Fitness Centre. Explain if his statement is true.</div></li></ol></li></ol>

The cumulative frequency curve below shows the amount of weight loss, measured to the nearest kg, of 200 men in Glow Fitness Centre.

The corresponding cumulative frequency table for this distribution is as shown below.

From Raw Test Scores to Box-and-Whisker Visualizations

<p>Eighty students took a Mathematics examination which consisted of two papers.</p>
<p>Each paper was marked out of 100. The diagram below shows the cumulative frequency curve for Paper 1.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767945445447-1v9j87hueu8i.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDU0NDU0NDctMXY5ajg3aHVldThpLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI2OTU4NTcsImV4cCI6MTc5MjY5NjE1N30.GRT0ubvLQQ7XQOkab5u-1ergUfcBjV1qcRyhTy4FRGU" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Cumulative frequency curve for Paper 1" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Estimate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the median mark,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the interquartile range,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the $80^{\text{th}}$ percentile.</span></div></li></ol></li><li><div class="list-item-paragraph">The box-and-whisker diagram below represents the result for Paper 2.</div><div class="content-block list-item-image" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767945567740-72oa6ecltow.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDU1Njc3NDAtNzJvYTZlY2x0b3cucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.ND5s4uaJQoRTHJaAZd_TZM7ehrtbdV_1pPThyhZcg2o" alt="List item image" class="aligncenter custom-width" style="max-width: min(350px, 100%); height: auto;" /></div><div class="list-item-paragraph">State, with a reason, which you think was the more difficult paper.</div></li></ol>

Eighty students took a Mathematics examination which consisted of two papers.

Each paper was marked out of 100. The diagram below shows the cumulative frequency curve for Paper 1.

Visualizing Data Spread and Central Tendency

<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767944223552-75yqyislj5x.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDQyMjM1NTItNzV5cXlpc2xqNXgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.kgLDjj5aYahJWgiWePuOAgctc_FSO_YrR3dbuzXmCSc" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Cumulative frequency graph for Mathematics and Science examinations" class="aligncenter custom-width" /></figure></div>
<p>The cumulative frequency graph shows the distribution of marks for&nbsp;$400$ students in a Mathematics and Science examination.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the median mark for the Mathematics examination.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the interquartile range for the Science examination.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If the distinctions for Mathematics were allocated to the top $25\%$ of the students, find the mark to achieve a distinction.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">One student is chosen at random. You may assume that the marks gained in the two examinations are independent. Expressing your answer as a fraction in its lowest terms, calculate the probability that the student gains</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">less than $20$ marks for both examinations,</span></div></li><li><div class="list-item-content"><span class="list-item-text">more than $60$ marks on one examination, but not on the other.</span></div></li></ol></li></ol>

The cumulative frequency graph shows the distribution of marks for $400$ students in a Mathematics and Science examination.

One student is chosen at random. You may assume that the marks gained in the two examinations are independent. Expressing your answer as a fraction in its lowest terms, calculate the probability that the student gains

Problems Involving Two Cumulative Frequency Curve

Regional Academic Standards: A Comparative Statistical Study

<p>The cumulative frequency curve below shows the marks obtained by $250$ pupils in a particular examination. The maximum mark is $80$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767942056268-l7ae7ybwak.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDIwNTYyNjgtbDdhZTd5Yndhay5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODU3LCJleHAiOjE3OTI2OTYxNTd9.v7fCGQR3rRvTOudZBrnaQMPVqIickCZNl4v_pGUbaAU" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Cumulative frequency curve of examination marks" class="aligncenter custom-width" /></figure></div>
<p>From the graph, estimate</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the median of this distribution,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the interquartile range,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the $80^{\mathrm{th}}$ percentile,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the minimum mark such that a pupil can be in the top $20\%$.</span></div></li></ol>

The cumulative frequency curve below shows the marks obtained by $250$ pupils in a particular examination. The maximum mark is $80$.

the minimum mark such that a pupil can be in the top $20\%$.

Data Representation: Frequency Tables and Cumulative Curves

<p>The cumulative frequency graph shows the distribution of money donated for a charity event by $400$ students in a school.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767941609060-k5puqff1ju.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDE2MDkwNjAtazVwdXFmZjFqdS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODU3LCJleHAiOjE3OTI2OTYxNTd9.vG3aD7bF5pT7OGWNbPmmPayWYrnF7vUadyvHt4xI1WA" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Cumulative frequency graph of donations" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the median amount.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the interquartile range.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Estimate the percentage number of students who had donated $\$35$ or more.</span></div></li></ol>

The cumulative frequency graph shows the distribution of money donated for a charity event by $400$ students in a school.

Estimate the percentage number of students who had donated $\$35$ or more.

Quantitative Evaluation of Student Achievement Metrics

<p>The cumulative frequency curve below illustrates the marks obtained, out of $100$, by $160$ students in a Mathematics test.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767941300867-s4l17lpqly9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDEzMDA4NjctczRsMTdscHFseTkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.WFacpDIqrwSaUooVULTnzhh-JmqXBxcIzZZ-sgl9LcQ" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Cumulative frequency curve of Mathematics test marks" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Use the graph to estimate the</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">median mark,</span></div></li><li><div class="list-item-content"><span class="list-item-text">interquartile range,</span></div></li><li><div class="list-item-content"><span class="list-item-text">$90^{\mathrm{th}}$ percentile,</span></div></li><li><div class="list-item-content"><span class="list-item-text">number of students who scored more than $60$.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Given that&nbsp;$40\%$ of the students failed the test, find the least marks a student must obtain to pass the test.</span></div></li></ol>

The cumulative frequency curve below illustrates the marks obtained, out of $100$, by $160$ students in a Mathematics test.

Given that $40\%$ of the students failed the test, find the least marks a student must obtain to pass the test.

Statistical Analysis of Continuous Frequency Distributions

<p>The cumulative frequency curve shows the marks obtained by $250$ pupils in school $X$ in a particular examination. The maximum mark is $80$.</p>
<div class="content-block"><figure class="question-figure aligncenter" style="text-align: center;"><img src="https://vocwkpgapvksartxhhor.supabase.co/storage/v1/object/sign/question-images/pasted/1767937951531-5z3aqm29xgk.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5Mzc5NTE1MzEtNXozYXFtMjl4Z2sucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTg1NywiZXhwIjoxNzkyNjk2MTU3fQ.2DQhkgG61cSAWLappSJbRmtoUvnCRpuBFKiok1QwlNg" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Cumulative frequency curve of marks obtained by pupils" class="aligncenter custom-width" /></figure></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">If $\frac{3}{5}$ of the pupils passed the examination, what is the passing mark?</span></div></li><li><div class="list-item-content"><span class="list-item-text">The cumulative frequency curve can be represented in the following frequency table.</span></div></li></ol>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr><th style="text-align: center; --cell-justify: center">Marks obtained by pupils, $x$</th><th style="text-align: center; --cell-justify: center">No. of pupils</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$0 \le x &lt; 20$</td><td style="text-align: center; --cell-justify: center">$30$</td></tr><tr><td style="text-align: center; --cell-justify: center">$20 \le x &lt; 40$</td><td style="text-align: center; --cell-justify: center">$p$</td></tr><tr><td style="text-align: center; --cell-justify: center">$40 \le x &lt; 60$</td><td style="text-align: center; --cell-justify: center">$q$</td></tr><tr><td style="text-align: center; --cell-justify: center">$60 \le x &lt; 80$</td><td style="text-align: center; --cell-justify: center">$20$</td></tr></tbody></table></div></div>
<p>Find the value of $p$ and of $q$.</p>

The cumulative frequency curve shows the marks obtained by $250$ pupils in school $X$ in a particular examination. The maximum mark is $80$.

If $\frac{3}{5}$ of the pupils passed the examination, what is the passing mark?

The cumulative frequency curve can be represented in the following frequency table.

Math Question Bank