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E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

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N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The number of minutes taken by four boys to complete a question is \(x - y\), \(8\), \(10\) and \(x + y\). The mean time taken is \(9\) minutes.</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Show that&nbsp;\(x = 9\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Given that the standard deviation is \(5\) minutes, find the value of \(y\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Four girls answered the same question and the mean time taken is also \(9\) minutes but the standard deviation is \(2\) minutes. Compare and comment on these two sets of data.</span></div></li></ol>

The number of minutes taken by four boys to complete a question is \(x - y\), \(8\), \(10\) and \(x + y\). The mean time taken is \(9\) minutes.

Given that the standard deviation is \(5\) minutes, find the value of \(y\).

Four girls answered the same question and the mean time taken is also \(9\) minutes but the standard deviation is \(2\) minutes. Compare and comment on these two sets of data.

Finding Unknowns from Averages and Spread

Comparing Distributions Using Centre and Spread

IP2U15 HOT Q3

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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA5ODIwMzktaTZ4eTN6cXUyMnEucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTkzNywiZXhwIjoxNzkyNjk2MjM3fQ.i-JsuTr5bsPK9b26XfJi9UGrhFxzO9eRNG6DQN-fiBE" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEwMDc2MzEtYjl6NGVkZDV4czUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTkzNywiZXhwIjoxNzkyNjk2MjM3fQ.xe843KZ6UnKiDb9cC7Pn8QqATPV5YsRKie6KK3_BJ3c" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEwMTg2ODktaXprZDlsc3FxZi5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1OTM3LCJleHAiOjE3OTI2OTYyMzd9.HP0jz6GsHh8EGekM0vm-Ut0bGQdvnxeSrmvm50LijLo" 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 \(e\).

Percentiles and Quartiles from Cumulative Frequency Curves

Probabilities of Successive Events

Box-and-Whisker Plots

Probability

<p>The mass \(x\) grams of each of \(90\) oranges of variety \(A\) were recorded. The data obtained was illustrated as shown in the table.</p>
<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">Mass \(x\) grams</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of oranges</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(60 &lt; x \le 70\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(1\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(70 &lt; x \le 80\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(3\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(80 &lt; x \le 90\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(9\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(90 &lt; x \le 100\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(28\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(100 &lt; x \le 110\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(37\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(110 &lt; x \le 120\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(8\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(120 &lt; x \le 130\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(4\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find the mean and standard deviation for the above data.</span></div></li><li><div class="list-item-content"><span class="list-item-text">In an another variety, variety \(B\), of the \(90\) oranges, the lightest orange has the mass \(55\) grams and the heaviest orange has the mass \(138\) grams. The median, lower quartile and upper quartile are \(102\) grams, \(93\) grams and \(109\) grams respectively. Draw the box-whisker diagram for this data.</span></div></li><li><div class="list-item-content"><span class="list-item-text">If you were to buy some oranges, which variety would you buy? Explain your answer.</span></div></li></ol>

The mass \(x\) grams of each of \(90\) oranges of variety \(A\) were recorded. The data obtained was illustrated as shown in the table.

Find the mean and standard deviation for the above data.

In an another variety, variety \(B\), of the \(90\) oranges, the lightest orange has the mass \(55\) grams and the heaviest orange has the mass \(138\) grams. The median, lower quartile and upper quartile are \(102\) grams, \(93\) grams and \(109\) grams respectively. Draw the box-whisker diagram for this data.

If you were to buy some oranges, which variety would you buy? Explain your answer.

Mean and Standard Deviation for Grouped Data

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA1MTgxODQtazdhd2oyb2d0MGgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTkzNywiZXhwIjoxNzkyNjk2MjM3fQ.h_vIc766C1BEGrdIJnQNE4ICPkFRaIR3wqIEOfRz-CM" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA2MDQwMzYtYjA4ZXNuNXA2OHUucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTkzNywiZXhwIjoxNzkyNjk2MjM3fQ.daaMCi8MhLYg596wGoTnsKA-cB3FwgU8sns1KLZBA24" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njk0OTkyMzk2NjMtbjV1M295dG90aC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1OTM3LCJleHAiOjE3OTI2OTYyMzd9.AYffn7UNHw_f390H2u-rJo_SyraLuQXFU4RElxsMrhc" 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 \(A\).

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzAxOTE4NDQtdjNzaXFqNXJxZy5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1OTM3LCJleHAiOjE3OTI2OTYyMzd9.Lj94vJJMld1n1_5PbgG3O-ofU3_O8gv3RrZeSLq2GJU" 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 frequency table below summarizes the volume of apple juice dispensed from vending machine \(A\) for&nbsp;\(50\) trials.</p>
<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">Volume from \(A\) (\(x\) ml)</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Frequency</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(246 &lt; x \le 248\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(4\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(248 &lt; x \le 250\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(7\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(250 &lt; x \le 252\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(28\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(252 &lt; x \le 254\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(6\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(254 &lt; x \le 256\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(5\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">For vending machine \(A\), calculate</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the mean,</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation.</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">Another vending machine \(B\) is used to dispense apple juice. The mean and standard deviation of the volumes of apple juice dispensed by vending machine \(B\) is \(251.36\) ml and \(2.22\) ml respectively. The nominal volume of each cup of apple juice dispensed from these vending machines is \(250\) ml. If you run a vending machine business, which machine would you buy? Explain your choice.</span><span class="inline-marks">[2]</span></div></li><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">What is the range of volume dispensed by machine \(A\)?</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Why is the standard deviation preferred over the range as a measure of the spread of data?</span><span class="inline-marks">[1]</span></div></li></ol></li></ol>

The frequency table below summarizes the volume of apple juice dispensed from vending machine \(A\) for \(50\) trials.

Another vending machine \(B\) is used to dispense apple juice. The mean and standard deviation of the volumes of apple juice dispensed by vending machine \(B\) is \(251.36\) ml and \(2.22\) ml respectively. The nominal volume of each cup of apple juice dispensed from these vending machines is \(250\) ml. If you run a vending machine business, which machine would you buy? Explain your choice.

Median, Quartiles, Range and Interquartile Range

<p>The waiting time in minutes for&nbsp;\(50\) patients at two hospitals were given as follows.</p>
<p>Hospital \(A\):</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr style="height: 43px;"><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Time (minutes)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(20 &lt; t \le 24\)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(24 &lt; t \le 28\)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(28 &lt; t \le 32\)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(32 &lt; t \le 36\)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(36 &lt; t \le 40\)</th></tr></thead><tbody><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Number of patients</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(4\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(14\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(23\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(6\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(3\)</td></tr></tbody></table></div></div>
<p>Hospital \(B\):</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 200px;"><colgroup><col style="width: 200px;"></colgroup><thead><tr><th style="text-align: left; --cell-justify: flex-start">Mean (minutes) = \(29.2\)</th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start">Standard Deviation (minutes) = \(3.5\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">For Hospital \(A\), calculate an estimate of</span></div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-content"><span class="list-item-text">the mean waiting time,</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-content"><span class="list-item-text">the standard deviation, correct to \(3\) significant figures.</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">Compare briefly, the waiting time for the two hospitals.</span></div></li></ol>

The waiting time in minutes for \(50\) patients at two hospitals were given as follows.

For Hospital \(A\), calculate an estimate of

Compare briefly, the waiting time for the two hospitals.

Technical Report: Variance Analysis of Biological Samples

<p>The test results of a class of students are shown below.</p>
<div class="content-block content-block--table"><div class="table-container"><table><thead><tr><th style="text-align: center; --cell-justify: center">Marks</th><th style="text-align: center; --cell-justify: center">\(1-5\)</th><th style="text-align: center; --cell-justify: center">\(6-10\)</th><th style="text-align: center; --cell-justify: center">\(11-15\)</th><th style="text-align: center; --cell-justify: center">\(16-20\)</th><th style="text-align: center; --cell-justify: center">\(21-25\)</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">Frequency</td><td style="text-align: center; --cell-justify: center">\(0\)</td><td style="text-align: center; --cell-justify: center">\(3\)</td><td style="text-align: center; --cell-justify: center">\(27\)</td><td style="text-align: center; --cell-justify: center">\(8\)</td><td style="text-align: center; --cell-justify: center">\(2\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Calculate an estimate of</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the mean marks,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation of the marks.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">If the mean and standard deviation for the entire school’s results in the same test are \(13.5\) marks and \(2.1\) marks respectively. Compare briefly, the class results against the entire school’s results.</span></div></li></ol>

The test results of a class of students are shown below.

If the mean and standard deviation for the entire school’s results in the same test are \(13.5\) marks and \(2.1\) marks respectively. Compare briefly, the class results against the entire school’s results.

Analytical Report: Comparative Class Performance Statistics

<p>Two brands of light bulbs were tested for their reliability. \(10\) light bulbs of each brand were randomly selected and they were switched on for a number of days until they blew out. The number of days that each bulb lasted was then noted. The results are given below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 605px;"><colgroup><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"></colgroup><thead><tr style="height: 43px;"><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Brand</th><th colspan="10" style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Number of days bulb lasted</th></tr></thead><tbody><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(A\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(101\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(100\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(99\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(105\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(102\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(101\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(98\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(100\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(97\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(97\)</td></tr></tbody></table></div></div>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 605px;"><colgroup><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"><col style="width: 55px;"></colgroup><thead><tr style="height: 43px;"><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Brand</th><th colspan="10" style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Number of days bulb lasted</th></tr></thead><tbody><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(B\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(95\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(101\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(100\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(103\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(92\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(107\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(106\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(94\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(98\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(104\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-content"><span class="list-item-text">For Brand \(A\), find</span></div><ol class="i-list"><li data-list-item-id="item-a-i" data-list-item-path="item-a/item-a-i"><div class="list-item-content"><span class="list-item-text">the mean number of days the light bulbs lasted,</span></div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-content"><span class="list-item-text">the standard deviation of the number of days, correct to \(3\) significant figures.</span></div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-content"><span class="list-item-text">For Brand \(B\), find</span></div><ol class="i-list"><li data-list-item-id="item-b-i" data-list-item-path="item-b/item-b-i"><div class="list-item-content"><span class="list-item-text">the mean number of days the light bulbs lasted,</span></div></li><li data-list-item-id="item-b-ii" data-list-item-path="item-b/item-b-ii"><div class="list-item-content"><span class="list-item-text">the standard deviation of the number of days, correct to \(3\) significant figures.</span></div></li></ol></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-content"><span class="list-item-text">Which brand of light bulb is more reliable? Give a reason for your answer.</span></div></li></ol>

Two brands of light bulbs were tested for their reliability. \(10\) light bulbs of each brand were randomly selected and they were switched on for a number of days until they blew out. The number of days that each bulb lasted was then noted. The results are given below.

Which brand of light bulb is more reliable? Give a reason for your answer.

Mean and Standard Deviation for Ungrouped Data

Statistical Properties of Ordered and Unordered Numerical Sets

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDc0MjAwMDQtOWdtY2JuMmp1bC5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1OTM3LCJleHAiOjE3OTI2OTYyMzd9.c9sh3CV6I3ALiU8CvBbyk_9mfKwJ4G5U1UMy2St3GaI" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDc2NzUxODQtNzJ4cGt2aDRlMHkucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTkzNywiZXhwIjoxNzkyNjk2MjM3fQ.OGoXHVvI3DjmWbxueRidxW1FlQmG45s3Dn7t7sfuGUo" 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.

Cumulative Frequency Table and Curve

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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDU0NDU0NDctMXY5ajg3aHVldThpLnBuZyIsInNjb3BlIjoiZG93bmxvYWQiLCJpYXQiOjE3OTI2OTU5MzcsImV4cCI6MTc5MjY5NjIzN30.-Nc7eeq1TMbIDtNm8_4BD2ooCbrK0KMWOcossMJLR2w" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDU1Njc3NDAtNzJvYTZlY2x0b3cucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTkzNywiZXhwIjoxNzkyNjk2MjM3fQ.2zvcxv9s5VmLUz4vT9bf2MG4dv5PLcGV_g6XRaeu5l4" 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

<p>The Mathematics Preliminary Examination results for students across schools \(X\) and \(Y\) were recorded. Schools \(X\) and \(Y\) have&nbsp;\(200\) and \(150\) math students respectively.</p>
<p>The following cumulative frequency curves show the distribution of the marks.</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/1767943223594-jdsp45ot1uc.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NDMyMjM1OTQtamRzcDQ1b3QxdWMucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTkzNywiZXhwIjoxNzkyNjk2MjM3fQ.AN67962X3sNmD7hp7XVTyt3jF7jN7VkZMy9ZU1H2F6Q" style="max-width: min(500px, 100%); height: auto; display: block;" alt="Cumulative frequency curve for student's marks in Math Prelim Examination in School X and Y" class="aligncenter custom-width" /></figure><div class="image-caption" style="text-align: center;">Cumulative frequency curve for student's marks in Math Prelim Examination in School X and Y</div></div>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Find the median mark for both schools \(X\) and \(Y\).</div></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">Calculate the interquartile range for both schools \(X\) and \(Y\).</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">A teacher analysed the data above and commented: "School \(X\) performs better than school \(Y\) because the cumulative frequency curve of \(X\) is always above \(Y\)." Use your results in <strong>(a)</strong> and <strong>(b)</strong> to argue if you agree with the teacher's analysis.</div></li><li data-list-item-id="item-d" data-list-item-path="item-d"><div class="list-item-paragraph">A distinction grade is given for students who scored \(80\) marks or more.<br><br>Which school has a greater percentage distinction for the Math preliminary examination?</div></li></ol>

The Mathematics Preliminary Examination results for students across schools \(X\) and \(Y\) were recorded. Schools \(X\) and \(Y\) have \(200\) and \(150\) math students respectively.

The following cumulative frequency curves show the distribution of the marks.

Find the median mark for both schools \(X\) and \(Y\).

Calculate the interquartile range for both schools \(X\) and \(Y\).

A distinction grade is given for students who scored \(80\) marks or more.

Which school has a greater percentage distinction for the Math preliminary examination?

Math Question Bank