A Math

A Math (IGCSE)

E Math

E Math (IGCSE)

H1 Math

H2 Math

IB AA HL Math

IB AA SL Math

IB AI HL Math

IB AI SL Math

IP Math

N(A) A Math

N(A) E Math

PSLE

PSLE Foundation Mathematics

<p>The following table gives the examination grades for&nbsp;\(120\) students.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 400px;"><colgroup><col style="width: 25.0000%;"><col style="width: 37.5000%;"><col style="width: 37.5000%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p style="text-align: center;"><strong>Grade</strong></p></th><th style="text-align: center; --cell-justify: center"><p style="text-align: center;"><strong>Number of students</strong></p></th><th style="text-align: center; --cell-justify: center"><p style="text-align: center;"><strong>Cumulative frequency</strong></p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>\(1\)</p></td><td style="text-align: center; --cell-justify: center"><p>\(9\)<br></p></td><td style="text-align: center; --cell-justify: center"><p>\(9\)<br></p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>\(2\)<br></p></td><td style="text-align: center; --cell-justify: center"><p>\(25\)<br></p></td><td style="text-align: center; --cell-justify: center"><p>\(34\)<br></p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>\(3\)<br></p></td><td style="text-align: center; --cell-justify: center"><p>\(35\)<br></p></td><td style="text-align: center; --cell-justify: center"><p>\(p\)<br></p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>\(4\)<br></p></td><td style="text-align: center; --cell-justify: center"><p>\(q\)<br></p></td><td style="text-align: center; --cell-justify: center"><p>\(109\)<br></p></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>\(5\)<br></p></td><td style="text-align: center; --cell-justify: center"><p>\(11\)<br></p></td><td style="text-align: center; --cell-justify: center"><p>\(120\)<br></p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-1779261690585-l4eti01xl" data-list-item-path="item-1779261690585-l4eti01xl"><div class="list-item-paragraph">Find the value of<span class="inline-marks">[4]</span></div><ol class="i-list"><li data-list-item-id="subitem-1779261693498-k0sciq428" data-list-item-path="item-1779261690585-l4eti01xl/subitem-1779261693498-k0sciq428"><div class="list-item-paragraph">\(p\);</div></li><li data-list-item-id="subitem-1779261693693-q48t7a3h0" data-list-item-path="item-1779261690585-l4eti01xl/subitem-1779261693693-q48t7a3h0"><div class="list-item-paragraph">\(q\).</div></li></ol></li><li data-list-item-id="item-1779261718434-negrc9qfi" data-list-item-path="item-1779261718434-negrc9qfi"><div class="list-item-paragraph">Find the mean grade.<span class="inline-marks">[2]</span></div></li><li data-list-item-id="item-1779261719794-7ua9230kl" data-list-item-path="item-1779261719794-7ua9230kl"><div class="list-item-paragraph">Write down the standard deviation.<span class="inline-marks">[1]</span></div></li></ol>

The following table gives the examination grades for \(120\) students.

<p style="text-align: center;"><strong>Grade</strong></p>

<p style="text-align: center;"><strong>Number of students</strong></p>

<p style="text-align: center;"><strong>Cumulative frequency</strong></p>

Mean and Standard Deviation for Ungrouped Data

Statistics

<p>The following frequency distribution of marks has mean \(4.5\).</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 450px;"><colgroup><col style="width: 100px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: left; --cell-justify: flex-start"><p><strong>Mark</strong></p></th><th style="text-align: center; --cell-justify: center"><p>\(1\)</p></th><th style="text-align: center; --cell-justify: center"><p>\(2\)</p></th><th style="text-align: center; --cell-justify: center"><p>\(3\)</p></th><th style="text-align: center; --cell-justify: center"><p>\(4\)</p></th><th style="text-align: center; --cell-justify: center"><p>\(5\)</p></th><th style="text-align: center; --cell-justify: center"><p>\(6\)</p></th><th style="text-align: center; --cell-justify: center"><p>\(7\)</p></th></tr></thead><tbody><tr><td style="text-align: left; --cell-justify: flex-start"><p><strong>Frequency</strong></p></td><td style="text-align: center; --cell-justify: center"><p>\(2\)</p></td><td style="text-align: center; --cell-justify: center"><p>\(4\)</p></td><td style="text-align: center; --cell-justify: center"><p>\(6\)</p></td><td style="text-align: center; --cell-justify: center"><p>\(9\)</p></td><td style="text-align: center; --cell-justify: center"><p>\(x\)</p></td><td style="text-align: center; --cell-justify: center"><p>\(9\)</p></td><td style="text-align: center; --cell-justify: center"><p>\(4\)</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Find the value of \(x\).<span class="inline-marks">[4]</span></div></li><li><div class="list-item-paragraph">Write down the standard deviation.<span class="inline-marks">[2]</span></div></li></ol>

The following frequency distribution of marks has mean \(4.5\).

<p>\(16\) pupils sat for a test. The sum of their marks is \(504\). The sum of the squares of their marks is \(16855\).</p>
<ol class="abc-list"><li data-list-item-id="item-a" data-list-item-path="item-a"><div class="list-item-paragraph">Calculate</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-paragraph">the mean,</div></li><li data-list-item-id="item-a-ii" data-list-item-path="item-a/item-a-ii"><div class="list-item-paragraph">the standard deviation, of these test marks.</div></li></ol></li><li data-list-item-id="item-b" data-list-item-path="item-b"><div class="list-item-paragraph">One of the marks was \(43\) but was incorrectly recorded as&nbsp;\(25\). Calculate the correct mean and standard deviation of these test marks.</div></li><li data-list-item-id="item-c" data-list-item-path="item-c"><div class="list-item-paragraph">Explain, stating your reasons clearly,</div><ol class="i-list"><li data-list-item-id="item-c-i" data-list-item-path="item-c/item-c-i"><div class="list-item-paragraph">why the mean has increased,</div></li><li data-list-item-id="item-c-ii" data-list-item-path="item-c/item-c-ii"><div class="list-item-paragraph">why the standard deviation has increased.</div></li></ol></li></ol>

\(16\) pupils sat for a test. The sum of their marks is \(504\). The sum of the squares of their marks is \(16855\).

One of the marks was \(43\) but was incorrectly recorded as \(25\). Calculate the correct mean and standard deviation of these test marks.

Effects of Changing Data on Statistical Measures

IP2U15 HOT Q2

<p><strong>Variance</strong></p>
<p>Variance measures the spread or dispersion of a set of data points around their mean (average). For a group of discrete data, the variance (\(\sigma^2\)) is calculated as the average of the squared differences between each data point and the mean.</p>
<p style="text-align: center">\[\sigma^2 = \frac{\sum(x - \bar{x})^2}{n}\]</p>
<p><strong>Standard Deviation</strong></p>
<p>The standard deviation is the square root of the variance. It gives a measure of the root-mean-square deviation from the mean, expressed in the same units as the data.</p>
<p style="text-align: center">\[\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}\]</p>
<p>Show in an algebraic way that, an alternative expression for standard deviation of a set of discrete data is given by \(\sigma = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}\).</p>

Variance measures the spread or dispersion of a set of data points around their mean (average). For a group of discrete data, the variance (\(\sigma^2\)) is calculated as the average of the squared differences between each data point and the mean.

\[\sigma^2 = \frac{\sum(x - \bar{x})^2}{n}\]

The standard deviation is the square root of the variance. It gives a measure of the root-mean-square deviation from the mean, expressed in the same units as the data.

\[\sigma = \sqrt{\frac{\sum(x - \bar{x})^2}{n}}\]

Show in an algebraic way that, an alternative expression for standard deviation of a set of discrete data is given by \(\sigma = \sqrt{\frac{\sum x^2}{n} - \bar{x}^2}\).

Variance and Standard Deviation

<p>One hundred students were asked how many hand phones they have in their home. The results of the survey are summarized in the table below.</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">No. of hand phones (\(x\))</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">No. of students (\(f\))</th><th style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(fx\)</th></tr></thead><tbody><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(1\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(15\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(c\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(2\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(25\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(50\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(3\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(b\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(d\)</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">\(a\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(84\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(5\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(12\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(60\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(6\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(5\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(30\)</td></tr><tr style="height: 23px;"><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">Total</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(100\)</td><td style="height: 23px; min-height: 23px; text-align: center; --cell-justify: center">\(e\)</td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">Find</span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">the values of \(a\), \(b\), \(c\), \(d\) and \(e\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">the mean and standard deviation of the number of hand phones in their home.</span></div></li></ol></li><li><div class="list-item-content"><span class="list-item-text">There are \(5\) red cards, \(6\) yellow cards and \(4\) blue cards in a box. Annie draws a card from the box at random. Write down the probability that she will not get a blue card.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Brandon draws \(2\) cards at random, one after the other, without replacement. Find the probability that the two cards are of different colour.</span></div></li><li><div class="list-item-content"><span class="list-item-text">Annie and Brandon take turn to draw one card at a time, without replacement, until either player gets a blue card. Assuming that Annie starts first, find the probability that she will get a blue card exactly on her second try.</span></div></li></ol>

One hundred students were asked how many hand phones they have in their home. The results of the survey are summarized in the table below.

There are \(5\) red cards, \(6\) yellow cards and \(4\) blue cards in a box. Annie draws a card from the box at random. Write down the probability that she will not get a blue card.

Brandon draws \(2\) cards at random, one after the other, without replacement. Find the probability that the two cards are of different colour.

Annie and Brandon take turn to draw one card at a time, without replacement, until either player gets a blue card. Assuming that Annie starts first, find the probability that she will get a blue card exactly on her second try.

Probabilities of Successive Events

Constructing and Interpreting Frequency Tables

Probability of a Single Event

Waiting-Time and Stopping-Rule Probabilities

Probability

<p>The thirty students in a class each took a Biology test. Their scores are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table><colgroup><col style="width: 120px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 60px;"><col style="width: 60px;"></colgroup><thead><tr style="height: 43px;"><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Score</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(4\)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(5\)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(6\)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(7\)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(8\)</th><th style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(10\)</th></tr></thead><tbody><tr style="height: 43px;"><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">Frequency</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(2\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(5\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(11\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(8\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(3\)</td><td style="height: 43px; min-height: 43px; text-align: center; --cell-justify: center">\(1\)</td></tr></tbody></table></div></div>
<p>Calculate</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the mean score,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation.</span></div></li></ol>

The thirty students in a class each took a Biology test. Their scores are shown in the table below.

Technical Report: Variance Analysis of Student Achievement

<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.

Comparing Distributions Using Centre and Spread

Statistical Properties of Ordered and Unordered Numerical Sets

<p>A class of students took a Mathematics test. Their scores are represented in the stem-and-leaf diagram as shown.</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/1767953771817-j6yj3lfba9.png?token=eyJraWQiOiJiZTM3ZjNhZS05OWM4LTQ5ZGQtYjRiNS1iMTBhYmJhNjY5MWYiLCJhbGciOiJIUzI1NiJ9.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3Njc5NTM3NzE4MTctajZ5ajNsZmJhOS5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1ODM0LCJleHAiOjE3OTI2OTYxMzR9.ADEEi-zJ3Tis86Uzms4mmc8Xu9Xabp6wJaDLKQsrftM" style="max-width: min(350px, 100%); height: auto; display: block;" class="aligncenter custom-width" /></figure></div>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the modal score,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the median score,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the mean score,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation of the scores, correct to \(3\) significant figures.</span></div></li></ol>

A class of students took a Mathematics test. Their scores are represented in the stem-and-leaf diagram as shown.

the standard deviation of the scores, correct to \(3\) significant figures.

Median

Mode

Stem-and-Leaf Diagrams

Evaluating Modal, Median, and Mean Values in Discrete Distributions

<p style="margin-left: 6.5rem">\(\mathrm{Mean} = \frac{\sum f x}{\sum f}\)</p>
<p style="margin-left: 2rem">\(\mathrm{Standard\ deviation} = \sqrt{\frac{\sum f x^2}{\sum f} - \left( \frac{\sum f x}{\sum f} \right)^2}\)</p>
<p>Find the mean and standard deviation for the following set of data.</p>
<p style="text-align: center">\( 28 \qquad 25 \qquad 25 \qquad 20 \qquad 21 \qquad 24 \qquad 22 \qquad 23 \)</p>

\(\mathrm{Mean} = \frac{\sum f x}{\sum f}\)

\(\mathrm{Standard\ deviation} = \sqrt{\frac{\sum f x^2}{\sum f} - \left( \frac{\sum f x}{\sum f} \right)^2}\)

Find the mean and standard deviation for the following set of data.

\( 28 \qquad 25 \qquad 25 \qquad 20 \qquad 21 \qquad 24 \qquad 22 \qquad 23 \)

Statistical Analysis of Ungrouped Numerical Observations

<p>The Mathematics marks of \(6\) girls are given as follows: \(42, 32, 41, 26, 39, 33\).</p>
<p>Find</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the mean,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation, correct to \(3\) significant figures.</span></div></li></ol>

The Mathematics marks of \(6\) girls are given as follows: \(42, 32, 41, 26, 39, 33\).

the standard deviation, correct to \(3\) significant figures.

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