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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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzExMzcxMjAtb3VidHdhYTRkbGYucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTczMiwiZXhwIjoxNzkyNjk2MDMyfQ.ziAIjKjzxJjaCV6PGbIgksf0QILPQhtGDLb4cuSK2_4" 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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzEyMjM0NDYtbWFiMmd1a2NodTgucG5nIiwic2NvcGUiOiJkb3dubG9hZCIsImlhdCI6MTc5MjY5NTczMiwiZXhwIjoxNzkyNjk2MDMyfQ.cae23POFTzJSxPrXSQ35Et33LEotYXqb8h1qEzlP4oQ" 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 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.

Comparing Distributions Using Centre and Spread

<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.eyJ1cmwiOiJxdWVzdGlvbi1pbWFnZXMvcGFzdGVkLzE3NjkzMzA0NTI5NDQtdTA5cm5zamFxei5wbmciLCJzY29wZSI6ImRvd25sb2FkIiwiaWF0IjoxNzkyNjk1NzMyLCJleHAiOjE3OTI2OTYwMzJ9.wAqdeeKugBBfNX2nGe20HQKMjPLWRZN_nWiPB9zK6Fc" 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

Probability

<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

<div class="content-block content-block--table"><div class="table-container"><table><thead><tr><th style="text-align: center; --cell-justify: center">Mass ($m$) in grams</th><th style="text-align: center; --cell-justify: center">Frequency</th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center">$40 &lt; m \le 50$</td><td style="text-align: center; --cell-justify: center">$15$</td></tr><tr><td style="text-align: center; --cell-justify: center">$50 &lt; m \le 60$</td><td style="text-align: center; --cell-justify: center">$20$</td></tr><tr><td style="text-align: center; --cell-justify: center">$60 &lt; m \le 70$</td><td style="text-align: center; --cell-justify: center">$5$</td></tr></tbody></table></div></div>
<p>The table shows the distribution of the mass of a number of eggs.</p>
<p>Calculate an estimate of</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">the mean mass of the eggs,</span></div></li><li><div class="list-item-content"><span class="list-item-text">the standard deviation of the mass of the eggs.</span></div></li></ol>

The table shows the distribution of the mass of a number of eggs.

the standard deviation of the mass of the eggs.

Technical Report: Variance Analysis of Mass Distribution

<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>In the recent Job Week carried out by a group of $30$ scouts, the amount of money collected are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 600px;"><colgroup><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"><col style="width: 100px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Amount collected ($\$w$)</strong></p></th><th style="text-align: center; --cell-justify: center"><p>$0&lt;w\le 20$</p></th><th style="text-align: center; --cell-justify: center"><p>$20&lt;w\le 40$</p></th><th style="text-align: center; --cell-justify: center"><p>$40&lt;w\le 60$</p></th><th style="text-align: center; --cell-justify: center"><p>$60&lt;w\le 80$</p></th><th style="text-align: center; --cell-justify: center"><p>$80&lt;w\le 100$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p><strong>Number of scouts</strong></p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$12$</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>$3$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li><div class="list-item-paragraph">Copy and complete the following table.</div><div class="table-container"><table style="table-layout: fixed; width: 600px;"><colgroup><col style="width: 150px;"><col style="width: 150px;"><col style="width: 150px;"><col style="width: 150px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Amount collected ($\$w$)</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Mid-value, $x$</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Frequency, $f$</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>$fx$</strong></p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$0&lt;w\le 20$</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$20&lt;w\le 40$</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$40&lt;w\le 60$</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$60&lt;w\le 80$</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$80&lt;w\le 100$</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"><p>$\sum f=$</p></td><td style="text-align: left; --cell-justify: flex-start"><p>$\sum fx=$</p></td></tr></tbody></table></div></li><li><div class="list-item-paragraph">Estimate the mean amount of money collected by each of the $30$ scouts.</div></li><li><div class="list-item-paragraph">State the class interval which contains the median.</div></li></ol>

In the recent Job Week carried out by a group of $30$ scouts, the amount of money collected are shown in the table below.

<p><strong>Amount collected ($\$w$)</strong></p>

Constructing and Interpreting Frequency Tables

Scouts’ Earnings in Job Week: Mean and Median Class

<p>The thickness of $50$ magazines from a bookstore measured in cm (correct to $2$ significant figures) are given in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 620px;"><colgroup><col style="width: 12.9032%;"><col style="width: 14.5161%;"><col style="width: 14.5161%;"><col style="width: 14.5161%;"><col style="width: 14.5161%;"><col style="width: 14.5161%;"><col style="width: 14.5161%;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Thickness, $x$ (cm)</strong></p></th><th style="text-align: center; --cell-justify: center"><p>$0.50\le x&lt;0.52$</p></th><th style="text-align: center; --cell-justify: center"><p>$0.52\le x&lt;0.54$</p></th><th style="text-align: center; --cell-justify: center"><p>$0.54\le x&lt;0.56$</p></th><th style="text-align: center; --cell-justify: center"><p>$0.56\le x&lt;0.58$</p></th><th style="text-align: center; --cell-justify: center"><p>$0.58\le x&lt;0.60$</p></th><th style="text-align: center; --cell-justify: center"><p>$0.60\le x&lt;0.62$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p><strong>Frequency</strong></p></td><td style="text-align: center; --cell-justify: center"><p>$6$</p></td><td style="text-align: center; --cell-justify: center"><p>$8$</p></td><td style="text-align: center; --cell-justify: center"><p>$12$</p></td><td style="text-align: center; --cell-justify: center"><p>$11$</p></td><td style="text-align: center; --cell-justify: center"><p>$7$</p></td><td style="text-align: center; --cell-justify: center"><p>$6$</p></td></tr></tbody></table></div></div>
<ol class="abc-list"><li data-list-item-id="item-1765426513651-9bmc9rpdv" data-list-item-path="item-1765426513651-9bmc9rpdv"><div class="list-item-paragraph">State the median class.</div></li><li data-list-item-id="item-1765426514334-9eu0q8ue6" data-list-item-path="item-1765426514334-9eu0q8ue6"><div class="list-item-paragraph">State the modal class.</div></li><li data-list-item-id="item-1765426516216-t7discddj" data-list-item-path="item-1765426516216-t7discddj"><div class="list-item-paragraph">Fill in the frequency table below.</div><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Thickness, $x$ (cm)</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>$f$</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Mid-Value ($x$)</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>$fx$</strong></p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$0.50\le x&lt;0.52$</p></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$0.52\le x&lt;0.54$</p></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$0.54\le x&lt;0.56$</p></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$0.56\le x&lt;0.58$</p></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$0.58\le x&lt;0.60$</p></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$0.60\le x&lt;0.62$</p></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p><strong>Total</strong></p></td><td style="text-align: center; --cell-justify: center"></td><td style="text-align: center; --cell-justify: center"><p><strong>Total</strong></p></td><td style="text-align: center; --cell-justify: center"></td></tr></tbody></table></div></li><li data-list-item-id="item-1765426689742-qvdgirlov" data-list-item-path="item-1765426689742-qvdgirlov"><div class="list-item-paragraph">Estimate the mean thickness of the magazines.</div></li></ol>

The thickness of $50$ magazines from a bookstore measured in cm (correct to $2$ significant figures) are given in the table below.

<p><strong>Thickness, $x$ (cm)</strong></p>

Magazine Thickness: Median Class, Modal Class, and Mean Estimate

<p>The grouped frequency table below shows the height of $15$ plants in an experiment.</p>
<p>Complete the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 600px;"><colgroup><col style="width: 150px;"><col style="width: 150px;"><col style="width: 150px;"><col style="width: 150px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Height ($H$cm)</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Mid-value ($x$)</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>Frequency ($f$)</strong></p></th><th style="text-align: center; --cell-justify: center"><p><strong>$fx$</strong></p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$30&lt;H\le 40$</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$40&lt;H\le 50$</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: center; --cell-justify: center"><p>$1$</p></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$50&lt;H\le 60$</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: center; --cell-justify: center"><p>$6$</p></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$60&lt;H\le 70$</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: left; --cell-justify: flex-start"></td></tr><tr><td style="text-align: center; --cell-justify: center"><p>$70&lt;H\le 80$</p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: center; --cell-justify: center"><p>$2$</p></td><td style="text-align: left; --cell-justify: flex-start"></td></tr></tbody></table></div></div>
<p>Find the estimated mean height of the $15$ plants.	</p>

The grouped frequency table below shows the height of $15$ plants in an experiment.

Find the estimated mean height of the $15$ plants.	

Grouped Data on Plant Heights

<p>The data below represent the distance in km travelled by $20$ people from home to their workplace.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 500px;"><colgroup><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;"><col style="width: 50px;"><col style="width: 50px;"><col style="width: 50px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p>$11$</p></th><th style="text-align: center; --cell-justify: center"><p>$5$</p></th><th style="text-align: center; --cell-justify: center"><p>$9$</p></th><th style="text-align: center; --cell-justify: center"><p>$35$</p></th><th style="text-align: center; --cell-justify: center"><p>$43$</p></th><th style="text-align: center; --cell-justify: center"><p>$16$</p></th><th style="text-align: center; --cell-justify: center"><p>$23$</p></th><th style="text-align: center; --cell-justify: center"><p>$27$</p></th><th style="text-align: center; --cell-justify: center"><p>$21$</p></th><th style="text-align: center; --cell-justify: center"><p>$12$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p>$3$</p></td><td style="text-align: center; --cell-justify: center"><p>$11$</p></td><td style="text-align: center; --cell-justify: center"><p>$16$</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$31$</p></td><td style="text-align: center; --cell-justify: center"><p>$35$</p></td><td style="text-align: center; --cell-justify: center"><p>$11$</p></td><td style="text-align: center; --cell-justify: center"><p>$23$</p></td><td style="text-align: center; --cell-justify: center"><p>$15$</p></td><td style="text-align: center; --cell-justify: center"><p>$24$</p></td></tr></tbody></table></div></div>
<ol class="i-list"><li><div class="list-item-paragraph">Complete the frequency table for the data.</div><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Distance ($d$</strong> <strong>km)</strong></p></th><th style="text-align: center; --cell-justify: center"><p>$0&lt;d\le 10$</p></th><th style="text-align: center; --cell-justify: center"><p>$10&lt;d\le 20$</p></th><th style="text-align: center; --cell-justify: center"><p>$20&lt;d\le 30$</p></th><th style="text-align: center; --cell-justify: center"><p>$30&lt;d\le 40$</p></th><th style="text-align: center; --cell-justify: center"><p>$40&lt;d\le 50$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p><strong>Number of people</strong></p></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td><td style="text-align: left; --cell-justify: flex-start"></td></tr></tbody></table></div></li><li><div class="list-item-paragraph">Using the frequency table in <strong>(i)</strong>, calculate an estimate for the mean of the distance 	travelled by the $20$ people.</div></li></ol>

The data below represent the distance in km travelled by $20$ people from home to their workplace.

Grouped Data: Distance Travelled by Individuals

<p>During a period of $60$ days a weather station recorded number of hours of sunshine each day. The results are given in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 610px;"><colgroup><col style="width: 100px;"><col style="width: 85px;"><col style="width: 85px;"><col style="width: 85px;"><col style="width: 85px;"><col style="width: 85px;"><col style="width: 85px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Time ($x$ hours)</strong></p></th><th style="text-align: center; --cell-justify: center"><p>$0\le x\le 2$</p></th><th style="text-align: center; --cell-justify: center"><p>$2&lt;x\le 4$</p></th><th style="text-align: center; --cell-justify: center"><p>$4&lt;x\le 6$</p></th><th style="text-align: center; --cell-justify: center"><p>$6&lt;x\le 8$</p></th><th style="text-align: center; --cell-justify: center"><p>$8&lt;x\le 10$</p></th><th style="text-align: center; --cell-justify: center"><p>$10&lt;x\le 12$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p><strong>Frequency</strong></p></td><td style="text-align: center; --cell-justify: center"><p>$10$</p></td><td style="text-align: center; --cell-justify: center"><p>$15$</p></td><td style="text-align: center; --cell-justify: center"><p>$12$</p></td><td style="text-align: center; --cell-justify: center"><p>$10$</p></td><td style="text-align: center; --cell-justify: center"><p>$8$</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td></tr></tbody></table></div></div>
<p>Calculate an estimate of the mean number of hours of sunshine each day.	</p>

During a period of $60$ days a weather station recorded number of hours of sunshine each day. The results are given in the table below.

Calculate an estimate of the mean number of hours of sunshine each day.	

Sunshine Duration: Grouped Frequency Analysis

<p>The weekly allowance of $150$ children are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="table-layout: fixed; width: 570px;"><colgroup><col style="width: 120px;"><col style="width: 90px;"><col style="width: 90px;"><col style="width: 90px;"><col style="width: 90px;"><col style="width: 90px;"></colgroup><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Weekly allowance ($)</strong></p></th><th style="text-align: center; --cell-justify: center"><p>$40&lt;x\le 45$</p></th><th style="text-align: center; --cell-justify: center"><p>$45&lt;x\le 50$</p></th><th style="text-align: center; --cell-justify: center"><p>$50&lt;x\le 55$</p></th><th style="text-align: center; --cell-justify: center"><p>$55&lt;x\le 60$</p></th><th style="text-align: center; --cell-justify: center"><p>$60&lt;x\le 65$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p><strong>Number of children</strong></p></td><td style="text-align: center; --cell-justify: center"><p>$23$</p></td><td style="text-align: center; --cell-justify: center"><p>$38$</p></td><td style="text-align: center; --cell-justify: center"><p>$45$</p></td><td style="text-align: center; --cell-justify: center"><p>$31$</p></td><td style="text-align: center; --cell-justify: center"><p>$13$</p></td></tr></tbody></table></div></div>
<p>Calculate an estimate of the mean weekly allowance of the children.</p>

The weekly allowance of $150$ children are shown in the table below.

<p><strong>Weekly allowance ($)</strong></p>

Calculate an estimate of the mean weekly allowance of the children.

Weekly Allowance of Children: Grouped Data Analysis

<p>In a donation drive carried out by a group of $45$ volunteers, the amount of money collected are shown in the table below.</p>
<div class="content-block content-block--table"><div class="table-container"><table style="width: 100%;"><thead><tr><th style="text-align: center; --cell-justify: center"><p><strong>Amount collected ($\$w$)</strong></p></th><th style="text-align: center; --cell-justify: center"><p>$0&lt;w\le 20$</p></th><th style="text-align: center; --cell-justify: center"><p>$20&lt;w\le 40$</p></th><th style="text-align: center; --cell-justify: center"><p>$40&lt;w\le 60$</p></th><th style="text-align: center; --cell-justify: center"><p>$60&lt;w\le 80$</p></th><th style="text-align: center; --cell-justify: center"><p>$80&lt;w\le 100$</p></th></tr></thead><tbody><tr><td style="text-align: center; --cell-justify: center"><p><strong>Number of volunteers</strong></p></td><td style="text-align: center; --cell-justify: center"><p>$8$</p></td><td style="text-align: center; --cell-justify: center"><p>$18$</p></td><td style="text-align: center; --cell-justify: center"><p>$9$</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td><td style="text-align: center; --cell-justify: center"><p>$5$</p></td></tr></tbody></table></div></div>
<p>Estimate the mean amount of money collected by each of the $45$ volunteers.		</p>

In a donation drive carried out by a group of $45$ volunteers, the amount of money collected are shown in the table below.

Estimate the mean amount of money collected by each of the $45$ volunteers.		

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