The following table shows the cumulative frequencies for values of x . x < 0 < 10 < 15 < 25 < 30 < 40 c f 0 12 30 90 102 120 Without drawing a cumulative frequency graph, find: the interquartile range

Given data: x < 0 < 10 < 15 < 25 < 30 < 40 c f 0 12 30 90 102 120 Let number of data values be n . ⇒ n = 120 We know that, for grouped continuous data the formula of lower quartile is Q 1 = Upper limit of the class whose cumulative frequency is greater than and nearest to n 4 and upper quartile is Q 3 = Upper limit of the class whose cumulative frequency is greater than and nearest to 3 n 4 . Lower quartile is Q 1 = Upper limit of the class whose cumulative frequency is greater than and nearest to 120 4 Thus lower quartile = 15 . Upper quartile is Q 3 = Upper limit of the class whose cumulative frequency is greater than and nearest to 3 × 120 4 . Thus, upper quartile = 25 Interquartile range = Q 3 - Q 1 = 90 th value - 30 th value = 25 - 15 = 10

The following table shows the cumulative frequencies for values of x . x < 0 < 10 < 15 < 25 < 30 < 40 c f 0 12 30 90 102 120 Without drawing a cumulative frequency graph, find: the 85 th percentile.

Given data: x < 0 < 10 < 15 < 25 < 30 < 40 c f 0 12 30 90 102 120 Let number of data values be n . ⇒ n = 120 We know that, for grouped continuous data the formula of 85 th percentile = Upper limit of the class whose cumulative frequency is greater than and nearest to 85 100 × n th value. Thus, 85 th percentile for the given data is 0 . 85 × 120 th value = 102 nd value = 30 .

Fifty 10 -gram samples of a particular type of mushroom are collected by volunteers at a university and tested. The following table shows the mass of toxins, in hundredths of a gram, in these samples. Mass 10 . 01 g 0 - 4 - 11 - 17 - 20 - 30 No. 10 g samples 2 19 23 3 3 It was found that toxins made up between 0 . 75 % and 2 . 25 % of the mass of n of these samples. Use your curve to estimate the value of n .

Given curve is, Cumulative frequency curve is plotted against the endpoints of data class on the x -axis and cumulative frequency on the y -axis. Cumulative frequency curves enable us to find the median and quartiles from a set of grouped data. Here, n = 50 , where n represents total number of data values. The range of middle 80 % is n = ( c f value at 22.5 g )-( c f value at 7.5 g ) , where c f is cumulative frequency. To determine the cf value at 22 . 5 g : Draw a line across from 22 . 5 value on the horizontol axis until it hits the curve and then draw a line horizontolly sidewatds until it touches the vertical axis. To detemine the cf value at 7 . 5 g : Draw a line across from 7 . 5 value on the horizontol axis until it hits the curve and then draw a line horizontolly sidewatds until it touches the vertical axis. Thus, n = ( c f value at 22 . 5 g ) - ( c f value at 7 . 5 g ) ≈ 48 - 8 = 40 .

Fifty 10 -gram samples of a particular type of mushroom are collected by volunteers at a university and tested. The following table shows the mass of toxins, in hundredths of a gram, in these samples. Mass 10 . 01 g 0 - 4 - 11 - 17 - 20 - 30 No. 10 g samples 2 19 23 3 3 Make an assessment of the variation in the percentage of toxic material in these samples. Can you suggest any possible reasons for such variation?

Given data: Mass 10 . 01 g 0 - 4 - 11 - 17 - 20 - 30 No. 10 g samples 2 19 23 3 3 The cumulative frequency is the sum of the frequencies for every class, up to and including the current one. A table which displays the manner in which cumulative frequencies are distributed over various classes is called a cumulative frequency distribution or cumulative frequency table. Cumulative frequency distribution table for the given data is Mass 10 . 01 g 0 - 4 4 - 11 11 - 17 17 - 20 20 - 30 No. 10 g samples (f) 2 19 23 3 3 Cumulative frequency 2 21 44 47 50 A curve that represents the cumulative frequency distribution of grouped data on a graph is called a Cumulative Frequency Curve or an Ogive Steps to draw less than type cumulative frequency curve: (i) Draw and mark the horizontal and vertical axes. (ii) Take the cumulative frequencies along the y -axis (vertical axis) and the upper-class limits on the x -axis (horizontal axis). (iii) Against each upper-class limit, plot the corresponding cumulative frequencies. (iv) Connect the points with a continuous curve. Plotted points are ( 0 , 0 ) , ( 4 , 2 ) , ( 11 , 21 ) , ( 17 , 44 ) , ( 20 . 47 ) , ( 30 , 50 ) Less than type cumulative frequency curve of the given data is Let number of data values be n . ⇒ n = 50 To determine the lower quartile Q 1 , we draw a horizontal line across from the frequency axis to the curve at the n 4 th value and then down to the x -axis. i.e., 50 4 th = 12 . 5 th value. Draw a line across from 12 . 5 on the vertical axis until it hits the curve and then draw a line vertically down until it touches the horizontal axis. This is the lower quartile. Thus, lower quartile of the given data is Q 1 = 0 . 0875 . To determine the upper quartile Q 3 , we draw a horizontal line across from the frequency axis to the curve at the 3 n 4 th value and then down to the x -axis. i.e., 3 × 50 4 th = 37 . 5 th value Draw a line across from 37 . 5 value on the vertical axis until it hits the curve and then draw a line vertically down until it touches the horizontal axis. This is the upper quartile. Thus, the upper quartile of the given data is Q 3 = 0 . 145 . Interquartile range = Q 3 - Q 1 ≈ 0 . 145 - 0 . 0875 = 0 . 06 g . Thus, from cumulative graph it is clear that Variation is quite dramatic (from 0 up to a possible 6 % of mass). Mushrooms are notoriously difficult to identify, so samples may not all be of the same type, and their toxicity varies by season.

Dean Chalmers and Julian Gilbey Solutions for Chapter: Measures of Variation, Exercise 3: EXERCISE 3A

Author:Dean Chalmers & Julian Gilbey

Dean Chalmers Mathematics Solutions for Exercise - Dean Chalmers and Julian Gilbey Solutions for Chapter: Measures of Variation, Exercise 3: EXERCISE 3A

Attempt the practice questions on Chapter 3: Measures of Variation, Exercise 3: EXERCISE 3A with hints and solutions to strengthen your understanding. Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book solutions are prepared by Experienced Embibe Experts.

Questions from Dean Chalmers and Julian Gilbey Solutions for Chapter: Measures of Variation, Exercise 3: EXERCISE 3A with Hints & Solutions

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3A>Q 11

The following table shows the cumulative frequencies for values of $x .$

$x$	$< 0$	$< 10$	$< 15$	$< 25$	$< 30$	$< 40$
$c f$	$0$	$12$	$30$	$90$	$102$	$120$

Without drawing a cumulative frequency graph, find:

the interquartile range

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3A>Q 11

The following table shows the cumulative frequencies for values of $x .$

$x$	$< 0$	$< 10$	$< 15$	$< 25$	$< 30$	$< 40$
$c f$	$0$	$12$	$30$	$90$	$102$	$120$

Without drawing a cumulative frequency graph, find:

the $85 th$ percentile.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3A>Q 12

Fifty $10$ -gram samples of a particular type of mushroom are collected by volunteers at a university and tested. The following table shows the mass of toxins, in hundredths of a gram, in these samples.

Mass $(10.01 g)$	$0 -$	$4 -$	$11 -$	$17 -$	$20 - 30$
No. $10 g$ samples	$2$	$19$	$23$	$3$	$3$

Draw a cumulative frequency curve to illustrate the data.

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3A>Q 12

Mass $(10.01 g)$	$0 -$	$4 -$	$11 -$	$17 -$	$20 - 30$
No. $10 g$ samples	$2$	$19$	$23$	$3$	$3$

Use your curve to estimate, correct to $2$ decimal places:

i. the interquartile range

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3A>Q 12

Mass $(10.01 g)$	$0 -$	$4 -$	$11 -$	$17 -$	$20 - 30$
No. $10 g$ samples	$2$	$19$	$23$	$3$	$3$

Use your curve to estimate, correct to $2$ decimal places:

Question Image

the range of the middle $80 % .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3A>Q 12

Mass $(10.01 g)$	$0 -$	$4 -$	$11 -$	$17 -$	$20 - 30$
No. $10 g$ samples	$2$	$19$	$23$	$3$	$3$

It was found that toxins made up between $0.75 %$ and $2.25 %$ of the mass of $n$ of these samples. Use your curve to estimate the value of $n .$

Question Image

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3A>Q 12

Mass $(10.01 g)$	$0 -$	$4 -$	$11 -$	$17 -$	$20 - 30$
No. $10 g$ samples	$2$	$19$	$23$	$3$	$3$

Make an assessment of the variation in the percentage of toxic material in these samples. Can you suggest any possible reasons for such variation?

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3A>Q 13

A $9$ -year study was carried out on the pollutants released when biomass fuels are used for cooking. Researchers offered nearly $1000$ people living in $12$ villages in southern China access to clean biogas and to improved kitchen ventilation. Some people took advantage of neither; some changed to clean fuels; some improved their kitchen ventilation; and some did both. The following diagram shows data on the concentrations of nitrogen dioxide in these people's homes at the end of the study.

Question Image

Study the data represented in the diagram and then write a brief analysis that summarises the results of this part of the study.

Dean Chalmers and Julian Gilbey Solutions for Chapter: Measures of Variation, Exercise 3: EXERCISE 3A

Dean Chalmers Mathematics Solutions for Exercise - Dean Chalmers and Julian Gilbey Solutions for Chapter: Measures of Variation, Exercise 3: EXERCISE 3A

Questions from Dean Chalmers and Julian Gilbey Solutions for Chapter: Measures of Variation, Exercise 3: EXERCISE 3A with Hints & Solutions

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