The masses, in kilograms, of the animals treated at a veterinary clinic in the past year are illustrated in a histogram. The histogram has four columns of equal height. The following table shows the class intervals and the number of animals in two of the classes. Mass kg 3 - 5 6 - 12 13 - 32 33 - 44 No. animals f a 371 1060 b Find the value of a and of b , and show that a total of 2226 animals were treated at the clinic. Calculate an estimate of the lower boundary of the masses of the heaviest 50 % of these animals.

Lower boundary of class interval = Lower limit - 1 2 gap between two intervals Upper boundary of class interval = Lower limit - 1 2 gap between two intervals Mass ( Kg ) 2 . 5 - 5 . 5 5 . 5 - 12 . 5 125 - 32 - 5 325 - 44 . 5 No. of arimaly ( f ) a 371 1060 b class width 3 7 20 12 Frequeray density a 3 53 53 b 12 Class width is the size of the class interval Frequency density is 371 7 = 1060 20 = 53 for all classes, so class frequency = 53 × c l a s s w i d t h . a = 53 × 3 = 159 and b = 53 × 12 = 636 Total frequency is 159 + 371 + 1060 + 636 = 2226 . 50 % of 2226 = 1113 , so the 1114 th mass is the lightest of the heaviest 50 % . 1114 - 159 - 371 = 584 mass in third class. Estimate is 12 . 5 + 584 1060 × ( 32 . 5 - 12 . 5 ) = 23 . 5 kg .

The percentage examination scores of 747 students are given in the following table. Score % p - 50 51 - 70 70 - 80 81 - q No. students f 165 240 195 147 Given that the frequency densities of the four classes of percentage scores are in the ratio 5 : 8 : 13 : 7 , find the value of p and of q .

Score p - 0 . 5 - 50 . 5 50 . 5 - 70 . 5 70 . 5 - 80 . 5 80.5- q + 0 . 5 f 165 240 195 147 width 51 - p 20 10 q - 80 frequency density 165 51 - ρ 12 19 . 5 147 q - 80 Therefore 165 51 - p : 12 : 19 . 5 : 147 q - 80 are equivalent percentage of 5 : 8 : 13 : 7 Now consider 165 51 - p 12 = 5 8 gives p = 29 and 19 . 5 147 q - 80 = 13 7 gives q = 94 . Therefore, 5 : 8 : 13 : 7 is equivalent to 7 . 5 : 12 : 19 . 5 : 10 . 5 and these are the frequency densities.

The reaction times, t seconds, of 66 participants were measured in an experiment and presented below. Time ( t seconds ) No. participants ( c f ) t < 1 . 5 0 t < 3 . 0 3 t < 4 . 5 8 t < 6 . 5 32 t < 8 . 5 54 t < 11 . 0 62 t < 13 . 0 66 Draw a cumulative frequency polygon to represent the data.

Take a graph paper and mark the time (t seconds) along the x -axis and the corresponding cumulative frequencies along the y -axis. Join these points successively by line segments, we will get a polygon, known as a cumulative frequency polygon.

The reaction times, t seconds, of 66 participants were measured in an experiment and presented below. Time ( t seconds ) No. participants ( c f ) t < 1 . 5 0 t < 3 . 0 3 t < 4 . 5 8 t < 6 . 5 32 t < 8 . 5 54 t < 11 . 0 62 t < 13 . 0 Draw a cumulative frequency polygon to represent the data. Use your graph to estimate: the number of participants with reaction times between 5 . 5 and 7 . 5 seconds

Take a graph paper and mark the time (t seconds) along the x -axis and the corresponding cumulative frequencies along the y -axis. Join these points successively by line segments, we will get a polygon, known as a cumulative frequency polygon. At 7 . 5 seconds the cumulative frequency is 43 At 5 . 5 seconds the cumulative frequency is 20 Therefore, the number of participants with the given reaction time is equal to the difference between above two cumulative frequencies = 43 - 20 = 23 .

The reaction times, t seconds, of 66 participants were measured in an experiment and presented below. Time ( t seconds ) No. participants ( c f ) t < 1 . 5 0 t < 3 . 0 3 t < 4 . 5 8 t < 6 . 5 32 t < 8 . 5 54 t < 11 . 0 62 t < 13 . 0 66 Use your graph to estimate: the lower boundary of the slowest 20 reaction times.

Take a graph paper and mark the time (t seconds) along the x -axis and the corresponding cumulative frequencies along the y -axis. Join these points successively by line segments, we will get a polygon, known as a cumulative frequency polygon. To find the lower boundary of the slowest 20 reaction times , we need to take the difference of slowest 20 reaction times and maximum participants The slowest 20 times are for participants who took longer than the 66 - 20 = 46 t h , and the 46 t h value is 7 . 8 seconds.

EASY

AS and A Level

IMPORTANT

Earn 100

The thicknesses, $k mm,$ of some steel sheets are represented in the histogram. It is given that $k < 0.4$ for $180$ sheets.

Find the value of $n,$ given that frequency density measures sheets per $n mm .$
The sheets are classified as thin, medium or thick in the ratio $1 : 3 : 1 .$
Estimate the thickness of a medium sheet, giving your answer in the form $a ⩽ k < b .$ How accurate are your values for $a$ and $b ?$

Important Questions on Representation of Data

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 1 - Representation of Data>EXERCISE 1B>Q 10

The masses, in kilograms, of the animals treated at a veterinary clinic in the past year are illustrated in a histogram. The histogram has four columns of equal height. The following table shows the class intervals and the number of animals in two of the classes.

Mass $(kg)$	$3 - 5$	$6 - 12$	$13 - 32$	$a$
No. animals $(f)$	$a$	$371$	$1060$	$b$

Find the value of

a

and of

b,

and show that a total of

2226

animals were treated at the clinic.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 1 - Representation of Data>EXERCISE 1B>Q 10

Mass $(kg)$	$3 - 5$	$6 - 12$	$13 - 32$	$33 - 44$
No. animals $(f)$	$a$	$371$	$1060$	$b$

Find the value of

a

and of

b,

and show that a total of

2226

animals were treated at the clinic.

Calculate an estimate of the lower boundary of the masses of the heaviest $50 %$ of these animals.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 1 - Representation of Data>EXERCISE 1B>Q 11

The minimum daily temperature at a mountain village was recorded to the nearest $0.5 ° C$ on $200$ consecutive days. The results are grouped into a frequency table and a histogram is drawn.

The temperatures ranged from $0.5 ° C$ to $2 ° C$ on $n$ days, and this class is represented by a column of height $h cm .$
The temperatures ranged from $- 2.5$ to $- 0.5 ° C$ on $d$ days. Find, in terms of $n, h$ and $d,$ the height of the column that represents these temperatures.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 1 - Representation of Data>EXERCISE 1B>Q 12

The frequency densities of the four classes in a histogram are in the ratio $4 : 3 : 2 : 1 .$ The frequencies of these classes are in the ratio $10 : 15 : 24 : 8 .$

Find the total width of the histogram, given that the narrowest class interval is represented by a column of width $3 cm .$

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 1 - Representation of Data>EXERCISE 1B>Q 13

The percentage examination scores of

747

students are given in the following table.

Score $(%)$	$p - 50$	$51 - 70$	$70 - 80$	$81 - q$
No. students $(f)$	$165$	$240$	$195$	$147$

Given that the frequency densities of the four classes of percentage scores are in the ratio

5 : 8 : 13 : 7,

find the value of

p

and of

q .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 1 - Representation of Data>EXERCISE 1C>Q 1

The reaction times, $t$ seconds, of $66$ participants were measured in an experiment and presented below.

Time $($ $t$ seconds $)$	No. participants $(c f)$
$t < 1.5$	$0$
$t < 3.0$	$3$
$t < 4.5$	$8$
$t < 6.5$	$32$
$t < 8.5$	$54$
$t < 11.0$	$62$
$t < 13.0$	$66$

Draw a cumulative frequency polygon to represent the data.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 1 - Representation of Data>EXERCISE 1C>Q 1

The reaction times, $t$ seconds, of $66$ participants were measured in an experiment and presented below.

Time $($ $t$ seconds $)$	No. participants $(c f)$
$t < 1.5$	$0$
$t < 3.0$	$3$
$t < 4.5$	$8$
$t < 6.5$	$32$
$t < 8.5$	$54$
$t < 11.0$	$62$
$t < 13.0$

Draw a cumulative frequency polygon to represent the data.

Use your graph to estimate:

the number of participants with reaction times between $5.5$ and $7.5$ seconds

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 1 - Representation of Data>EXERCISE 1C>Q 1

The reaction times, $t$ seconds, of $66$ participants were measured in an experiment and presented below.

Time $($ $t$ seconds $)$	No. participants $(c f)$
$t < 1.5$	$0$
$t < 3.0$	$3$
$t < 4.5$	$8$
$t < 6.5$	$32$
$t < 8.5$	$54$
$t < 11.0$	$62$
$t < 13.0$	$66$

Use your graph to estimate:

the lower boundary of the slowest $20$ reaction times.

Important Questions on Representation of Data

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