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.

The following table shows the widths of the 70 books in one section of a library, given to the nearest centimetre. Width ( c m ) 10 - 14 15 - 19 20 - 29 30 - 39 40 - 44 No. books ( f ) 3 13 25 24 5 Given that the upper boundary of the first class is 14 . 5 cm , write down the upper boundary of the second class.

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 Here, the gap between two interval is equal to 15 - 14 = 20 - 19 = 30 - 29 = 40 - 39 = 1 The upper boundary of the second class is 19 + 0 . 5 = 19 . 5 cm

The following table shows the widths of the 70 books in one section of a library, given to the nearest centimetre. Width ( c m ) 10 - 14 15 - 19 20 - 29 30 - 39 40 - 44 No. books ( f ) 3 13 25 24 5 Draw up a cumulative frequency table for the data and construct a cumulative frequency graph.

To write a cumulative frequency table Find the frequency for each class interval Locate the endpoint for each class interval calculate the cumulative frequency by adding the numbers in the Frequency column i.e, First, add the number from the Frequency column to its predecessor. record all results in the table Width (cm) < 9 . 5 < 14 . 5 < 19 . 5 < 29 . 5 < 39 . 5 < 44 . 5 No. books ( c . f . ) 0 3 16 41 65 70 Take a graph paper and mark the width (cm) 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.

EASY

AS and A Level

IMPORTANT

Earn 100

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.

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 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.

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 2

The following table shows the widths of the $70$ books in one section of a library, given to the nearest centimetre.

Width ( $c m$ )	$10 - 14$	$15 - 19$	$20 - 29$	$30 - 39$	$40 - 44$
No. books ( $f$ )	$3$	$13$	$25$	$24$	$5$

Given that the upper boundary of the first class is $14.5 cm$ , write down the upper boundary of the second class.

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 2

The following table shows the widths of the $70$ books in one section of a library, given to the nearest centimetre.

Width ( $c m$ )	$10 - 14$	$15 - 19$	$20 - 29$	$30 - 39$	$40 - 44$
No. books ( $f$ )	$3$	$13$	$25$	$24$	$5$

Draw up a cumulative frequency table for the data and construct a cumulative frequency graph.

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

Important Questions on Representation of Data

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