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.

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 Use your graph to estimate: the number of books that have widths of less than 27 cm

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. Cumulative frequency value at width 27 c m is 34 or 35 .

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 Use your graph to estimate: the widths of the widest 20 books.

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. Cumulative frequency value at width 27 c m is 34 or 35 . The widest 20 books are wider than the 50 t h book. The 50 t h value is 33 . 25 c m and the 70 t h (last) value is 44 . 5 c m , So the widths are from ≈ 33 . 25 to 44 . 5 cm .

Measurements of the distances, x mm , between two moving parts inside car engines were recorded and are summarised in the following table. There were 156 engines of type A and 156 engines of type B . Distance ( x mm ) x < 0 . 10 x < 0 . 35 x < 0 . 60 x < 0 . 85 x < 1 . 20 Engine A ( c f ) 0 16 84 144 156 Engine B ( c f ) 0 8 62 120 156 Draw and label two cumulative frequency curves on the same axes.

Take a graph paper and mark the distance (x mm) along the x -axis and the corresponding cumulative frequencies along the y -axis. Join these points successively by line segments,

EASY

AS and A Level

IMPORTANT

Earn 100

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 $q .$

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 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.

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 ( $f$ )	$3$	$13$	$25$	$24$	$5$

Use your graph to estimate:
the number of books that have widths of less than $27 cm$

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 ( $f$ )	$3$	$13$	$25$	$24$	$5$

Use your graph to estimate:
the widths of the widest $20$ books.

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 3

Measurements of the distances, $x mm$ , between two moving parts inside car engines were recorded and are summarised in the following table. There were $156$ engines of type $A$ and $156$ engines of type $B$ .

Distance $(x mm)$	$x < 0.10$	$x < 0.35$	$x < 0.60$	$x < 0.85$	$x < 1.20$
Engine $A (c f)$	$0$	$16$	$84$	$144$	$156$
Engine $B (c f)$	$0$	$8$	$62$	$120$	$156$

Draw and label two cumulative frequency curves on the same axes.

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

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.

Important Questions on Representation of Data

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