The following table illustrates the heights, in centimetres, of 150 children. Height cm No. children f 140 up to 144 a 144 up to 150 b 150 up to 160 69 160 up to 165 28 Given that a calculated estimate of the mean height is exactly 153 . 14 cm , show that 142 a + 147 b = 7726 , and evaluate a and b .

Given: Height cm No. children f 140 up to 144 a 144 up to 150 b 150 up to 160 69 160 up to 165 28 Class mid-values are 142 , 147 , 155 and 162 . 5 . Let the class mid- values be x . We know that, mean x ¯ = ∑ x f ∑ f Now, 142 a + 147 b + ( 155 × 69 ) + ( 162 . 5 × 28 ) 150 = 153 . 14 ⇒ 142 a + 147 b + 15245 = 22971 ⇒ 142 a + 147 b = 7726 .......(i) Hence showed. Here, total number of children = 150 ⇒ a + b + 69 + 28 = 150 ⇒ a + b = 53 ⇒ b = 53 - a .....(ii) Substitute equation (ii) in (i) ⇒ 142 a + 147 ( 53 - a ) = 7726 ⇒ a = 13 and b = 40 .

The following table illustrates the heights, in centimetres, of 150 children. Height cm No. children f 140 up to 144 a 144 up to 150 b 150 up to 160 69 160 up to 165 28 Given that a calculated estimate of the mean height is exactly 153 . 14 cm , show that 142 a + 147 b = 7726 , and evaluate a and b . Calculate an estimate of the standard deviation of the heights.

Given: Height cm No. children f 140 up to 144 a 144 up to 150 b 150 up to 160 69 160 up to 165 28 Class mid-values are 142 , 147 , 155 and 162 . 5 . Let the class mid- values be x . We know that, mean x ¯ = ∑ x f ∑ f Now, 142 a + 147 b + ( 155 × 69 ) + ( 162 . 5 × 28 ) 150 = 153 . 14 ⇒ 142 a + 147 b + 15245 = 22971 ⇒ 142 a + 147 b = 7726 .......(i) Here, total number of children = 150 ⇒ a + b + 69 + 28 = 150 ⇒ a + b = 53 ⇒ b = 53 - a .....(ii) Substitute equation (ii) in (i) ⇒ 142 a + 147 ( 53 - a ) = 7726 ⇒ a = 13 and b = 40 . Now, mean of squares = ∑ x 2 f ∑ f = 3623692 150 = 24157 . 946 We know that, standard deviation SD = Mean of squares - Mean 2 Therefore, standard deviation SD = 3523692 150 - 22971 150 2 = 6 . 23 cm .

The mass of waste produced by a school during its three 13 -week terms is given in tonnes, correct to 2 decimal places, in the following table Mass of waste (tonnes) 0 . 15 – 0 . 29 0 . 30 – 0 . 86 0 . 87 – 1 . 35 1 . 36 – 2 . 00 No. weeks f 5 8 20 6 Calculate estimates of the mean and standard deviation of the mass of waste produced per week, giving both answers correct to 2 decimal places

Given: First convert discrete class into continuous classes by subtracting 0 . 005 from the lower limit and adding 0 . 005 to the upper limit. Mass of waste (tonnes) 0 . 145 – 0 . 295 0 . 295 – 0 . 865 0 . 865 – 1 . 355 1 . 355 – 1 . 995 No. weeks f 5 8 20 6 Class mid-values are 0 . 22 , 0 . 58 , 1 . 11 and 1 . 68 . Let the class mid- values be x . We know that, mean x ¯ = ∑ x f ∑ f Now, mean ≈ 0 . 22 × 5 + 0 . 58 × 8 + 1 . 11 × 20 + 1 . 68 × 6 39 = 38 . 02 39 = 0 . 97 tonnes Now, Mean of squares = ∑ x 2 f ∑ f = 0 . 22 2 × 5 + 0 . 58 2 × 8 + 1 . 11 2 × 20 + 1 . 68 2 × 6 39 We know that, SD = Mean of squares - Mean 2 Therefore, standard deviation S D ≈ 0 . 22 2 × 5 + 0 . 58 2 × 8 + 1 . 11 2 × 20 + 1 . 68 2 × 6 39 - 38 . 02 39 8 . = 44 . 5096 39 - 38 . 02 39 2 = 0 . 44 tonnes

EASY

AS and A Level

IMPORTANT

Earn 100

For the dataset denoted by $x$ in the following table, $k$ is a constant.

$x$ $15$ $16$ $17$ $18$ $19$ $20$

$f$ $2 k$ $k + 5$ $k - 3$ $10$ $8$ $3$

Find the value of $k$ and calculate the variance of $x,$ given that $\bar{x} = 17 .$

Important Questions on Measures of Variation

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3B>Q 8

The following table illustrates the heights, in centimetres, of $150$ children.

Height $(cm)$	No. children $(f)$
$140$ up to $144$	$a$
$144$ up to $150$	$b$
$150$ up to $160$	$69$
$160$ up to $165$	$28$

Given that a calculated estimate of the mean height is exactly $153.14 cm,$ show that $142 a + 147 b = 7726,$ and evaluate $a$ and $b .$

MEDIUM

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3B>Q 8

The following table illustrates the heights, in centimetres, of $150$ children.

Height $(cm)$	No. children $(f)$
$140$ up to $144$	$a$
$144$ up to $150$	$b$
$150$ up to $160$	$69$
$160$ up to $165$	$28$

Given that a calculated estimate of the mean height is exactly $153.14 cm,$ show that $142 a + 147 b = 7726,$ and evaluate $a$ and $b .$

Calculate an estimate of the standard deviation of the heights.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3B>Q 9

Kristina plans to raise money for charity. Her plan is to walk

217 km

7

days so that she walks

k + 2 n - n^{2} km

on the

n

th day. Find the standard deviation of the daily distances she plans to walk, and compare this with 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 3B>Q 10

The mass of waste produced by a school during its three $13$ -week terms is given in tonnes, correct to $2$ decimal places, in the following table

Mass of waste (tonnes)	$0.15 - 0.29$	$0.30 - 0.86$	$0.87 - 1.35$	$1.36 - 2.00$
No. weeks $(f)$	$5$	$8$	$20$	$6$

Calculate estimates of the mean and standard deviation of the mass of waste produced per week, giving both answers correct to $2$ decimal places

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 3 - Measures of Variation>EXERCISE 3B>Q 10

The mass of waste produced by a school during its three $13$ -week terms is given in tonnes, correct to $2$ decimal places, in the following table

Mass of waste (tonnes)	$0.15 - 0.29$	$0.30 - 0.86$	$0.87 - 1.35$	$1.36 - 2.00$
No. of weeks $(f)$	$5$	$8$	$20$	$6$

Calculate estimates of the mean and standard deviation of the mass of waste produced per week, giving both answers correct to $2$ decimal places

No waste is produced in the $13$ weeks of the year that the school is closed. If this additional data is included in the calculations, what effect does it have on the mean and on the standard deviation?

EASY

AS and A Level

IMPORTANT

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

The ages, in whole numbers of years, of a hotel's $50$ staff are given in the following table. Calculated estimates of the mean and variance are $37.32$ and $69.1176,$ respectively.

Age (years)	$23 - 30$	$31 - 37$	$38 - 45$	$46 - 59$
No. staff $(f)$	$14$	$x$	$y$	$6$

Exactly $1$ year after these calculations were made, Gudrun became the $51$ st staff member and the mean age became exactly $38$ years. Find Gudrun's age on the day of her recruitment, and determine what effect this had on the variance of the staff's ages. What assumptions must be made to justify your answers?

EASY

AS and A Level

IMPORTANT

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

Refer to the following diagram. In position $1,$ a $10$ -metre rod is placed $10$ metres from a fixed point, $P .$ Six small discs, $A$ to $F,$ are evenly spaced along the length of the rod. The rod is rotated anti-clockwise about its centre by $α = 30 °$ to position $2 .$ The distances from $P$ to the discs are denoted by $x .$

Question Image

What effect does the $30 °$ rotation have on values of $x$ ? Investigate this by first considering the effect on the average distance from $P$ to the discs.

EASY

AS and A Level

IMPORTANT

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

Question Image

Find two values that can be used as measures of the change in the variation of $x$ caused by the rotation.

For the dataset denoted by x in the following table, k is a constant. x 15 16 17 18 19 20 f 2k k+5 k–3 10 8 3 Find the value of k and calculate the variance of x, given that x¯=17.

Important Questions on Measures of Variation

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For the dataset denoted by $x$ in the following table, $k$ is a constant.

$x$ $15$ $16$ $17$ $18$ $19$ $20$

$f$ $2 k$ $k + 5$ $k - 3$ $10$ $8$ $3$

Find the value of $k$ and calculate the variance of $x,$ given that $\bar{x} = 17 .$