120 people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table. Time (seconds) 1 – 25 26 – 35 36 – 45 46 – 55 56 – 90 Number of people 4 24 38 34 20 Calculate estimates of the mean and standard deviation of the reading times.

First convert discrete class into continuous classes by subtracting 0 . 5 from the lower limit and adding 0 . 5 to the upper limit. Time (seconds) 0 . 5 – 25 . 5 25 . 5 – 35 . 5 35 . 5 – 45 . 5 45 . 5 – 55 . 5 55 . 5 – 90 . 5 Number of people f 4 24 38 34 20 Class mid-values of continuous intervals are 13 , 30 . 5 , 40 . 5 , 50 . 5 and 73 . Let the class mid- values be x . We know that, mean x ¯ = ∑ x f ∑ f Now, Mean ≈ Σ x f Σ f = 5500 120 = 45 . 8 seconds. Now, mean of squares = ∑ x 2 f ∑ f We know that, SD = Mean of squares - Mean 2 Therefore, standard deviation SD ≈ Σ x 2 f Σ f - Σ x f Σ f 2 = 278620 120 - 5500 120 2 = 14 . 9 seconds.

The weights, in kilograms, of the 15 basketball players in each of two squads, A and B , are shown below. Squad A 97 98 104 84 100 109 115 99 122 82 116 96 84 107 91 Squad B 75 79 94 101 96 77 111 108 83 84 86 115 82 113 95 Represent the data by drawing a back-to-back stem-and-leaf diagram with squad A on the left-hand side of the diagram and squad B on the right-hand side.

Given data: Squad A 97 98 104 84 100 109 115 99 122 82 116 96 84 107 91 Squad B 75 79 94 101 96 77 111 108 83 84 86 115 82 113 95 A back-to-back stem-and-leaf plot is a method for comparing two data distributions by attaching two sets of 'leaves' to the same 'stem' in a stem-and-leaf plot The leaf numbers running from least (at the top) to highest (at the bottom). Numbers in the right leaf run lowest to highest, with the lowest number placed closest to the stem. The data on the left side of the stem runs in the opposite direction (like a mirror) with the numbers running from lowest to highest starting at the stem.A back-to-back stem-and-leaf plot is a method for comparing two data distributions by attaching two sets of 'leaves' to the same 'stem' in a stem-and-leaf plot. The stem represents the category figure, the leaves represent the final digit(s) of each data point and the key tells us how to read the values.

The weights, in kilograms, of the 15 basketball players in each of two squads, A and B , are shown below. Squad A 97 98 104 84 100 109 115 99 122 82 116 96 84 107 91 Squad B 75 79 94 101 96 77 111 108 83 84 86 115 82 113 95 Find the interquartile range of the weights of the players in squad A .

Given data: Squad A 97 98 104 84 100 109 115 99 122 82 116 96 84 107 91 Squad B 75 79 94 101 96 77 111 108 83 84 86 115 82 113 95 A back-to-back stem-and-leaf plot is a method for comparing two data distributions by attaching two sets of 'leaves' to the same 'stem' in a stem-and-leaf plot The leaf numbers running from least (at the top) to highest (at the bottom). Numbers in the right leaf run lowest to highest, with the lowest number placed closest to the stem. The data on the left side of the stem runs in the opposite direction (like a mirror) with the numbers running from lowest to highest starting at the stem.A back-to-back stem-and-leaf plot is a method for comparing two data distributions by attaching two sets of 'leaves' to the same 'stem' in a stem-and-leaf plot. The stem represents the category figure, the leaves represent the final digit(s) of each data point and the key tells us how to read the values. Let number of data values be N for squad A . ⇒ N = 15 We know that, lower quartile Q 1 is the median of the lower 50 % of the data. Formula to find lower quartile is Q 1 = N + 1 4 th data value, where N is number of data values. ⇒ Lower quartile, Q 1 = 15 + 1 4 th = 4 th data value. Hence, the lower quartile of the given data is Q 1 = 91 . Now, upper quartile Q 3 is the median of the upper 50 % of the data. Formula to find upper quartile is Q 3 = 3 N + 1 4 th data value, where N is number of data values. ⇒ Upper quartile Q 3 = 3 15 + 1 4 th = 12 th data value. Thus, the upper quartile of the given data is Q 3 = 109 . Interquartile range is IQR = Q 3 - Q 1 Therefore, IQR = Q 3 - Q 1 = 109 - 91 = 18 kg

EASY

AS and A Level

IMPORTANT

Earn 100

The heights, $x cm,$ of a group of $28$ people were measured. The mean height was found to be $172.6 cm$ and the standard deviation was found to be $4.58 cm .$ A person whose height was $161.8 cm$ left the group.

Find $Σ x^{2}$ for the original group of $28$ people. Hence find the standard deviation of the heights of the remaining group of $27$ people.

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>END-OF-CHAPTER REVIEW EXERCISE 3>Q 9

$120$ people were asked to read an article in a newspaper. The times taken, to the nearest second, by the people to read the article are summarised in the following table.

Time (seconds)	$1 - 25$	$26 - 35$	$36 - 45$	$46 - 55$	$56 - 90$
Number of people	$4$	$24$	$38$	$34$	$20$

Calculate estimates of the mean and standard deviation of the reading times.

EASY

AS and A Level

IMPORTANT

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

The weights, in kilograms, of the $15$ basketball players in each of two squads, $A$ and $B$ , are shown below.

Squad $A$	$97$	$98$	$104$	$84$	$100$	$109$	$115$	$99$	$122$	$82$	$116$	$96$	$84$	$107$	$91$
Squad $B$	$75$	$79$	$94$	$101$	$96$	$77$	$111$	$108$	$83$	$84$	$86$	$115$	$82$	$113$	$95$

Represent the data by drawing a back-to-back stem-and-leaf diagram with squad $A$ on the left-hand side of the diagram and squad $B$ on the right-hand side.

EASY

AS and A Level

IMPORTANT

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

The weights, in kilograms, of the $15$ basketball players in each of two squads, $A$ and $B$ , are shown below.

Squad $A$	$97$	$98$	$104$	$84$	$100$	$109$	$115$	$99$	$122$	$82$	$116$	$96$	$84$	$107$	$91$
Squad $B$	$75$	$79$	$94$	$101$	$96$	$77$	$111$	$108$	$83$	$84$	$86$	$115$	$82$	$113$	$95$

Find the interquartile range of the weights of the players in squad $A .$

EASY

AS and A Level

IMPORTANT

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

The weights, in kilograms, of the $15$ basketball players in each of two squads, $A$ and $B$ , are shown below.

Squad $A$	$97$	$98$	$104$	$84$	$100$	$109$	$115$	$99$	$122$	$82$	$116$	$96$	$84$	$107$	$91$
Squad $B$	$75$	$79$	$94$	$101$	$96$	$77$	$111$	$108$	$83$	$84$	$86$	$115$	$82$	$113$	$95$

A new player joins squad $B .$ The mean weight of the $16$ players in squad $B$ is now $93.9 kg .$ Find the weight of the new player.

EASY

AS and A Level

IMPORTANT

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

The heights, $x cm,$ of a group of $82$ children are summarised as follows.

$Σ (x - 130) = - 287,$ standard deviation of $x = 6.9$

Find the mean height.

EASY

AS and A Level

IMPORTANT

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

The heights, $x cm,$ of a group of $82$ children are summarised as follows.

$Σ (x - 130) = - 287,$ standard deviation of $x = 6.9$

Find $Σ {(x - 130)}^{2} .$ [Answer up to two decimal values]

EASY

AS and A Level

IMPORTANT

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

A sample of $36$ data values, $x,$ gave $Σ (x - 45) = - 148$ $Σ {(x - 45)}^{2} = 3089 .$

i. Find the mean and standard deviation of the $36$ values.

EASY

AS and A Level

IMPORTANT

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

A sample of $36$ data values, $x,$ gave $Σ (x - 45) = - 148$ $Σ {(x - 45)}^{2} = 3089 .$

One extra data value of $29$ was added to the sample. Find the standard deviation of all $37$ values.

Important Questions on Measures of Variation

Learn and Prepare for any exam you want !