As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest 0 . 1 hour. The results are displayed in the following table. Time spent ( t hours) 0 . 1 ⩽ t ⩽ 0 . 5 0 . 6 ⩽ t ⩽ 1 . 0 1 . 1 ⩽ t ⩽ 2 . 0 2 . 1 ⩽ t ⩽ 3 . 0 3 . 1 ⩽ t ⩽ 4 . 5 Frequency 11 15 18 30 21 Draw, on graph paper, a histogram to illustrate this information and find the difference between frequencies of the classes 2 . 1 ⩽ t ⩽ 3 . 0 and 3 . 1 ⩽ t ⩽ 4 . 5 .

Given that, the data to the nearest 0 . 1 hr is Time spent ( t hours) 0 . 1 ⩽ t ⩽ 0 . 5 0 . 6 ⩽ t ⩽ 1 . 0 1 . 1 ⩽ t ⩽ 2 . 0 2 . 1 ⩽ t ⩽ 3 . 0 3 . 1 ⩽ t ⩽ 4 . 5 Frequency 11 15 18 30 21 Now plotting the histogram Thus the difference between frequencies of the classes 2 . 1 ⩽ t ⩽ 3 . 0 and 3 . 1 ⩽ t ⩽ 4 . 5 is 9

A delivery of 150 boxes, each containing 20 items, is made to a retailer. The numbers of damaged items in the boxes are shown in the following table. No. damaged items 0 1 2 3 4 5 6 or more No. boxes f 100 10 10 10 10 10 0 Find the mode, the mean and the median number of damaged items.

Given data is No. damaged items 0 1 2 3 4 5 6 or more No. boxes f 100 10 10 10 10 10 0 Formula to calculate mean is x ¯ = ∑ f x N , where N = ∑ f . Mean x ¯ = 0 × 100 + 1 × 10 + 2 × 10 + 3 × 10 + 4 × 10 + 5 × 10 + 6 + 7 + 8 + . . . . × 0 150 = 1 Hence, the mean of the given data is 1 . We know that, mode is the value of the data with the most frequency. ⇒ The mode is 0 . We know that, median is the middle term of the set of values sorted in ascending or descending order. Since, N = ∑ f = 150 ⇒ Median is the mean of the 75 t h and 76 t h value, which is 0 . Thus the value of median is 0 .

A delivery of 150 boxes, each containing 20 items, is made to a retailer. The numbers of damaged items in the boxes are shown in the following table. No. damaged items 0 1 2 3 4 5 6 or more No. boxes f 100 10 10 10 10 10 0 Which of the three measures of central tendency would be the most appropriate to use as the average in this case? Explain why using the other two measures could be misleading.

Given data is No. damaged items 0 1 2 3 4 5 6 or more No. boxes f 100 10 10 10 10 10 0 Formula to calculate mean is x ¯ = ∑ f x N , where N = ∑ f . Mean x ¯ = 0 × 100 + 1 × 10 + 2 × 10 + 3 × 10 + 4 × 10 + 5 × 10 + 6 + 7 + 8 + . . . . × 0 150 = 1 Hence, the mean of the given data is 1 . We know that, mode is the value of the data with the most frequency. ⇒ The mode is 0 . We know that, median is the middle term of the set of values sorted in ascending or descending order. Since, N = ∑ f = 150 ⇒ Median is the mean of the 75 t h and 76 t h value, which is 0 . Thus the value of median is 0 . Mean is the most appropriate; the median and mode are 0 may suggest that none of the items are damaged.

EASY

AS and A Level

IMPORTANT

Earn 100

The numbers of items returned to the electrical department of a store on each of $100$ consecutive days are given in the following table.

No. items $0$ $1$ $2$ $3$ $4$ $5$ $6 - p$

No. days $49$ $16$ $10$ $9$ $7$ $5$ $4$

Sketch a curve that shows the shape of this set of data, and mark onto it the relative positions of the mode, the mean and the median.

Important Questions on Measures of Central Tendency

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>END-OF-CHAPTER REVIEW EXERCISE 2>Q 10

As part of a data collection exercise, members of a certain school year group were asked how long they spent on their Mathematics homework during one particular week. The times are given to the nearest $0.1$ hour. The results are displayed in the following table.

Time spent ( $t$ hours)	$0.1 ⩽ t ⩽ 0.5$	$0.6 ⩽ t ⩽ 1.0$	$1.1 ⩽ t ⩽ 2.0$	$2.1 ⩽ t ⩽ 3.0$	$3.1 ⩽ t ⩽ 4.5$
Frequency	$11$	$15$	$18$	$30$	$21$

Draw, on graph paper, a histogram to illustrate this information and find the difference between frequencies of the classes $2.1 ⩽ t ⩽ 3.0$ and $3.1 ⩽ t ⩽ 4.5$ .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>END-OF-CHAPTER REVIEW EXERCISE 2>Q 10

Time spent ( $t$ hours)	$0.1 ⩽ t ⩽ 0.5$	$0.6 ⩽ t ⩽ 1.0$	$1.1 ⩽ t ⩽ 2.0$	$2.1 ⩽ t ⩽ 3.0$	$3.1 ⩽ t ⩽ 4.5$
Frequency	$11$	$15$	$18$	$30$	$21$

Calculate an estimate of the mean time spent on their Mathematics homework by members of this year group.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>END-OF-CHAPTER REVIEW EXERCISE 2>Q 11

For

150

values of

x,

it is given that

Σ (x - 1) + Σ (x - 4) = 4170 .

Find

\bar{x} .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>END-OF-CHAPTER REVIEW EXERCISE 2>Q 12

On Monday, a teacher asked eight students to write down a number, which is denoted by

x .

On Tuesday, when one of these students was absent, they asked them to add

1

to yesterday's number and write it down. Find the number written down on Monday by the student who was absent on Tuesday, given that

\bar{x} = 30 \frac{1}{4},

and that the mean of Monday's and Tuesday's numbers combined was

27 \frac{1}{3} .

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>END-OF-CHAPTER REVIEW EXERCISE 2>Q 13

A delivery of $150$ boxes, each containing $20$ items, is made to a retailer. The numbers of damaged items in the boxes are shown in the following table.

No. damaged items	$0$	$1$	$2$	$3$	$4$	$5$	$6$ or more
No. boxes $(f)$	$100$	$10$	$10$	$10$	$10$	$10$	$0$

Find the mode, the mean and the median number of damaged items.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>END-OF-CHAPTER REVIEW EXERCISE 2>Q 13

A delivery of $150$ boxes, each containing $20$ items, is made to a retailer. The numbers of damaged items in the boxes are shown in the following table.

No. damaged items	$0$	$1$	$2$	$3$	$4$	$5$	$6$ or more
No. boxes $(f)$	$100$	$10$	$10$	$10$	$10$	$10$	$0$

Which of the three measures of central tendency would be the most appropriate to use as the average in this case? Explain why using the other two measures could be misleading.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>END-OF-CHAPTER REVIEW EXERCISE 2>Q 14

The monthly salaries, $w$ dollars, of $10$ women are such that $Σ (w - 3000) = - 200 .$

The monthly salaries, $m$ dollars, of $20$ men are such that $Σ (m - 4000) = 120 .$

Find the difference between the mean monthly salary of the women and the mean monthly salary of the men.

EASY

AS and A Level

IMPORTANT

Cambridge International AS & A Level Mathematics : Probability & Statistics 1 Course Book>Chapter 2 - Measures of Central Tendency>END-OF-CHAPTER REVIEW EXERCISE 2>Q 14

The monthly salaries, $w$ dollars, of $10$ women are such that $Σ (w - 3000) = - 200 .$

The monthly salaries, $m$ dollars, of $20$ men are such that $Σ (m - 4000) = 120 .$

Find the mean monthly salary of all the women and men together.

No. items	$0$	$1$	$2$	$3$	$4$	$5$	$6 - p$
No. days	$49$	$16$	$10$	$9$	$7$	$5$	$4$

Important Questions on Measures of Central Tendency

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