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 Is the mode a good value to use as the average in this case? Give a reason for your answer.

Given data is No. items 0 1 2 3 4 5 6 - p No. days 49 16 10 9 7 5 4 We know that, mode is the value with the highest frequency. Here the value with highest frequency that is 49 is 0 . So, the mode is 0 . The mode is not a good average to use, since it is the smallest value and is not at all central.

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.

Given data is No. items 0 1 2 3 4 5 6 - p No. days 49 16 10 9 7 5 4 Data values along with cumulative frequency: No. items 0 1 2 3 4 5 6 - p No. days 49 16 10 9 7 5 4 Cumulative frequency 49 65 75 84 91 96 100 Here, number of data values is n = ∑ f = 100 , then formula to find median = n + 1 2 th data value, where n is the number of data values in the set. ⇒ Median = 100 + 1 2 th = 50 . 5 data value. So, the median is mean of 50 t h and 51 s t value. Here, the 50 t h and 51 s t values are 1 only. ⇒ Median = 1 + 1 2 = 1 Hence, the median of the given data is 1 . We know that, mode is the value with the highest frequency. Here the value with highest frequency that is 49 is 0 . So, the mode is 0 . We know that, mean = Σ x · f Σ f ⇒ Mean = 0 × 49 + 1 × 16 + 2 × 10 + 3 × 9 + 4 × 7 + 5 × 5 + 6 + p 2 × 4 100 = 1 . 5 = 116 + 4 × 6 + p 2 100 = 1 . 5 [since, the frequency of data values from 6 to p is 4 ] ⇒ 116 + 2 p + 12 = 150 ⇒ p = 11 Thus the value of p is 11 . Here, the values of mean, mode and median are 11 , 0 and 1 respectively. The data is positively skewed : m o d e < m e d i a n < m e a n

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

EASY

AS and A Level

IMPORTANT

Earn 100

The following histogram illustrates the masses, $m$ kilograms, of the $216$ sales of hay that a farmer made to customers last year.

Show that a calculated estimate of the mean is equal to 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 7

The following histogram illustrates the masses, $m$ kilograms, of the $216$ sales of hay that a farmer made to customers last year.

Question Image

Estimate the price per kilogram at which the hay was sold, given that these sales generated exactly $$ 1944 .$ Why is it possible that none of the customers actually paid this amount per kilogram for the hay?

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 8

An Internet service provider wants to know how customers rate its services. A questionnaire asks customers to tick one of the following boxes excellent, good, average, poor, very poor

How might the company benefit from knowing each of the available average responses of its customers?

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 8

An internet service provider wants to know how customers rate its services. A questionnaire asks customers to tick one of the following boxes.

excellent, good, average, poor, very poor

What additional benefit could the company obtain by using the following set of tick boxes instead?

excellent $= 5,$ good $= 4,$ average $= 3,$ poor $= 2,$ very poor $= 1$

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 9

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	$49$	$16$	$10$	$9$	$7$	$5$	$4$

Write down the median.

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 9

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	$49$	$16$	$10$	$9$	$7$	$5$	$4$

Is the mode a good value to use as the average in this case? Give a reason for your answer.

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 9

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	$49$	$16$	$10$	$9$	$7$	$5$	$4$

Find the value of $p,$ given that a calculated estimate of the mean is $1.5 .$

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 9

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.

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 $3.1 ⩽ t ⩽ 4.5$ .

The following histogram illustrates the masses, m kilograms, of the 216 sales of hay that a farmer made to customers last year. Show that a calculated estimate of the mean is equal to the median.

Important Questions on Measures of Central Tendency

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The following histogram illustrates the masses, $m$ kilograms, of the $216$ sales of hay that a farmer made to customers last year.

Show that a calculated estimate of the mean is equal to the median.