Chapter 2 – Part 1 – Measures of central tendency – practice problems

1. Find the mean, median and mode for the following data: 25, 37, 31, 32, 32, 29, 30, 32, 29, 34(Ans: Mean: 31.1 units, Median: 31.5 units, Mode: 32 units)

2. Find the mean, median and mode for the following data:

No. of vehicles 0 1 2 3 4 5No. of families 3 13 18 10 5 1

(Ans: Mean: 2.04 vehicles per family, Median: 2 vehicles per family, Mode: 2 vehicles per family)

3. Given below is the data about the fruit boxes received by a fruit vendor. Find arithmetic mean.

Weight in Kg. 10 11 12 14 15No. of boxes 4 6 6 5 2

(Ans: 12.08 Kg.)

4. The average monthly wage of all the workers in a factory is Rs. 444/-. If the average wages paid to male and female workers are Rs. 480/- and Rs. 360/- respectively, find the percentage of male and female workers employed in the factory.

(Ans: Male: 70%, Female: 30%)

5. Income of the employees in an industrial firm is given below. Total income of 10 employees in the class over Rs. 250 is Rs. 3000. Compute the mean income. Each employee belonging to top 25% of the earners is required to pay 5% of his income to workers’ relief fund. Estimate the contribution to this fund.

Income (Rs.) Below 50 50-100 100-150 150-200 200-250 250 and overFrequency 90 150 100 80 70 10

(Ans: Mean: Rs. 117.5, Contribution: Rs. 1331.25)

6. From the result of two colleges, A and B below, state which of them is better and why?

Name of exam

College A College BAppeared Passed Appeared Passed

M.A. 300 250 1000 800M.Com. 500 450 1200 950

B.A. 2000 1500 1000 700B.Com. 1200 750 800 500Total 4000 2950 4000 2950

(Ans: By simple arithmetic mean, college A is better. By weighted arithmetic mean, both the colleges are equally good)

7. In a batch of 15 students, 5 failed in a test. The marks of the remaining 10 students who have passed are 9, 7, 5, 6, 7, 8, 8, 9, 7, 6. What is the median of all the 15 students?

(Ans: 6 marks)

8. Shriram Autocomponents Ltd. has collected the data about the output produced on a machine per hour as follows: 12, 19, 21, 30, 13, 19, 22, 31, 17, 20, 24, 31, 18, 21, 27, 31. Compute the arithmetic mean, median and mode for the above data.

(Ans: Mean: 22.25 units, Median: 21 units, Mode: 31 units)

9. Calculate the arithmetic mean and median for the following data:

Marks 0-10 10-30 30-60 60-100No. of students 5 12 25 8

(Ans: Mean: 40.6 marks, Median: 39.6 marks)

10. A factory pays workers on piece rate basis and also a bonus to each worker on the basis of individual output in each quarter. The rate of bonus payable is as follows:

Output in units 70-74 75-79 80-84 85-89 90-94 95-99 100-104Bonus in Rs. 40 45 50 60 70 80 100

The individual output of a batch of 50 workers is given below: 94, 83, 78, 76, 88, 86, 93, 80, 91, 82, 89, 97, 92, 84, 92, 80, 85, 83, 98, 103, 87, 88, 88, 81, 95, 86, 99, 81, 87, 90, 84, 97, 80, 75, 93, 101, 82, 82, 89, 72, 85, 83, 75, 72, 83, 98, 77, 87, 71, 80.By suitable classification, find:a. Average bonus per worker for the quarterb. Total quarterly bonus paidc. Average output per workerd. Median value of the bonus

(Ans: a: Rs. 59.7, b: Rs. 2985, c: 86.1 units, d: Rs. 60)

11. Following table shows the age distribution of people in a particular region. Find the median.

Age (Years)Below

10Below

20Below

30Below

40Below

50Below

60Below

7070 and

overNo. of people

(in thousands)2 5 9 12 14 15 15.5 15.6

(Ans: 27 years)

12. Find the missing frequency if the arithmetic mean is 28 for the data given below. Also find its median.

Profits per shop 0-10 10-20 20-30 30-40 40-50 50-60No. of shops 12 18 27 ? 17 6

(Ans: Missing frequency: 20, Median: 27.41 profits)

13. The arithmetic mean, the mode and the median of a group of 75 observations were calculated to be 27, 34 and 29 respectively. It was later discovered that one observation was wrongly read as 43 instead of correct value 53. Examine the correct value of three averages.

(Ans: Mean: 27.13, Median: 34, Mode: 29)

14. In the frequency distribution of 200 students given below:

c 10-20 20-30 30-40 40-50 50-60 60-70 70-80Frequency 10 28 ? 60 ? 20 16

Find the missing frequencies, if the median value of the data is 45. Also, find the arithmetic mean of the entire data. If 10% of the students below the A.M. are considered above the pass marks, find the pass marks. If top 10% of the students are awarded the ‘distinction grade’, find the minimum marks required to score a distinction.

(Ans: Missing frequencies: 32 and 34, Mean: 45.2 marks, Pass marks: 42, Distinction marks: 68)

15. For the following data about the marks obtained in an entrance exam by all the students admitted in 50 colleges in a state, find the mean, median and mode.

Marks obtainedMore

than 3530-35 25-30 20-25 15-20

Less than 15

No. of colleges 7 10 15 9 5 4Average no. of students in a college 200 250 300 200 150 100

(Ans: Mean: 27.85 marks, Median: 28.03 marks, Mode: 27.87 marks)

16. Find the median and mode for the following data:

Weight in Kg. 11-20 21-30 31-40 41-50 51-60 61-70 71-80No. of persons 42 38 120 84 48 36 31

Also find these values graphically. (Ans: Median: 40.46 Kg, Mode: 37.45 Kg)

17. Calculate the mean, median and mode for the following data:

Class Frequency Class Frequency0-2 8 25-30 45

2-4 12 30-40 604-10 20 40-50 20

10-15 10 50-60 1315-20 16 60-80 1520-25 25 80-100 4

(Ans: Mean: 30.1 units, Median: 28.29 units, Mode: 28.15 units)

18. The median and mode for the following data are 33.5 and 34 respectively. Find the missing frequencies.

Value 0-10 10-20 20-30 30-40 40-50 50-60 60-70 TotalFrequency 4 16 ? ? ? 6 4 230

(Ans: Missing frequencies: 60, 100 and 40)

19. Given below is the distribution of 140 candidates obtaining marks ‘x’ or higher in a certain examination. Find the mean, median and mode of the distribution.

Marks 10 20 30 40 50 60 70 80 90 100No. of students 140 133 118 100 75 45 25 9 2 0

(Ans: Mean: 51.21 marks, Median: 51.66 marks, Mode: 53.33 marks)

20. Calculate mean, median and mode for the following distribution:

Class % Relative frequency Class % Relative frequency0-8 0.60 32-40 23.80

8-16 5.00 40-48 21.4016-24 10.80 48-56 17.2024-32 15.80 56-64 5.40

(Ans: Mean: 37.37 units, Median: 37.98 units, Mode: 38.15 units)

21. Calculate the arithmetic mean and median of the distribution given below:

Class Limits 130-134 135-139 140-144 145-149 150-154 155-159 160-164Frequency 5 15 28 24 17 10 1

(Ans: Mean: 145.45 units, Median: 144.91 units)

22. The regional transport authority is concerned about the speed of motorbikes, college students are driving on a section of the road. Following are the speeds of 45 drivers in km/hr: 15, 31, 44,

56, 38, 32, 48, 42, 58, 29, 45, 49, 38, 48, 62, 46, 68, 56, 48, 52, 62, 47, 64, 49, 61, 42, 47, 52, 69, 55, 58, 52, 29, 69, 55, 39, 18, 39, 61, 58, 48, 37, 55, 18, 49.a. Classify the data considering classes 15-24, 25-34, 35-44 etc.b. Draw histogram and Ogive curves for the data. Read the modal value from the graph. c. State report says that not more than 10% of the students exceed 55 km/hr. State whether

collected data support the given report or not. (Ans: Mode: 52, collected data does not support the findings as 33.33% exceed the speed limit)

23. The management of a hotel has employed 2 managers, 5 cooks and 8 waiters. The monthly salaries of the manager, the cook and the waiter are Rs. 3000, Rs. 1200 and Rs. 1000 respectively. Find the mean salary of the employees.

(Ans: Rs. 1333.33)24. Prepare frequency distribution table. Present the information graphically to determine median

and mode of the distribution. 26, 35, 61, 29, 36, 48, 57, 67, 69, 50, 48, 40, 42, 47, 41, 37, 62, 51, 47, 64, 63, 33, 31, 32, 35, 40, 38, 37, 60, 51, 54, 56, 37, 46, 42, 38, 61, 59, 58, 44, 39, 57, 18, 44, 45, 45, 47, 38, 44, 47.

(Ans: Median: 45 units, Mode: 43 units)

25. From the following data of income distribution, calculate the arithmetic mean. It is given that: a. The total income of persons in the highest group is Rs. 435. b. None is earning less than Rs. 20 (Ans: Mean: Rs. 48)

Income (Rs.)Below

30Below

40Below

50Below

60Below

70Below

8080 and

overNo. of persons 16 36 61 76 87 95 5

26. A student was examined in four subjects. Examination was conducted as an oral and written test. Oral test was of 25 marks and written test was of 75 marks. Given the following information, calculate weighted average of marks obtained in the written test considering percentage of marks in oral test as weights.

Subject W X Y ZMarks in written test 40 50 35 60

Marks in oral test 10 15 18 20

(Ans: Weighted Mean: 47.3 Marks)

27. A student appeared in three tests of value 20, 50 and 30 marks. He obtained 75% in the first and 60% in the second test. What should be his percentage in the third test if his aggregate is 60%? (Ans: 50%)

28. Age distribution of 180 persons is as below:

Age (in years) Below 40 40-50 50-60 60-70 70 and aboveNo. of persons 25 35 52 40 28

Find an appropriate measure of central tendency. (Ans: Median: 55.77 years)

29. Calculate median for the following data:

Sales (Rs. ‘000) 0-10 10-30 30-60 60-80 80-90No. of salesmen 5 15 30 8 2

(Ans: Median: Rs. 40,000)

30. Given below is the frequency distribution of the weights of 60 students in a class in school.

Weight in Kg 30-34 35-39 40-44 45-49 50-54 55-59 60-64No. of students 3 5 12 18 14 6 2

a. Draw histogram and find the modal value. b. Draw the ‘less than’ and ‘more than’ Ogive curve and find the median value. c. Use these values to find the arithmetic mean by using the empirical formula.

(Ans: Mode: 48 Kg, Median: 47.8 Kg, Arithmetic Mean: 47.7 Kg)

31. Find the mean, median and mode in the following series:

Size (above) 0 5 10 15 20 25 30Frequency 38 37 35 25 21 11 2

(Ans: Mean: 19.14 units, Median: 21 units, Mode: 24.3 units)

32. A.M. and Median of 50 items are 100 and 95 respectively. At the time of calculation, 2 items 180 and 90 were wrongly taken as 100 and 10. What are the correct values of mean and median? (Ans: Mean: 103.2 units, Median: 95 units)

33. A cyclist paddles from his house to his college at a speed of 10 km/hr and back from the college to his house at 15 km/hr. Find the average speed. (Ans: 12.5 km/hr)

34. For the distribution of daily wages of workers in a certain factory, the median and the mode are Rs. 12 and Rs. 13 respectively. Find the missing frequencies.

Daily wages 0-5 5-10 10-15 15-20 20-25No. of workers 7 ? 15 ? 3

(Ans: Missing frequencies are 12 and 13)

35. A machine depreciates by 40% in the first year, 25% in the second year and by 10% per annum for the next three years, each percentage being calculated on the diminishing value. What is the average percentage of depreciation for the entire period? (Ans: 13.44%)

36. Consider the following frequency distribution for late arrival of the Pragati Express during the months of July and August.

Late (in minutes) 0-9 10-19 20-29 30-39 40-49 50-59No. of days 25 15 8 4 3 7

a. Find the proportion of times the train is late by more than 20 minutes. b. Find the proportion of times the train is late by at most 30 minutes. c. Average late arrival time of the train.

(Ans: a: 11/31, b: 24/31, c: 19.02 min)

37. A contractor employs three categories of workers – men, women and children and pays daily wages of Rs. 10, Rs. 8 and Rs. 6 respectively. He employs 30 men, 15 women and 5 children. Since the average wage of the three categories is Rs. 8, the total wages will be Rs. 400 daily. Is this value correct? Comment.

(Ans: We must consider weighted average here. Hence total wages paid should be Rs. 450)38. The distribution of sizes of an article is given below:

Length (cm) 0-20 20-40 40-60 60-80 80-100No. of articles 12 28 35 30 15

Find the percentage of articles with size between 45 and 65 cm. (Ans: 28.33%)

39. The quantity of a shampoo in the bottles should be 500 ml. However, following variations were observed.

Quantity (ml) 485-489 490-494 495-499 500-504 505-509 510-514 515-519No. of bottles 2 7 16 23 20 9 3

a. What is the modal value? b. Find the percentage of bottles having the quantity less than the modal value.

(Ans: a: 503 ml, b: 51.38%)

40. The following numbers give the weights (in Kg.) of 55 students in the class. Prepare a suitable frequency table. 63, 42, 74, 40, 60, 82, 56, 77, 78, 65, 67, 50, 80, 81, 66, 49, 77, 90, 78, 86, 51, 96, 93, 52, 95, 98, 84, 94, 41, 68, 76, 61, 69, 42, 75, 94, 64, 83, 80, 69, 95, 79, 70, 53, 79, 71, 80, 54, 72, 76, 73, 50, 84, 59, 79.a. Draw a histogram and find modal value from it. b. Draw a less than ogive curve and find the median value.

(Ans: a: 74 Kg, b: 72 Kg)

41. Draw a histogram, frequency polygon, frequency curve and ogive curves for the following distribution. Also read median and mode from these graphs.

Class limits 0-10 10-20 20-30 30-40 40-50 50-60Frequency 4 16 15 20 7 5

(Ans: Median: 29 units, Mode: 33 units)

42. A contractor employs three categories of workers – skilled, semi skilled and unskilled. He pays daily wages of Rs. 100, Rs. 60 and Rs. 40 to them respectively. He employs 30 skilled, 15 semi skilled and 5 unskilled workers. Calculate the average daily wage per worker.

(Ans: Rs. 82)

43. Find the arithmetic mean for the following data:

Classes 1-10 11-20 21-30 31-40 41-50Frequency 8 15 25 10 7

(Ans: 24.42 units)

44. Before constructing a dam on a river, the central water research institute performed a series of tests to measure the water flow past the proposed location of the dam during the period of 246 days when there was sufficient flow of water. The results of testing were used to construct the following frequency distribution.

River flow (Thousand cubic meters per min)

1001-1050

1051-1100

1101-1150

1151-1200

1201-1250

1251-1300

1301-1350

1351-1400

No. of days 7 21 32 49 58 41 27 11

a. Draw ogive curves for the above data. b. Calculate mode for above data.c. From the ogive curves, estimate the proportion of days for which flow occurs at less than

1300 thousand cubic meters per minute. (Ans: b: 1217.8 cmpm, c: 0.8415)

45. The time required in minutes for each of the 50 students to read 20 pages of a book is recorded below: 43, 52, 47, 49, 36, 48, 41, 47, 50, 32, 45, 48, 40, 43, 48, 36, 51, 44, 49, 53, 37, 34, 42, 45, 47, 47, 44, 50, 31, 48, 43, 45, 44, 36, 49, 51, 43, 53, 46, 39, 50, 42, 42, 47, 38, 51, 46, 40, 38, 45. a. Classify the data by considering classes 30-34, 35-39 etc. b. Draw histogram and ogive curves for the data. Also read the value of median and mode

from the graph. (Ans: Median: 45 min, Mode: 47 min)

46. Following table gives frequency distribution of weekly wages of 100 workers in a factory. Find the median by calculation and also graphically. (Ans: Rs. 354.677)

Weekly wages (Rs.)

200-240 250-290 300-340 350-390 400-440 450-490 500-540

No. of workers

9 12 23 31 10 8 7

47. Find the modal age of workers using graphical method and by calculation. (Ans: 48.33 years)

Age (more than)(in years) 15 25 35 45 55 65 75No. of workers 115 111 96 70 40 18 3

48. 30 households in downtown New York were surveyed regarding the monthly mortgage payments made them. Figures are recorded in ‘00 as follows: 750, 529, 760, 640, 685, 775, 560, 678, 1130, 650, 1120, 940, 1170, 620, 989, 880, 765, 930, 720, 900, 865, 850, 900, 980, 832, 850, 1056, 920, 1050 and 1075. Prepare frequency distribution table and find mean, median and mode for this distribution.

(Ans: Mean: Rs. 85333.3, Median: Rs. 86000, Mode: Rs. 93333.3)