ARITHMETIC MEAN

MEASURES OF CENTRAL TENDENCY-ARITHMETIC MEAN

Q.1-WHAT DO YOU MEAN BY MEASURE OF CENTRAL TENDENCY? DISCUSS THE OBJECTIVES FOR MEASURING CENTRAL TENDENCY.

ANSWER:

(A) MEANING OF MEASURE OF CENTRAL TENDENCY

  • It is a single value or figure that represents the entire set of data.
  • It is a value to which most of the observations are closer.

(B) FOLLOWING ARE THE OBJECTIVES OF AVERAGES:

(1) TO PRESENT A BRIEF PICTURE OF DATA

  • Averages summarises data into a single figure, which makes it easier to understand and remember.

(2) TO MAKE COMPARISON EASIER

  • Averages are very helpful for making comparative studies as they reduce the bulky statistical data to a single figure.

(3) TO HELP IN DECISION-MAKING

  • Most of the decisions in research or planning are based on the average value of certain variables.

(4) TO HELP IN FORMULATION OF POLICY

  • It is very useful in policy formulation.
  • For example: For the removal of poverty from India, government takes into consideration per capita income.

Q.2-WHAT IS MEANT BY ARITHMETIC MEAN? WHAT ARE ITS TYPES?

ANSWER:

(A) ARITHMETIC MEAN

  • Arithmetic Mean is defined as “the sum of the values of all observations divided by the number of observations”.
  • It is also known as ‘Mean’ or ‘Average’ by the common man.
  • It is generally denoted by .

(B) TYPES

  1. Simple arithmetic mean
  2. Weighted arithmetic mean

Q.3-BRIEFLY EXPLAIN THE MERITS AND DEMERITS OF ARITHMETIC MEAN.

ANSWER:

(A) FOLLOWING ARE SOME OF THE MERITS OF ARITHMETIC MEAN:

(1) EASY TO COMPUTE

  • Its calculation is very easy because it requires knowledge of only simple mathematics i.e. addition, multiplication and division of numbers.

(2) SIMPLE TO UNDERSTAND

  • It is also simple to understand the meaning of arithmetic mean i.e., the value per unit or cost per unit, etc.

(3) BASED ON ALL ITEMS

  • It takes into consideration all the values of data.
  • It is considered to be more representative of the distribution.

(4) RIGIDLY DEFINED

  • Its value is always definite because it is rigidly defined.

(5) GOOD BASIS OF COMPARISON

  • It provides a sound basis of comparison of two or more group of data.

(6) ALGEBRAIC TREATMENT

  • It is capable of further algebraic treatment. So, it is widely used in advance statistical analysis.

(B) FOLLOWING ARE SOME OF THE DEMERITS OF ARITHMETIC MEAN:

(1) COMPLETE DATA IS REQUIRED

  • It cannot be computed unless all the items of a series are available.

(2) AFFECTED BY EXTREME VALUES

  • Since arithmetic average is calculated from all the items of a series, it can be unduly affected by extreme values i.e. very small or very large items.

(3) ABSURD RESULT

  • Sometimes arithmetic mean gives absurd results. For example, if a teacher says that average number of students in a class is 28.75, it sounds illogical.

(4) CALCULATION OF MEAN BY OBSERVATION NOT POSSIBLE

  • Arithmetic mean cannot be computed by simply observing the series like median or mode.

(5) NO GRAPHIC REPRESENTATION

  • Arithmetic Mean cannot be represented or depicted on graph paper.

(6) NOT POSSIBLE IN CASE OF OPEN ENDED FREQUENCY DISTRIBUTION

  • In case of open ended class frequency distribution, it is not possible to compute arithmetic mean without making assumption about the class size.

(7) NOT POSSIBLE IN CASE OF QUALITATIVE CHARACTERISTICS

  • It cannot be computed for a qualitative data; like data on intelligence, honesty, smoking habit, etc.

Q.4-WHAT ARE THE ESSENTIALS OF A GOOD AVERAGE?

OR

DISCUSS THE REQUIREMENTS OF A GOOD MEASURE OF CENTRAL TENDENCY.

ANSWER:

(1) EASY TO UNDERSTAND

  • It should be easy to understand so that a layman can use it.

(2) EASY TO COMPUTE

  • It should be easy to compute.
  • Its calculation should not involve mathematical complexities.

(3) BASED ON ALL OBSERVATIONS

  • Average should be calculated by taking into consideration each and every item of the series.

(4) RIGIDLY DEFINED

  • It should have a definite and fixed value irrespective of method of calculations.

(5) CAPABLE OF FURTHER ALGEBRAIC TREATMENT

  • It should be capable of further algebraic treatment so that it can be used advance analysis.

(6) NOT AFFECTED MUCH BY EXTREME VALUES

  • The value of an average should not be affected much by extreme values.
  • One or two very small or very large values, should not affect the value of the average significantly.

Practice Questions:

INDIVIDUAL SERIES

Q.1 FOLLOWING ARE THE MARKS OF A GROUP OF FIVE STUDENTS. CALCULATE THE AVERAGE MARKS OF THE GROUP USING ARITHMETIC MEAN.

S. No.

1

2

3

4

5

Marks (OUT OF 80)

80

72

58

40

50

Q.2 FOLLOWING WERE THE PRICES OF ONION (PER KG), IN THE LAST 10 DAYS, IN A RETAIL MARKET OF MUMBAI. CALCULATE THE AVERAGE PRICE USING ARITHMATIC MEAN. (USE DIRECT METHOD).

Day

1

2

3

4

5

6

7

8

9

10

Price (`)

22

25

28

29

30

30

31

32

33

38

Q.3 FOLLOWING WERE THE PRICES OF ONION (PER KG), IN THE LAST 10 DAYS, IN A RETAIL MARKET OF MUMBAI. CALCULATE THE AVERAGE PRICE USING ARITHMATIC MEAN. (USE SHORT CUT METHOD).

Day

1

2

3

4

5

6

7

8

9

10

Price (`)

22

25

28

29

30

30

31

32

33

38

Q.4 FOLLOWING ARE THE DAILY WAGES OF 10 WORKERS IN A FACTORY. CALCULATE THE AVERAGE WAGES USING ARITHMATIC MEAN. USE STEP DEVIATION METHOD.

S. No.

1

2

3

4

5

6

7

8

9

10

Wages (`)

200

250

300

350

400

450

500

550

600

650

DISCRETE SERIES

Q.1 FROM THE FOLLOWING CALCULATE THE AVERAGE TIME SPENT BY STUDENTS ON PHYSICAL EXERCISE. (USING ARITHMETIC MEAN).

Time Spent on Physical

Exercise (In Minutes)

10

20

30

40

50

60

No. of Student

2

5

13

15

3

2

Q.2 FIND MEAN WAGES FROM THE FOLLOWING. USE DIRECT METHOD.

Wages (`)

200

220

240

260

280

300

No. of Workers

1

6

10

15

5

3

Q.3 FIND MEAN WAGES FROM THE FOLLOWING. USE SHORT CUT METHOD.

Wages (`)

200

220

240

260

280

300

No. of Workers

1

6

10

15

5

3

Q.4 FIND MEAN WAGES FROM THE FOLLOWING. USE SHORT STEP DEVIATION METHOD.

Wages (`)

200

220

240

260

280

300

No. of Workers

1

6

10

15

5

3

CONTINUOUS SERIES

Q.1 FIND ARITHMETIC MEAN FROM THE FOLLOWING. USE DIRECT METHOD.

CLASS

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

FREQUENCY

3

5

12

10

8

2

Q.2 FIND ARITHMETIC MEAN FROM THE FOLLOWING. USE SHORT CUT METHOD.

CLASS

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

FREQUENCY

3

5

12

10

8

2

Q.3 FIND ARITHMETIC MEAN FROM THE FOLLOWING. USE STEP DEVIATION METHOD.

CLASS

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

FREQUENCY

3

5

12

10

8

2

INCLUSIVE Series

Age No. of Students

0 – 9 3

10-19 5

20-29 12

30-39 6

40-49 4

MID VALUE Series

Mid Values Frequency

5 3

15 5

25 12

35 6

45 4

INVERTED Series

Age No. of Students

50-50 3

40-40 5

30-30 12

20-20 6

10- 0 4

UNEQUAL GAP Series

Age No. of Students

0- 5 3

5-15 5

15-35 12

35-40 6

40-50 4

EXCLUSIVE CONTINUOUS SERIES

Age No. of Students

0-10 3

10-20 5

20-30 12

30-40 6

40-50 4

CUMULATIVE FREQ Series

Age No. of Students

Less than 10 3

Less than 20 8

Less than 30 20

Less than 40 26

Less than 50 30

CUMULATIVE FREQ Series

Age No. of Students

More than 0 30

More than 10 27

More than 20 22

More than 30 10

More than 40 4

OPEN ENDED Series

Age No. of Students

Below 10 3

10-20 5

20-30 12

30-40 6

50 and Above 4

a. Arithmetic Mean from Exclusive Continuous Series

Q.1 FIND ARITHMETIC MEAN FROM THE FOLLOWING. USE DIRECT METHOD.

CLASS

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

FREQUENCY

3

5

12

10

8

2

b. Arithmetic Mean from Inclusive Continuous Series

Q.1 FROM THE FOLLOWING, FIND AVERAGE ELECTRICITY CONSUMPTION USING ARITHMETIC MEAN.

Power (Units)

10 – 19

20 – 29

30 – 39

40 – 49

50 – 59

60 – 69

No. of Families

5

10

11

12

8

4

c. Arithmetic Mean from Mid Value Series

Q.2 FIND ARITHMETIC MEAN FROM THE FOLLOWING.

Mid Values

8

16

24

32

40

48

Frequency

3

8

16

15

10

8

d. Arithmetic Mean from Inverted/Descending Ex Continuous Series

Q.3 COMPUTE AVERAGE AGE USING ARITHMETIC MEAN FROM THE FOLLOWING.

Age (in Years)

50 – 60

40 – 50

30 – 40

20 – 30

10 – 20

0 – 10

Number of Persons

9

12

15

7

5

2

e. Arithmetic Mean from Continuous Series with Unequal Intervals

Q.4 FIND ARITHMETIC MEAN FROM THE FOLLOWING.

Class

0 – 2

2 – 10

10 – 30

30 – 50

50 – 80

80 – 100

Frequency

20

5

2

3

12

8

f. Arithmetic Mean from Less Than Cumulative Frequency Distribution

Q.5 FIND AVERAGE AGE USING ARITHMETIC MEAN FROM THE FOLLOWING DATA.

Age (in Years)

Less than 10

Less than 20

Less than 30

Less than 40

Less than 50

Less than 60

No. of Persons

5

25

50

80

95

100

g. Arithmetic Mean from More than Series/Frequency Distribution

Q.6 FIND AVERAGE MARKS USING ARITHMETIC MEAN FROM THE FOLLOWING.

Marks

More than 10

More than 20

More than 30

More than 40

More than 50

More than 60

No. of Students

100

75

45

25

10

0

h. Arithmetic Mean from Open end Series

Q.7 CALCULATE AVERAGE WEIGT OF CLASS XII STUDENTS USING ARITHMETIC MEAN FROM THE FOLLOWING DATA.

WEIGHT (In Kg)

Below 40

40 – 50

50 – 60

60 – 70

70 – 80

80 and Above

Number of Students

3

5

12

10

6

4

MISSING FIGURE

INDIVIDUAL SERIES

Q.1 RAHUL REMEMBERS HIS MARKS IN FIVE SUBJECTS OUT OF SIX. HE ALSO KNOWS HIS AVERAGE MARKS WHICH IS 90 BUT HE IS UNABLE TO RECALL HIS MARKS IN ONE SUBJECT. CAN YOU HELP HIM TO FIND MARKS IN ECONOMICS?

Subjects

English

Mathematics

Economics

Accountancy

B. St

P. Ed

Marks (Out of 100)

95

58

95

94

100

DISCRETE SERIES

Q.2 FIND THE MISSING FREQUENCY IF MEAN OF THE GIVEN SERIES IS 17.

X

4

10

15

20

30

50

f

5

8

20

5

2

CONTINUOUS SERIES

Q.3 FIND THE MISSING VALUE IF MEAN IS 28.

CLASS

0 – 10

10 – 20

20 – 30

30 – 40

40 – 50

50 – 60

FREQUENCY

12

18

27

17

6

CORRECTED/AMENDED MEAN

Q.1 ‘XYZ COMPANY’ HAS 100 WORKERS AND THEIR AVERAGE SALARY IS `35,000. AFTER COMPUTING THE AVERAGE IT WAS DISCOVERED THAT SALARY OF A WORKER, ‘HONEY SINGH’ WAS TAKEN AS `2,50,000 INSTEAD OF 25,000. CALCULATE THE CORRECT AVERAGE SALARY.

Q.2 THE AVERAGE MARKS OF 40 STUDENTS IN A CLASS WERE 75. LATER ON IT WAS DISCOVERED THAT MARKS OF TWO STUDENTS WERE MISREAD AS 15 AND 19 INSTEAD OF 50 AND 90. CALCULATE THE CORRECT AVERAGE MARKS.

Q.3 ‘RAHUL’ APPEARED IN SENIOR SECONDARY EXAMS IN THE YEAR 2016-17. HE HAD SIX SUBJECTS INCLUDING ONE OPTIONAL (PHYSICAL EDUCATION). TODAY HE HAS ARRIVED TO DELHI UNIVERSITY TO FILL THE ADMISSION FORM.

IN THAT FORM HE IS REQUIRED TO MENTION HIS “AVERAGE MARKS IN ANY FOUR MAIN SUBJECTS INCLUDING ONE LANGUAGE”.

HE IS NOT HAVING HIS DOCUMENTS WITH HIM AND IT IS THE LAST DAY TO FILL THE FORM. BUT HE COULD RECALL THE FOLLOWING:

  • HIS “AVERAGE OF SIX SUBJECTS”, WHICH IS 90 MARKS;
  • MARKS IN “PHYSICAL EDUCATION” 100, BEING THE HIGHEST; &
  • MARKS IN ‘MATHEMATICS’ 58 BEING THE LOWEST.

CAN YOU HELP HIM TO FIND THE REQUIRED CORRECT AVERAGE MARKS TO BE FILLED IN ADMISSION FORM?

COMBINED MEAN

Q.1 IN XYZ SCHOOL, THERE ARE TWO SECTIONS OF ECONOMICS IN CLASS XI. SECTION A HAS 50 STUDENTS AND ITS AVERAGE MARKS IN ECONOMICS ARE 48, SECTION B HAS 30 STUDENTS AND ITS AVERAGE MARKS IN ECONOMICS ARE 80. YOU ARE REQUIRED TO COMPUTE THE AVERAGE MARKS OF CLASS XI IN ECONOMICS:

Q.2 IN A FACTORY THERE ARE 100 WORKERS AND THEIR AVERAGE (MEAN) WAGES PER DAY ARE `350. OUT OF 100 WORKERS, 60 ARE MALE AND THEIR AVERAGE WAGES ARE `500. FIND THE AVERAGE WAGES OF FEMALE WORKERS.

Q.3 IN A SURVEY PEOPLE OF INDIA WERE DIVIDED IN THREE CATEGORIES VIZ. HIGH INCOME, MIDDLE INCOME & LOW INCOME. FROM THE FOLLOWING DATA REGARDING THEIR POPULATION AND ANNUAL INCOME, COMPUTE PER CAPITA INCOME OF THE COUNTRY (i.e. THEIR COMBINED AVERAGE):

High Income

Middle Income

Low Income

Population (In Crores)

10

60

30

Annual PCI (in `)

15,00,000

1,00,000

20,000

Q.4 IN A SURVEY IT IS FOUND THAT PER CAPITA INCOME OF THE COUNTRY IS `2,16,000 AND ITS POPULATION IS 100 CRORE. OUT OF THESE 70 CRORE ARE ABOVE POVERTY LINE AND THEIR PCI IS `3,00,000 PER ANNUM. SPOKESPERSON OF THE GOVERNMENT SAYS THAT IT IS A GOOD SIGN BECAUSE THERE IS A DIFFERENC OF MERE `84,000. DO YOU AGREE WITH THE STATEMENT? JUSTIFY YOUR ANSWER WITH FACTS.

WEIGHTED MEAN

Q.1 “A DRUGGIST SAYS THAT EVEN RETAIL BUSINESS OF MEDICINES REQUIRES HEAVY INVESTMENT, BECAUSE SOME MEDICINES ARE VERY EXPENSIVE”. ANALYSE THE FOLLOWING INFORMATION PROVIDED BY HIM AND USING SUITABLE TOOL TO COMPUTE AVERAGE, COMMENT OVER HIS STATEMENT.

Medicine

A

B

C

D

E

F

Price (in `)

5

10

100

400

1,000

20,000

Q.1 “A DRUGGIST SAYS THAT EVEN RETAIL BUSINESS OF MEDICINES REQUIRES HEAVY INVESTMENT, BECAUSE SOME MEDICINES ARE VERY EXPENSIVE”. ANALYSE THE FOLLOWING INFORMATION PROVIDED BY HIM AND USING SUITABLE TOOL TO COMPUTE AVERAGE, COMMENT OVER HIS STATEMENT.

Medicine

A

B

C

D

E

F

Price (in `)

5

10

100

400

1,000

20,000

Quantity (Units)

500

250

200

40

9

1

Q.2 PARAG, OWNER OF GOLDEN BOOK DEPOT SAYS, “DURING EXAM DAYS CUSTOMERS MAINLY DEMAND REFRESHERS BOOKS (SAMPLE PAPERS)”. HE WANT YOUR HELP TO COMPUTE HIS AVERAGE COLLECTION FROM ONE BOOK FROM THE FOLLOWING DATA:

REFRESHER BOOK

ENGLISH

ECONOMICS

ACCOUNTANCY

B. St

MATHS

Price (in `)

150

250

200

200

300

Units Sold

12

10

14

10

4

Q.2 COMPUTE THE WEIGHTED ARITHMETIC AVERAGE FROM THE FOLLOWING DATA:

BOOKS

ENGLISH

ACCOUNTANCY

ECONOMICS

BUSINESS

MATHS

Price (in `)

150

250

200

200

300

Units Sold

12

10

14

10

4

PROPERTIES OF ARITHMETIC MEAN:

Q.1 SUM OF DEVIATION FROM ACTUAL MEAN I.E. ARITHMETIC MEAN IS ALWAYS EQUAL TO ZERO.

Example-Following are the marks of 5 students in Economics:

STUDENTS

A

B

C

D

E

MARKS (Out of 20)

16

12

18

10

14

Q.2 SUM OF SQUARE OF DEVIATION FROM ACTUAL MEAN I.E. ARITHMETIC MEAN IS ALWAYS LEAST.

Example-Following are the marks of 5 students in Economics:

STUDENTS

A

B

C

D

E

MARKS (Out of 20)

16

12

18

10

14

Q.3 IF EVERY VALUE OF THE SERIES IS MULTIPLIED BY A CONSTANT, THEN MEAN OF THE SERIES GETS MULTIPLIED BY IT.

Example-Following are the marks of 5 students in Economics:

STUDENTS

A

B

C

D

E

MARKS (Out of 20)

16

12

18

10

14

Q.4 IF A CONSTANT VALUE IS ADDED IN EACH VALUE OF THE SERIES, THEN MEAN OF THE SERIES GETS ADDED BY IT.

Example-Following are the marks of 5 students in Economics:

STUDENTS

A

B

C

D

E

MARKS (Out of 20)

16

12

18

10

14