NCERT Solutions For Statistics Class 11

Extramarks offers NCERT solutions for Statistics Class 11, wherein, all the topics are thoroughly explained according to grade 11. The material of each topic in the entire chapter is described in NCERT Solutions Class 11 Statistics

Statistics is the concept of Mathematics that gives out a collection of interpretation, analysis, and also the presentation of numerical data and masses. NCERT Solutions Class 11 Statistics presents a comprehensive overview of statistics, beginning with its origins. Statistics is a singular word with multiple senses, according to the Oxford dictionary. It signifies that it is a collection of values for many things. It is also a way of quantifying tasks in order to improve their quality. The word “Statistics” is derived from the Latin word ‘Status’ which means a collection of figures or numbers, to find out some data on the basis of human interest.

Statistics are also called observational data. It can be found and experienced in day-to-day life, such as on TV or in other information found in papers, books, or newspapers. The motive of statistics is to grant sets of data to disagree with so that the analysts can focus on consequential changes and trends. Through this knowledge, the analysts can reach conclusions with regard to the meaning of the given information. In the beginning, statistics was highly preferred by the kings to gather details about their supreme states and other information required about their residence, yield, and revenue of the country.

Statistics class 11 solutions provide an extensive large range of exemplifying problems to be solved under the illustration of the syllabus. It comprises 9 chapters. The subject of statistics has now been renamed as ‘Statistics for Economics’. These books are written by experts who are specialists in their own field of study.

NCERT SOLUTION FOR CLASS 11 STATISTICS

NCERT Solution for Class 11

 Units           Name of the Chapter

Chapter 1    Introduction

Chapter 2    Collection of Data

Chapter 3    Organisation of Data

Chapter 4    Presentation of Data

Chapter 5    Measures of Central Tendency

Chapter 6    Measures of Dispersion

Chapter 7    Correlation

Chapter 8    Index Numbers

Chapter 9    Use of Statistical Tools

The above mentioned is the syllabus and study materials of NCERT solutions for class 11 Statistics.

NCERT Solution for Class 11 Commerce Statistics Chapter 1 – Introduction provides us with everything including data on all the concepts. As students have to learn the basic fundamentals of the subject of statistics in Class 11, this syllabus for Class 11 is an extensive study material, which explains the concepts in the best way possible.

Various concepts covered in this chapter are 

  1. Why Economics?
  2. What are statistics?
  3. Statistics in economics
  4. Consumption, Production, and Distribution
  5. Conclusion

The use of Statistical Tools furnishes us with all-inclusive data on all the concepts. As the students would have to learn the basic fundamentals of the subject of statistics in class 11, this curriculum for class 11 is a comprehensive study material, which explains the concepts in a great way.

-Concepts covered in this chapter –

-Introduction

-Identifying a problem or an area of study

-Choice of Target Group

-Collection of Data

-Organisation and Presentation of Data

-Analysis and Interpretation

-Conclusion

-Project

Concepts covered in this chapter –

  1. Introduction
  2. What are the sources of data?
  3. How do we collect the data?
  4. Preparation of Instrument
  5. Mode of Data Collection
  6. Pilot Survey
  7. Census or Complete Enumeration
  8. Population and Sample
  9. Random Sampling
  10. Non-Random Sampling
  11. Sampling and non-sampling errors
  12. Sampling Errors
  13. Non-Sampling Errors
  14. Census of India and NSSO

Q.1 Frame at least four appropriate multiple-choice options for the following questions:

    1. Which of the following is the most important when you buy a new dress?
    2. How often do you use computers?
    3. Which of the newspaper do you read regularly?
    4. Rise in the price of petrol is justified.
    5. What is the monthly income of your family?

Ans.

i. Which of the following is the most important when you buy a new dress?

  1. price
  2. quality
  3. style
  4. size

ii. How often do you use computers?

  1. every day
  2. 5-6 times a week
  3. 3-4 times a week
  4. less than 3 times a week

iii. Which of the newspaper do you read regularly?

  1. The Hindu
  2. The Times of India
  3. The Economics Times
  4. Hindustan Times

iv. Rise in the price of petrol is justified

  1. Because of increase in supply for petrol
  2. Because of abundance of petrol
  3. Because of hike in excise duty on petrol
  4. Because of appreciation of rupee

v. What is the monthly income of your family?

  1. Less than ₹ 30,000
  2. ₹ 30,000 to ₹ 50,000
  3. ₹ 50,000 to ₹ 1,00,000
  4. More than ₹ 1,00,000

Q.2 Frame five two-way questions (with ‘Yes’ or ‘No’).

Ans.

1. Do you have a car in your family?

2. Do you like economics?

3. Have you ever been out of your country?

4. Have you ever visited Taj Mahal?

5. Have to contribute towards Swaach Baharat Abhiyan

Q.3 State whether the following statements are True or False.

(i) There are many sources of data.

(ii) Telephone survey is the most suitable method of collecting data, when the population is literate and spread over a large area.

(iii) Data collected by investigator is called the secondary data.

(iv)There is certain bias involved in the non-random selection of samples.

(v) Non-sampling errors can be minimized by taking large samples.

Ans.

(i) There are many sources of data. – True

Reason: There are many sources of data collection which can be categorized under two heads primary data sources and secondary data sources.

Primary data sources consists of Direct personal interviews, Indirect oral investigations, Information through mailed Questionnaire or Information through questionnaire filled by enumerators.

The sources of secondary data includes published sources like books, magazine, government survey reports etc. and unpublished sources like research papers, company documents, etc.

(ii) Telephone survey is the most suitable method of collecting data, when the population is literate and spread over a large area. – False

Reason: Telephone survey is most suitable method of collecting data when population is illiterate. When population is literate and spread over a large area the most suitable method is mailed questionnaire method.

(iii) Data collected by investigator is called the secondary data. – False

Reason: Data collected by the investigator himself is called primary data; the data collected by some other agency and which is already available for furthers studies are called secondary data.

(iv) There is certain bias involved in the non-random selection of samples. – True

Reason: Non-random sampling is done on the basis of judgement and convenience of the investigator hence it is not free of certain bias like personal prejudice.

(v) Non-sampling errors can be minimized by taking large samples – False

Reason: Non-sampling errors are more serious than sampling errors and it is difficult to minimize them, ever by taking a large sample. Some of the non-sampling errors are: errors in data Acquisition, non-response errors, sampling bias, processing errors, etc.

Q.4 What do you think about the following questions? Do you find any problems with these questions? Describe.

    1. How far do you live from the closest market?
    2. If plastic bags are only 5 percent of our garbage, should it be banned?
    3. Wouldn’t you be opposed to increase in price of petrol?
    4. Do you agree with the use of chemical fertilisers?
    5. Do you use fetilisers in your fields?
    6. What is the yield per hectare in your field?

Ans.

(i) How far do you live from the closest market?

  • This question is quite ambiguous and respondent might find it difficult to answer it with certainty.

(ii) If plastic bags are only 5 percent of our garbage, should it be banned?

  • This question is too long; it might discourage respondent to answer. It also gives a clue about how the respondent should answer which can influence the respondents answer, it should be avoided.

(iii) Wouldn’t you be opposed to increase in price of petrol?

  • This question contains double negatives. The questions starting with “wouldn’t you” or “don’t you” should be avoided as they may create confusion and lead to biased responses.

(iv) Do you agree with the use of chemical fertilisers?

(v) Do you use fetilisers in your fields?

(vi) What is the yield per hectare in your field?

  • The above three questions (iv, v and vi) are correct but the order in which they have been asked is not correct. One should always ask the general questions first and then should move towards more specific ones. This helps the respondents to feel comfortable.
  • The correct order to ask these questions would have been.
  1. What is the yield per hectare in your field?
  2. Do you use fetilisers in your fields?
  3. Do you agree with the use of chemical fertilisers?

Q.5 You want to do a research on the popularity of Vegetable Atta Noodles among children. Design a suitable questionnaire for collecting this information.

Ans.

  • Name: ………………
  • Age: …………………
  • Gender: □ Male □ Female

1. Do you like eating noodles?

□ Yes □ No

2. How often do you eat noodles in a month?

□ Less than twice □ Less than three time

□ More than three times a month.

3. Have you eaten Vegetable Atta Noodles?

□ Yes □ No

4. Do you like Vegetable Atta Noodles?

□ Yes □ No

5. Do you prefer Vegetable Atta Noodles over other Masala noodles made of maida?

□ Yes □ No

6. In what meal time would you like to have Vegetable Atta Noodles?

□ Breakfast □ lunch

□ Evening snack □ dinner

7. Does your parents approve of you eating Vegetable Atta Noodles?

□ Yes □ No

Q.6 In a village of 200 farms, a study was conducted to find the cropping pattern. Out of the 50 farms surveyed, 50% grew only wheat. What is the population and the sample size?

Ans.

Population or universe in statistics refers to the totality of the items under study. Since, here the study was conducted to find out the cropping method of a village comprising 200 farms, the population size is 200 farms.

Sample in statistics refers to a group or section of population from which the information is to be obtained or collected. Since, here the information is collected from only 50 farms, the sample size for the study is 50 farms.

Q.7 Give two examples each of sample, population and variable.

Ans.

Sample in statistics refers to a group or section of population from which the information is to be obtained or collected. Or that part of population or universe which is selected for the purpose of study as representative of the universe.

Population or universe in statistics refers to the totality of the items under study.

A variable refers to the characteristics of the sample or population under study that can be expressed in numbers and are generally represented by the letters X, Y, Z. Each value of a variable is an observation.

Example 1: To find out the average weight of students 10 students from a class of 50 students are selected.

Here,

  • Sample is 10 students
  • Population is 50 students
  • Variable is height

Example 2: To find out the average income of workers in a factory comprising 250 workers, 50 workers are selected at random.

Here,

  • Sample is 50 workers
  • Population is 250 workers
  • Variable is income

Q.8 Which of the following methods give better results and why?

(a) Census

(b) Sample

Ans.

Census method refer to inclusion of all items in the field of statistical enquiry and sample methods refer to selection of few items as representatives of all the items.

Sample method gives better results than census method because of following reasons:

  1. Economical method: Census method requires huge expenditure. As a large number of enumerators have to be employed. They have to be trained. Their work has to be coordinated and supervised. There will be expenditure on traveling, food etc in case of census method. The cost of the survey is much smaller if we a sample method because of less efforts involved in it.
  2. Less time and efficiency consumption: The collection of data, the tabulation, and analysis take much less time in case of sampling method compared to census method. In fact the population census of India takes so much time that it takes place only once in ten years.
  3. Accuracy: In sampling method it is possible to check the extent or errors and take corrective actions. While in census method, it is almost impossible to detect errors, owing to its large magnitude.
  4. Less non-sampling errors: The magnitude of non-sampling errors is also much smaller in case of sampling method because of smaller size of data.
  5. More reliable: In sampling method scientific methods and trained investigators are employed for the collection of data, which make it more reliable than census method.

Q.9 Which of the following errors is more serious and why?

(a) Sampling error

(b) Non- sampling error

Ans.

Non-sampling errors are more serious than the sampling errors.

Sampling errors arise due to drawing of inferences about the population on the basis of a few observations. For example the estimate of average income of people in a certain region on the basis of a small set of observations may not be equal to the actual average. The difference between the sample estimate and the true average income in the region is called the sampling error. Such kind of errors can be minimised by increasing the size of sample.

Non sample errors are those which may arise due to error in collection or measurement of data. The enumerators or respondents may misinterpret the questions, or some enumerators may not be very efficient. They may not record the data correctly. It is difficult to minimise such non-sampling errors even by increasing the size of sample.

Thus, non-sampling errors are more serious than sampling errors.

Q.10 Suppose there are 10 students in your class. You want to select three out of them. How many samples are possible?

Ans.

Population =10

Sample size = 3

Number of possible samples can be calculated using combination method, 10C3:

Number of possible samples = 10! / 3! X (10-3)!

= 10 X 9 X 8 X 7! / 3 X 2 X 1 X 7!

= 120 samples

Q.11 Discuss how you would use the lottery method to select 3 students out of 10 in your class.

Ans.

To select 3 students out of 10 in a class by using lottery method we will follow the following 4 steps.

STEP 1: We will assign one number to each 10 student.

STEP 2: We will right down all these numbers on the identical chits. (Make sure no two chits contain the same number).

STEP 3: We will put all these chits in a box and mix them well.

STEP 4: Then we will pick out the 3 chits randomly out of the box one by one. (Without replacement)

The three students whose number is written on the drawn 3 chits will consider selected. This way every student has an equal chance of getting selected.

Q.12 Does the lottery method always give you a random sample? Explain.

Ans.

Yes, lottery method always gives a random sample. In random sampling, the items which get selected are beyond the control of the investigator, it depends entirely on chance. Each and every unit in the population has an equal chance of being selected. Similarly in lottery method, each individual unit is selected at random from the population and has equal opportunity of being selected.

For example, if there are 50 students, out of which we have to select 5, we will put 50 slips containing the names of all the students in a bowl and mix them well. Then five slips are selected from the bowl one by one without replacement. The students corresponding to these 5 slips will constitute our sample.

Here, each student has an equal opportunity of being selected same as in case of random selection. Hence, lottery method always gives a random sample.

Q.13 Explain the procedure for selecting a random sample of 3 students out of 10 in your class by using random number tables.

Ans.

For selecting a simple random sample, random number tables can be used. Several standard tables of random numbers are available. For using these tables, the population units comprising N units should be numbered from 1 to N, which makes it possible to determine the range of numbers to be selected.

Since our population size in 10 students, two digit random tables are selected.

To select 3 students out of ten, we first identify 10 students with a number from 1 to 10, like 01, 02, 03, 04, 05, 06, 07, 08, 09, and 10.

Then we take any two digit random number table and choose any page from the table. Starting at any row or column, we select 3 two digit numbers one by one, discarding the numbers greater than ten.

Let us assume the first selected number is 04.

The two numbers successive to the selected number (04) either horizontally or vertically comprises the remaining two sample say 06 or 08.

Finally the students bearing these selected numbers will constitute our sample.

Q.14 Do samples provide better results than surveys? Give reasons for your answer.

Ans.

Yes, samples do provide better results than surveys. The reasons behind the same are as follows:

  • Minimised costs- Expenditure is expected to be smaller if data collected is a small fraction of the total population using samples then if complete survey is done. In situation of limited financial resources at disposal, sample study is economical and viable choice.
  • Quick results –Data collection and summarisation is quicker in a sample method than in a survey method with complete count. It is a feasible choice when information is required on urgent basis.
  • Greater Scope- In certain circumstance or certain type of enquiry, to acquire and analyse data, highly trained personnel, specific equipments and expertise are required which are very limited in availability in such cases sampling methods are more practical.
  • Detailed enquiry and greater accuracy- Due to limited number of units under study in case of samples; it is possible to collect more detailed information by conducting intensive enquiries. Even highly trained and efficient people can be employed to achieve accurate results.
  • Highly convenient- The administration and organization of samples are easy in comparison to surveys. Samples can provide reasonably reliable and accurate information in shorter time and at a lower cost.

Q.15 The unit of correlation coefficient between height in feet and weight in kgs is

    1. Kg/feet
    2. Percentage
    3. non-existent

Ans.

iii. Non-existent

Explanation: Correlation coefficient has no unit. It is pure numeric term used to measure the degree of association between variables.

Q.16 The range of simple correlation coefficient is

i. 0 to infinity

ii. Minus one to plus one

iii. Minus infinity to infinity

Ans.

ii. Minus one to plus one

Explanation: The value of correlation coefficient lies between minus one to plus one, if the value lies outside this range it indicates error in calculation.

Q.17 If rxy is positive the relation between X and Y is of the type

i. When Y increases X increases

ii. When Y decreases X increases

iii. When Y increases X does not change

Ans.

i. When Y increases X increases

Explanation: A positive correlation implies that both the variables move in a similar direction. If there is a increase in X, Y also increases in same direction.

Q.18 If rxy = 0 the variable X and Y are

i. Linearly related

ii. Not linearly related

iii. Independent

Ans.

ii. Not linearly related

Explanation: If rxy = 0, it means that there is a absence of linear relation between the variables, but there may exist a non linear relation between variables.

Q.19 Of the following three measures which can measure any type of relationship

i. Karl Pearson’s coefficient of correlation

ii. Spearman’s rank correlation

iii. Scatter diagram

Ans.

ii. Spearman’s rank correlation

Explanation: Spearman’s rank correlation is the measure which can measure any type of relationship. But, Karl Pearson’s coefficient is the most widely used method due to the preciseness of the ‘r’ value.

Q.20 If precisely measured data are available the simple correlation coefficient is

i. More accurate than rank correlation coefficient

ii. Less accurate than rank correlation coefficient

iii. as accurate as the rank correlation coefficient

Ans.

iii. As accurate as the rank correlation coefficient

Q.21 Why is r preferred to co-variance as a measure of association?

Ans.

“r” i.e. correlation coefficient is preferred to co-variance as a measure of association because

  • r is independent of change in scale and origin.
  • r has a specific range (-1 to +1 ), so it comes handy to interpret the results quickly.
  • Co-variance is a part of correlation coefficient.

Q.22 Can r lie outside the -1 and 1 range depending on the type of data?

Ans.

No, value of r cannot lie outside the range -1 and 1 depending on the type of data. If r = 1 or -1 that means there is a perfect positive or perfect negative relation between variables and if r = 0 that means there isn’t any correlation. But, if value of r lies outside the range -1 and 1, then it indicates that there is an error in calculation.

Q.23 Does correlation imply causation?

Ans.

No, correlation does not imply causation. Correlation only implies association between two variables. It doesn’t represent any cause and effect relation between the two variables. Correlation between two variables only means that two variables are either positively or negatively or neither related at all. It only measures the degree and intensity of the relation between the variables. For example marks scored by a student in exams is correlated to number of days the student went to school but the marks scored by him doesn’t depend on the number of days he went to school.

Q.24 When is rank correlation more precise than simple correlation coefficient?

Ans.

Rank correlation is more precise than simple correlation coefficient in situations when it is required quantify qualities. Ranking is a better alternative for quantification of qualities, which can’t be done in simple correlation. It is also useful when correlation coefficient between two variables with extreme values is quite different from the coefficient without the extreme values. For example, in a beauty contest judges may have to prepare a list of participants in order of their beauty. There is no procedure or numerical system which judges beauty, so in order to prepare a list judges rank the participants based on their features. In this manner rank correlation can be used which gives more precise value than the simple correlation.

Q.25 Does zero correlation mean independence?

Ans.

No, zero correlation doesn’t mean independence. Zero correlation only indicates absence of a linear relation between the variables. There might exist a non-linear relation between them, thus zero correlation necessarily doesn’t mean independence.

Q.26 Can simple correlation coefficient measure any type of relationship?

Ans.

No, simple correlation is not able to measure all the types of relationship. It can only measure linear relationships between the variables. It is not able to interpret the non-linear relationships between the variables. In case correlation coefficient returns the value zero it either means there is no correlation or there is non-linear relationship which cannot be measured.

Q.27 List some variables where accurate measurement is difficult.

Ans.

Some of the variables for which accurate measurement is difficult are:

  • Beauty
  • Honesty
  • Intelligence
  • Ability
  • Bravery
  • Fairness, etc

Q.28 Interpret the values of r as 1,-1 and 0

Ans.

r = 1 implies there is a perfect positive correlation between the variables.

r = -1 implies there is a perfect negative correlation between the variables

r = 0 implies there is no correlation between the variables.

Q.29 Why does rank correlation coefficient differ from Pearsonian correlation coefficient?

Ans.

Rank coefficient differs from pearsonian correlation coefficient because in rank coefficient, specific ranks are assigned to the data which leads to loss of information and all the information regarding the data is not utilised. Only if the data in the rank coefficient method is ranked precisely, it leads to similar values as the pearsonian coefficient. . The value of rank coefficient also differs due to the first differences of the value of items in the series arranged are almost never constant. Generally, both the methods result in same value of ‘r’; but pearsonian coefficient is more widely used than rank coefficient as it utilizes information from the whole series of frequency distribution.

Q.30 An index number which accounts for the relative importance of the items is known as

    1. Weighted index
    2. Simple aggregative index
    3. Simple average index

Ans.

i. Weighted index

Weighted index number is an index number in which different items are given different importance in terms of different weights.

Q.31 In most of the weighted index numbers the weights pertains to

    1. Base year
    2. Current year
    3. both base and current year

Ans.

ii. Current year

In most of the weighted index numbers the weights pertains to current year because we want to know how the price/quantity has changed in current year from the level of base year.

Q.32 The impact of change in the price of the commodity with little weight in the index will be

    1. Small
    2. Large
    3. Uncertain

Ans.

i. small

Whether the change in price will be reflected in a price index number or not depends on the weight associated with the item. Lower or negligible weight reflects lower or negligible change in the index number and higher weight reflects higher change in the index number.

Q.33 A consumer price index measures changes in

    1. Retail prices
    2. Wholesale prices
    3. Producers prices

Ans.

ii. Wholesale prices

From 2014, the Reserve Bank of India (RBI) has adopted the new Consumer Price Index (CPI) as the key measure of inflation.

Q.34 The item having the highest weight in consumer price index for industrial workers is

    1. Food
    2. Housing
    3. Clothing

Ans.

i. Food

Food is the most essential commodity for living. Industrial workers have to spend more on food because they are engaged in physical labour.

Q.35 In general, inflation is calculated by using

    1. Wholesale price index
    2. Consumer price index
    3. Producer’s price index

Ans.

i. Wholesale price index

In general, inflation is calculated by using the wholesale price index. The formula used is:
Rate of inflation = [(WPI of current year/WPI of previous year) X 100] – 100

Q.36 Why do we need an index number?

Ans.

Index numbers are the statistical tools which measure the changes in the magnitude of a variable with respect to chosen base year. They measure average change in a group of related variables over two different situations. The changes measured are from time to time or from place to place. They are usually expressed in percentages.
Index number is one of the most widely used statistical tools. Index numbers are now-a-days called economic barometers. The importance of index numbers can be judged from the following points:

1. Helpful in measuring changes in value of money: Index numbers are widely used in the measurement of changes in the value of money. The value of money depends on its purchasing power which in turn depends on the prices of the commodities.
Purchasing power of money = 1/ consumer price index
Real income or wages= (Money income or wages/Consumer price index) × 100
The change in price inversely affects the value of money. Thus, the price index number throws light on the change of the value of money.
2. Helpful to policy makers: Index numbers serve as a useful guide to the business community in planning their decisions. The employers depend upon the cost of living index for deciding the increase in the dearness allowances of their employees. Index numbers play the role of essential guide in policy formulation related to inflation, unemployment, agricultural, industrial production etc.

3. There are certain changes whose measurements are not possible without index numbers. Index numbers make possible the measurement of relative changes of a group of related variables.

4. With the help of index numbers, the comparative study of changes in two variables becomes easy.

5. Index numbers help us to study the general trend of a phenomenon so as to draw important conclusions.

6. Index numbers help to determine real rise or fall in per capita income.

Q.37 What are the desirable properties of the base period?

Ans.

While constructing index numbers, we have to select two years, i.e. the base year and the year of comparison. Base year is the period against which comparisons are to be made. The selection of the base year is very important.

The following points need to be considered while selecting the base year:
i. The base year should be a normal one. The year selected as base year should be free from abnormal conditions like war, droughts, famines, earthquakes, booms, depressions etc.
ii. It should not be too near or too far from the current year.

While selecting the base year, choice has to be made between fixed base and chain base methods. In the fixed base method, we take a particular year as the base year and express changes in items on the basis of this year. Whereas in the chain based method, the base year goes on changing. For every successive year, the preceding year is taken as base year.

Q.38 Why is it essential to have different CPI for different categories of consumers?

Ans.

CPI, Consumer price index is a measure of average change in retail prices. It indicates the average change in the price paid by the final consumer for specified quantity of goods and services over a period of time. Different classes are getting affected differently by a change in the price level because different people consume different types of goods, consumer’s habit differs from individual to individual, place to place, strata to strata, etc. So, it is essential to have different CPI for different categories of consumers because the nature of consumption basket of consumers from different economic status varies hugely.

Q.39 What does a consumer price index for industrial workers measure?

Ans.

A consumer price index number for industrial workers measures the impact of price rise on the cost of living of common people i.e. general inflation. Over time, it is increasingly considered as the appropriate indicator of general inflation because it shows the most accurate impact of price rise on the cost of living.

Q.40 What is the difference between price index and quantity index?

Ans.

Price Index

Quantity Index

A price index is related to price and measures the change in prices in two periods. A quantity index is related to quantity and measures the changes in production, sales, consumption, etc. in two periods.
It indicates the changes in monetary value. It indicates the changes in the physical volume of production.

Q.41 Is the change in any price reflected in a price index number?

Ans.

The change in any price may or may not be reflected in a price index number. An index number shows changes in terms of average. For example, when it is said that index number in 2012-13 has risen to 110, it means that prices of all goods and services have increased by 10% on average. However, it does not mean that prices of all goods and services have uniformly risen by 10%. The price of a particular good might have risen by more than 10% or less than 10% or even might have not change. Whether the change in price will be reflected in a price index number or not depends on the weight associated with the item. Lower or negligible weight reflects lower or negligible change in the index number and higher weight reflects higher change in the index number.

Q.42 Can the CPI for urban non-manual employees represent the changes in the cost of living of the President of India?

Ans.

An index number calculated for the employees who are deriving 50% or more of his or her income from gainful employment on non-manual work in the urban non-agricultural sector, is termed as consumer price index for non-manual employees. Since the president of India is residing in urban area and is doing non-manual work in non-agricultural sector, so CPI for urban non-manual employees represents the changes in the cost of living of the President of India.

Q.43 Read the following table carefully and give your comments.

Index of industrial production base 1993-94
Industry Weight in % 1996-97 2003-04
General Index 100 130.8 189.0
Mining and quarrying 10.73 118.2 146.9
Manufacturing 79.58 133.6 196.6
Electricity 10.69 122.0 172.6

Ans.

The following comments can be made from the given table:

  • In the given table highest weight is given to manufacturing i.e. around 79.58 % as compared to mining and quarrying and electricity whose weights are 10.73 % and 10.69 % respectively.
  • As compared to the year 1993-94, general production has increased from 30.8 % to 89 % in 1996-97 and 2003-04 respectively.
  • The manufacturing sector’s performance is the best in both the years as compared to other two sectors. It has risen around 63 % from 1996-97 to 2003-04.
  • The mining and quarrying sector is the least growing sector as compared to other sectors.

Q.44 Try to list the important items of consumption in your family.

Ans.

The following are the important items of consumption in our family:

  • Food
  • Cloth
  • Electricity
  • Education
  • School bag
  • Books
  • Utensils
  • Household appliances, etc.

Q.45 If the salary of a person in the base year is ₹ 4000 per annum and the current year salary is ₹ 6000, by how much should his salary be raised to maintain the same standard of living if the CPI is 400?

Ans.

Salary of base year = ₹ 4000

Salary of current year = ₹ 6000

CPI = 400

If the current year CPI is 400 thus, to maintain the base year standard of living the salary should be raised to {(CPI X Salary of base year)/100 = (400 X 4000)/100 =} ₹ 16000

Thus, person’s salary should be ₹ 16000 in current year, but it has increased to ₹ 6000 only. Therefore, his salary should be raised to maintain the same standard of living if the CPI is 400, by (16000 – 6000 =) ₹ 10000.

Q.46 The consumer price index for June, 2005 was 125. The food index was 120 and that of other items 135. What is the percentage of the total weight given to food?

Ans.

The price index for June 2005 = 125

Food index = 120

Index for other items = 135

Assume weight assign to food is W1 and to other items is W2.

We know the sum of total weights is 100 i.e. W1+ W2 = 100.

Items

Index, I

Weights, W

WI

Food

120

W1

120 W1

other items

135

W2

135 W2

W1+ W2 = 100

120 W1+ 135 W2

CPI= ∑WI/∑W

125 = 120 W1+ 135 W2/ W1+ W2

125 W1+ 125 W2 = 120 W1+ 135 W2

125 W1 – 120 W1= 135 W2 – 125 W2

5 W1 = 10 W2

W1 = 2 W2 —–(1)

W1+ W2 = 100 —–(2)

By putting the value of W1 the equation (2), we get,

2 W2+ W2 = 100

3 W2 = 100

W2 = 100/3 = 33.33

W1 = 2 X(100/3) = 66.67

Thus, the weights assign to food is 66.67 and to other items are 33.33.

Q.47 Record the daily expenditure, quantities bought and prices paid per unit of the daily purchases of your family for two weeks. How has the price change affected your family?

Ans.

Answer may vary.

Week I
Tomato Potato Total Expenditure
Price (₹ /Kg) Quantity Purchased Expenditure Price (₹ /Kg) Quantity Purchased Expenditure
Monday 20 0.5 10 20 1 20 30
Tuesday 10 2 20 25 .5 12.5 32.5
Wednesday 15 1 15 15 1.5 7.5 22.5
Thursday 10 1 10 25 1 25 35
Friday 20 1 20 30 1.5 45 65
Saturday 15 .5 7.5 20 .5 10 17.5
Sunday 25 .25 6.25 25 1 25 31.5
Week II
Tomato Potato Total Expenditure
Price (₹ /Kg) Quantity Purchased Expenditure Price (₹ /Kg) Quantity Purchased Expenditure
Monday 25 0.5 12.5 20 1.5 30 42.5
Tuesday 20 1 20 20 1 20 40
Wednesday 20 .5 10 15 2 30 40
Thursday 15 1 15 20 1.5 30 45
Friday 10 2 20 25 .5 12.5 32.5
Saturday 15 1 15 15 2 30 45
Sunday 20 .5 10 20 1 20 30

We have observed that when prices are high my family is purchasing lesser quantity of goods. It will result in lesser expenditure made on the goods as compared to the expenditure made when prices are lower. As we can see that on Monday (first week), the price of tomato was ₹ 20, the quantity purchased was 0.5 kg and the total expenditure made on tomato was ₹ 10. When the price of tomato fell down to ₹ 10 on Tuesday, the quantity purchased increased to 2 Kg, hence expenditure made was ₹ 20. It shows the negative relation between price of the goods and demand for the goods. The same is observed in week II across commodities.

Q.48 Given the following data.

Year CPI of industrial workers

(1982 = 100)

CPI of urban non-manual employee

(1982 = 100)

CPI of agricultural labourers

(1986-87 = 100)

WPI

(1993-94=100

1995–96 313 257 234 121.6
1996–97 342 283 256 127.2
1997–98 366 302 264 132.8
1998–99 414 337 293 140.7
1999–00 428 352 306 145.3
2000–01 444 352 306 155.7
2001–02 463 390 309 161.3
2002–03 482 405 319 166.8
2003–04 500 420 331 175.9

Source: Economic Survey, 2004-2005, Government of India

(i) Comment on the relative values of the index numbers.

(ii) Are they comparable?

Ans.

(i) A consumer price index number for industrial workers measures the impact of price rise on the cost of living of common people i.e. general inflation.

An index number calculated for the employees who are deriving 50 % or more of his or her income from gainful employment on non-manual work in the urban non-agricultural sector is termed as consumer price index for non-manual employees.

Index Number of Agricultural Production (IAP) records the changes in agricultural production. It is used to study the rise and fall of the yield of crops from one period to other period.

Wholesale price index number refers to index number that measures the average changes in the wholesale price. It is an indicator of change in the general price level.

(ii) The given data of each category i.e. CPI of industrial workers, CPI of urban non-manual employee, CPI of agricultural labourers and WPI are comparable with themselves over given time period.

CPI of industrial workers and CPI of urban non-manual employee can be compared with each other because they have the same base year but rests of the categories are not comparable.

Q.49 Mark the following statements as true or false.

(i) Statistics can only deal with Quantitative data.

(ii) Statistics solve economic problems.

(iii) Statistics is of no use to Economics without Data.

Ans.

(i) False;

Reason: By statistic we mean both qualitative and quantitative facts that are used in economics. Statistics reveals both quantitative and qualitative aspects of data. For example: the statement “Literacy rate in India has gone up from 65.38% in 2001 to 74.4% in 2011” is a quantitative fact. But the chief characteristic of such information is that it describes the attributes of a group of people this is qualitative information which is often used in economics and other social sciences.

(ii) True;

Reason: Statistics is an indispensable tool for economists that help them in understanding economics problems. Economists use various methods of statistics, to find out the causes behind economic problems, with help of qualitative and quantitative facts relating to economic problems. After identifying the causes of the problem, it becomes easier to formulate policies to tackle it.

(iii) True;

Reason: Data help economists to present economic facts in a proper and definite form that further helps in proper understanding of what is stated. When economic facts are stated in statistical terms or in form of data, they become more exact and more convincing than vague statement hence we can say that statistics is of no use to economics without data.

Q.50 Make a list of activities you find in a bus stand or a market place. How many of them are economic activities?

Ans.

The list of activities we observe at a bus stand or in a market place is as follows:

  • Buying of goods
  • Selling of goods
  • People rendering different kind of services such as: rickshaw puller taking people from place to place, waiter working at a nearby eating joint, etc.
  • Production process carried out by some producers.

Above mentioned all activities are economic activities because they all are undertaken for a monetary gain. All these activities involve the use of scarce resources to carry out the task of production, consumption and distribution.

Q.51 ‘The government and policy makers use statistical data to formulate suitable policies of economic development.’ Illustrate with two examples.

Ans.

Statistics is used in finding relationships between different economic factors using data and verifies them. Statistical data and tools are used in predictions of future trends. Hence, statistical data is of key importance to the government and policy makers in formulating suitable policies of economic development.

For example: An economic planner might be interested in knowing the impact of today’s investment on national income in future. To carry on such an exercise economic planner needs to acquire proper knowledge of statistics and appropriate statistical data.

Other examples: Ministry of Finance requires statistical data of revenue and expenditure to prepare annual budget of the country. Banking organisations also need statistical data for fixing interest rates, advancement of loans and formation of credit policy.

Q.52 “You have unlimited wants and limited resources to satisfy them”. Explain this statement by giving two examples.

Ans.

The statement “You have unlimited wants and limited resources to satisfy them” refers to a universal problem of decision making or problem of choice arising at all levels of the economy due to scarcity of resources. Problem of choice arises due to unlimited human wants, scarcity of resources and alternative uses of resources.

The basic concern of an economy and also of an individual is to allocate the available scare resources to the best possible use confronted with unlimited wants. An economy as well as individual has to make a choice between various ways of allocating the available scarce resources according to his/her priority.

For example: An individual with ₹ 10,000 has to choose between buying a new T.V. set and paying tuition fee of his/her child according to his/her priority.

Similarly, an economy endowed with a given level of resources has to choose between the production of civil goods (sugar, rice, cycles etc.) or defence / military goods (guns, bombs, tanks etc.) depending upon the need of the economy.

Q.53 How will you choose the wants to be satisfied?

Ans.

An individuals’ choice of fulfilling a want depends upon the need of the hour, level of satisfaction and priority attached to the wants. An individual will fulfill those wants first which are on his top priority and yield him/her the highest satisfaction. His/her decision to satisfy a particular want is also guided by the need of the hour, the availability of good he/she want and availability of the resources (money) to acquire the want.

Q.54 What are your reasons for studying Economics?

Ans.

The need to study economics evolve from the basic problem of scarcity and choice that everyone has to face. Economics is the study of how people use available scarce resources to meet their needs. It deals with choices and the impact of these choices on each other. Economics involve the study of human beings engaged in various economic activities.

The following are the reasons that make the study of economics certain:

  • The study of consumption: Economics deals with the study of consumer behavior in different market. The study of consumption is concerned with how a rational consumer decides, given his income and many alternative goods to choose from, what to buy when he knows the prices.
  • The study of production: It deals with how the producer chooses what to produce for market when he knows the costs and prices.
  • The study of distribution: Economics is also studied to know how the national income or the total income arising of what has been produced in the country is distributed through wages, rent, interest and profits.
  • The study of basic macroeconomics problems in an economy: Economics also helps us to understand the basic economic problems faced by an economy like unemployment, poverty, etc. and also the underlined causes behind them so that effective and appropriate measures can be taken to correct them.

Q.55 Statistical methods are no substitute for the common sense. Comment with examples from your daily life.

Ans.

By the statement “Statistical methods are no substitute for the common sense” we mean that statistical data should not be believed upon blindly as it can be misused and misinterpreted. Statistical data can be deliberately manipulated. Statistic may prove the wrong as right. Statistical laws are probabilistic in nature. Results based on statistics are only approximate and not exact. Lack of knowledge or inculcation of adverse opinions may lead to misuse of statistical facts.

For example: In a glass manufacturing company, profits earned during five consecutive years are ₹ 5000, ₹ 4000, ₹ 3000, ₹ 2000, and ₹ 1000 respectively on the other hand in a plastic manufacturing company, profits earned during five consecutive years are ₹ 1000, ₹ 2000, ₹ 3000, ₹ 4000, and ₹ 5000 respectively. Thus both companies have made same average profit i.e. ₹ 3000. On the basis of this average (statistical data) one may conclude that both companies have faced same economics conditions. But it would be wrong. As one can clearly see that plastic manufacturing company is progressing and the glass manufacturing company is on decline.

Q.56 Which average would be suitable in following cases?

    1. Average size of readymade garments.
    2. Average intelligence of students in a class.
    3. Average production in a factory per shift.
    4. Average wage in an industrial concern.
    5. When the sum of absolute deviations from average is least.
    6. When quantities of the variable are in ratios.
    7. In case of open-ended frequency distribution.

Ans.

i. Average size of readymade garments.

Mode

Explanation: Mode is suitable average for average size of readymade garments because it gives the most frequent occurring value.

ii. Average intelligence of students in a class.

Median

Explanation: Median is a suitable average in case of a qualitative nature of the data.

iii. Average production in a factory per shift.

Mean

Explanation: Production can be measured on a quantitative scale so Arithmetic mean is suitable in this case.

iv. Average wage in an industrial concern.

Mean

Explanation: Wage can be measured on a quantitative scale so arithmetic mean is suitable in this case.

v. When the sum of absolute deviations from average is least.

Mean

Explanation: Mean shall be used because sum of deviations from mean is always zero or least than the other averages.

vi. When quantities of the variable are in ratios.

Mean

Explanation: Ratios are quantitative, so it is suitable to use arithmetic mean.

vii. In case of open-ended frequency distribution.

Median

Explanation: Median is used because there is no need to adjust class size or magnitude for using median.

Q.57 Indicate the most appropriate alternative from the multiple choices provided against each question.

i. The most suitable average for qualitative measurement is

  1. Arithmetic mean
  2. Median
  3. Mode
  4. Geometric mean
  5. None of the above

ii. Which average is affected most by the presence of extreme items?

  1. Median
  2. Mode
  3. Arithmetic mean
  4. None of the above

iii. The algebraic sum of deviation of a set of n values from A.M is

    1. N
    2. 0
    3. 1
    4. none of the above

Ans.

  1. The most suitable average for qualitative measurement is Median.
  2. Arithmetic mean is the average affected by the presence of the extreme values.
  3. 0 is the sum of deviations of a set of n values from AM.

Q.58 Comment whether the following statements are true or false.

    1. The sum of deviation of items from median is zero.
    2. An average alone is not enough to compare series.
    3. Arithmetic mean is a positional value.
    4. Upper quartile is the lowest value of top 25% of items.
    5. Median is unduly affected by extreme observations.

Ans.

i. The sum of deviation of items from median is zero.

False

Explanation: Generally, sum of deviations from mean is zero; but only in the case of symmetric distribution (mean=median=mode) above statement is true.

ii. An average alone is not enough to compare series.

True

Explanation: Averages are very rigid values, they don’t say anything about the variability of the series, and thus they are not enough to compare series.

iii. Arithmetic mean is a positional value.

False

Explanation: Arithmetic mean is not a positional value because it is calculated on the basis of all the observations.

iv. Upper quartile is the lowest value of top 25% of items.

True

Explanation: Quartile refers to a quarter, so when the frequency is arranged in a ascending order the upper quartile refers to the first 25% of the items.

v. Median is unduly affected by extreme observations.

False

Explanation: Median doesn’t get affected by extreme observations because it only takes the median class to calculate it. It is mean which gets affected by extreme observations.

Q.59 A measure of dispersion is a good supplement to the central value in understanding a frequency distribution. Comment.

Ans.

Measure of dispersion is better supplement to the central value in understanding a frequency distribution because it provides much clearer interpretation of the distribution. Measures of central tendency do not highlight variability present in the data; whereas measures of dispersion provide clear interpretation about the variability. They also explain the spread of individual values around the central value and also about the scatter and expansion of data in the series.

Q.60 Which measure of dispersion is the best and how?

Ans.

Standard deviation is the best measure of dispersion. It is based on all values of the distribution, thus a change in value of even one variable affects it. It is most widely used measure of dispersion:

  1. It is independent of origin but not of scale.
  2. It can be used if the samples are combined in various ways.
  3. It is very useful in advanced statistical works and further algebraic treatment is also possible.
  4. The square of deviations from mean is least in case of standard deviation.
  5. It is the best method to compare variability between two or more distributions.
  6. It plays a central role in normal distribution which in turn is very useful.

Q.61 Some measures of dispersion depend upon the spread of values whereas some are estimated on the basis of the variation of values from a central value. Do you agree?

Ans.

Yes, some measures of dispersion such as range, quartile deviation and inter-quartile range depend upon the spread of values; and some measures like mean deviation and standard deviation are estimated on the basis of a variation of values from a central value. Mean deviation, standard deviation takes into account all the values of the data through variation between them and a central value; whereas range, Q.D and inter-quartile range takes into account only some positional values. Other than these, Lorenz curve is also used to represent variation graphically.

Q.62 Which of the following alternatives is true?

a. The class midpoint is equal to:

  1. The average of the upper class limit and the lower class limit.
  2. The product of upper class limit and the lower class limit.
  3. The ratio of the upper class limit and the lower class limit
  4. None of the above.

b. The frequency distribution of two variables is known as

  1. Univariate Distribution
  2. Bivariate Distribution
  3. Multivariate Distribution
  4. None of the above

c. Statistical calculations in classified data are based on

  1. the actual value of observations
  2. the upper class limits
  3. the lower class limits
  4. the class midpoints

d. Range is the

    1. difference between the largest and the smallest observation
    2. difference between the smallest and the largest distribution
    3. average of the largest and the smallest observations
    4. ratio of the largest to the smallest observation

Ans.

a. (1)The average of the upper class limit and the lower class limit.

Explanation: The class mid value lies half way between the lower and the upper class limits. It is calculated by dividing the sum of upper class limit and lower class limit by 2.

b. (2) Bivariate Distribution

Explanation: A frequency distribution where two variables are measured for a same set of items through cross classification is called bivariate or two way distribution.

c. (4) the class midpoints

Explanation: Class mid points are the most important values, as they are representatives of the class in case of classified continuous series data and are take for the use in further statistical calculations.

d. (1) difference between the largest and the smallest observation

Explanation: Range is equal to largest value minus the smallest value. i.e. R= L-S

Q.63 “The quick brown fox jumps over the lazy dog”.

Examine the above sentence carefully and note the numbers of letters in each word. Treating the number of letters as a variable, prepare a frequency array for this data.

Ans.

Frequency array for the letters of the sentence given in the question:

Letters

Number of Letters

a

1

b

1

c

1

d

1

e

3

f

1

g

1

h

2

i

1

j

1

k

1

l

1

m

1

n

1

o

4

p

1

q

1

r

2

s

1

t

2

u

2

v

1

w

1

x

1

y

1

z

1

Q.64 Can there be any advantage in classifying things? Explain with an example from your daily life.

Ans.

Classification is a technique of arranging collected data into different groups or classes that have some common characteristics to facilitate analysis and interpretation.

There are various advantages of classifying data which are as follows:

1. It presents the data in a simple form: Classification process eliminates unnecessary details and makes the mass of complex data simple, brief and logical so that a normal person can understand data at a glance. For Example: Data collected in population census is huge and fragmented. It is very difficult to draw any conclusion from these data. But when Data is classified according to some common characteristic e.g. age, occupation, etc. it can be easily understood.

2. Classification of data facilitates comparison: Classified data enable a person to do comparisons and draw inferences about facts. For Example: You are given marks obtained by 60 students and you have been told to compare their intelligence level. It is a difficult task, but, when we classify students into 1st, 2nd and 3rd divisions on the basis of marks obtained, it makes comparison easy.

3. Classification of data highlights points of similarity and dissimilarity. For example population data can be classified as Employed and Unemployed, educated and uneducated, male and female, etc.

4. Classification of data prepares the basis for tabulation and statistical study of data.

5. Classification helps in finding out cause effect relationship in the data if there is any in the data. For example: Data of unemployed workers can help in finding out whether unemployed workers are more in educated or uneducated population.

Q.65 What is a variable? Distinguish between a discrete and a continuous variable.

Ans.

A variable refers to the characteristics of the sample or population under study that can be expressed in numbers and are generally represented by the letters X, Y, Z. Each value of a variable is an observation. For example: weight of students, income of workers, number of members in a family etc.

Continuous variable are those variable which can take all possible values in a given range. For example: heights and weights of individuals, prices of commodities. They can take any value. Integral values (1,2,3,4…) fractional values (1/2, 2/3, 4/5 ….) or values which are not rational numbers (√2, √3 , √7 √8 … )

Height does not change from 5ft 2” to 5ft 3”. But it will pass all possible values between these. Thus the variable height is capable of manifesting in any conceivable value.

Discrete variables on the other hand can take only some particular values. For example: members of a household. We can have 1 member, 2 members, 3 members and so on. So it cannot have fractional values. The value will jump from 1 to 2, from 2 to 3 and so on.

But it does not mean that a discrete variable can take only whole numbers as values. We can have a variable ‘x’ which may take values like 1/2, 1/4, 1/8, 1/16 etc. provided there is a uniform difference from one variable to other variable. It means it jumps from one value to the other and cannot take values in between those two fractions.

Q.66 Explain the ‘exclusive’ and ‘inclusive’ methods used in classification of data.

Ans.

There are two methods of classifying data into class intervals.

1. Exclusive method: Under this method the upper limits of one class is the lower limits of the next class. In this way continuity of the data is maintained. E.g. class intervals are 0-5, 5-10, 10-15 and so on. Now, 5 is coming twice so is 10 and 15. So, the upper limit of the class is excluded, means if a student has obtained 5 marks he is not included in the first group i.e. 0-5, but in the second i.e. 5-10.

2. Inclusive method: Under this method upper limits of the class interval are also included in that class. The class interval will be made like 0-4, 5-9, 10–14 and so on. This does not exclude the upper class limit in a class interval. Both class limits upper and lower limits are parts of the class interval.

Q.67 Use the data in Table 3.2 that relate to monthly household expenditure (in ₹) on food of 50 households and

(i) Obtain the range of monthly household expenditure on food.

(ii) Divide the range into appropriate number of class intervals and obtain the frequency distribution of expenditure.

(iii) Find the number of households whose monthly expenditure on food is

(a) less than ₹ 2000

(b) more than ₹ 3000

(c) between ₹ 1500 and ₹ 2500

Table 3.2

Monthly Household Expenditure (in Rupees) on food of 50 households

1904 1559 3473 1735 2760
2041 1612 1753 1855 4439
5090 1085 1823 2346 1523
1211 1360 1110 2152 1183
1218 1315 1105 2628 2712
4248 1812 1264 1183 1171
1007 1180 1953 1137 2048
2025 1583 1324 2621 3676
1397 1832 1962 2177 2575
1293 1365 1146 3222 1396

Ans.

Table 3.2

Monthly Household Expenditure (in Rupees) on food of 50 households

1904 1559 3473 1735 2760
2041 1612 1753 1855 4439
5090 1085 1823 2346 1523
1211 1360 1110 2152 1183
1218 1315 1105 2628 2712
4248 1812 1264 1183 1171
1007 1180 1953 1137 2048
2025 1583 1324 2621 3676
1397 1832 1962 2177 2575
1293 1365 1146 3222 1396

(i) The range of monthly household expenditure on food is obtained by subtracting the lowest expenditures from the highest expenditure i.e.

Range = 5090 – 1007

Range = ₹ 4083

(ii) Let, to class interval be equal to ₹ 500.

Dividing the range by class interval of we get, Number of classes = 4083/500 = 8.166 ~ 9 classes.

Now, these 9 classes will include all the given values of expenditures of households of food. We will prepare a continuous frequency distribution by exclusion method.

The frequency distribution of expenditure is as follows:

(iii)

a. the number of households whose monthly expenditure on food is less than ₹ 2000 is 20 + 13 = 33 households.

b. the number of households whose monthly expenditure on food is more than ₹ 3000 =2 +1+2+0+1 = 6 households.

c. the number of households whose monthly expenditure on food is between ₹ 1500 and ₹ 2500 is 13 + 6 = 19 households.

Q.68 In a city 45 families were surveyed for the number of cell phones they used. Prepare a frequency array based on their replies as recorded below.

1

3

2

2

2

2

1

2

1

2

2

3

3

3

3

3

3

2

3

2

2

6

1

6

2

1

5

1

5

3

2

4

2

7

4

2

4

3

4

2

0

3

1

4

3

Ans.

The frequency array of cell phones used by 45 families is as follows:

Thus from the above frequency array we can conclude that, out of 45 families, one family is not using any cell phone, 7 families are using 1 cell phone, 15 families are using 2 cell phones, 12 families are using 3 cell phones, 5 families are using 4 cell phones, 2 families are using 5 cell phones, 2 are using 6 cell phones and only 1 family is using 7 cell phones.

Q.69 What is ‘loss of information’ in classified data?

Ans.

Once the data is grouped into classes, an individual observation has no significance in further statistical calculations. For example, we have a class interval as 20-30. There are 6 observations in it i.e. 24, 25, 25, 27, 22, 28. When that data is grouped,the class 20-30 will show 6 frequencies. All values in this class will be assumed to be equal to the mid-value or the class mark. i.e. 25. Further, calculations are based on 25 only.

Thus, the use of class mark instead of actual values involves considerable loss of information. While classification summarises the data it does not show the details.

Q.70 Do you agree that classified data is better than raw data? why?

Ans.

Yes, classified data is better than raw data.

After data is collected the next task is to arrange it in such a way that becomes easy to handle and convenient for further analyses.

Classification means arranging things in an appropriate order and putting them in some homogeneous groups or classes. For example in a library, books are kept in some order, according to the subjects or authors, or in alphabetical order.

Classified data is certainly better than unclassified data for many reasons:

1. Classification saves our time by making it easier to find an item. If we want to find the highest mark of a student from marks of 1000 students we will have to go through the whole unclassified data. It would be a very tedious task. But if we had it classified under different classes in ascending or descending order we could easily find it.

2. Classification compresses data into groups of similar types. So it is easier to look at it and make comparisons with the other groups.

3. Proper organisation is needed before any further systematic statistical analyses can be undertaken.

Q.71 Distinguish between univariate and bivariate frequency distribution.

Ans.

The frequency distribution of a single variable is called Univariate distribution. If we collect figures about marks in mathematics of 100 students, it is a univariate distribution.

A bivariate frequency distribution is a frequency distribution of two variables. If we have data of students’ heights and weights in the same table that is bivariate distribution. The values of heights are classed in columns and values of weights are classed in rows. Each cell shows the frequency of the corresponding row and column. For example 3 children are 6 feet tall and weigh 55 kg.

Q.72 Prepare a frequency distribution by inclusive method taking class interval of 7 from the following data.

28 17 15 22 29 21 23 27 18 12
7 2 9 4 1 8 3 10 5 20
16 12 8 4 33 27 21 15 3 36
27 18 9 2 4 6 32 31 29 18
14 13 15 11 9 7 1 5 37 32
28 26 24 20 19 25 19 20 6 9

Ans.

Frequency distribution using (inclusive method and taking the class interval of 7 is as follows:

Q.73 Bar diagram is a

    1. one-dimensional diagram
    2. two-dimensional diagram
    3. diagram with no dimension
    4. none of the above

Ans.

(i) one-dimensional diagram

Explanation: Bar diagrams are called one-dimensional because it is only the height of the the bars that matters not the width.

Q.74 Data represented through a histogram can help in finding graphically the

    1. mean
    2. mode
    3. median
    4. all the above

Ans.

(ii) mode

Explanation: Histogram consists of a set of rectangles adjacent to each other in which each rectangle represents the class interval with the frequency.

Q.75 Ogives can be helpful in locating graphically the

    1. mean
    2. mode
    3. median
    4. none of the above

Ans.

(iii) median

Explanation: Ogive or cumulative frequency curve are constructed by plotting cumulative frequency data on the graph paper, in the form of a smooth curve. The intersection point of two ogive curves i.e. ‘less than’ and ‘more than’ ogive curves is called median.

Q.76 Data represented through arithmetic line graph help in understanding

    1. Long term trend
    2. Cyclicity in data
    3. seasonality in data
    4. all the above

Ans.

(i) long term trend

Explanation: Arithmetic line graphs are also called time series graph. These graphs are used to present the information which is given over a period of time. Here time is plotted on X axis and value of variable on Y axis.

Q.77 Median of a frequency distribution cannot be known from the ogives. (True/False)

Ans.

Median of a frequency distribution cannot be known from the ogives. – False

Reason: The intersection point of two ogive curves i.e. ‘less than’ and ‘more than’ ogive curves gives the value of median.

Q.78 What kinds of diagrams are more effective in representing the following?

    1. Monthly rainfall in a year.
    2. Composition of the population of Delhi by religion
    3. Components of cost in a factory

Ans.

i. Simple bar diagram

Reason: Monthly rainfall in a year is a discrete variable hence it can be best represented by a simple bar diagram with rainfall on Y-axis and months in the year on the X-axis.

ii. Component bar diagram

Reason: Component bar diagrams or sub-divided bar diagrams are to be used, if the total values of the given data are divided into various sub-parts or components. In this case the total population of Delhi can be sub-divided in terms of religion and can be presented by a component bar diagram.

iii. Pie diagram or pie chart

Reason: A pie diagram or circular diagram also know as pie chart represents a circle whose area is proportionally divided into the components which it represents. The components of cost in a factory can be easily represented by the pie chart. The circle will represent the total cost and different sectors or portions of the circle will represent the various components of cost according to their respective percentage share in total cost.

Q.79 Suppose you want to emphasise the increase in the share of urban non-workers and lower level of urbanisation in India as shown in Example 4.2. How would you do it in the tabular form?

Ans.

To showcase the increase in the share of urban non-workers and lower level of urbanisation in India, the data given in the example 4.2 (Case 2, page no.41) can be tabulated as follows:

Urban areas Rural areas Total
Worker population 9 31 40
Non-workers population 19 43 62
Total 28 74 102

From the above table we can see that out of total population of 102 crore, 74 crore people are residing in rural India and only 28 crore people are living in urban areas. This clearly indicates the low levels of urbanisation in the country.

Higher share of urban non-workers can be indicated from the fact that 19 crores people out of total 28 crore population in urban areas are non-working (i.e. 67.85% urban population is non-working), as compared to rural India where 43 crore out of 74 crore people are non-working (i.e. 58.1% of rural population is non-working).

Q.80 How does the procedure of drawing a histogram differ when class intervals are unequal in comparison to equal class intervals in a frequency table?

Ans.

Histogram consists of a set of rectangles adjacent to each other in which each rectangle represents the class interval with the frequency.

If the class intervals are of equal width, the area of the rectangles is proportional to their respective frequencies.

However, for some type of data, it is suitable, at times essential, to use varying width of class intervals.

For graphical illustration of such data, height for area of a rectangle is the quotient of height (i.e. frequency) and base (i.e. width of the class interval).

When intervals are equal, i.e. when all rectangles have the same base, area can easily be represented by the frequency of any interval for purposes of comparison.

But when class intervals are unequal i.e., bases vary in width, the frequencies (height of the rectangle) must be adjusted by calculating frequency density i.e. (class frequency divided by the width of the class interval) to yield comparable measurements.

Q.81 The Indian Sugar Mills Association reported that, ‘Sugar production during the first fortnight of December 2001 was about 3,87,000 tonnes as against 3,78,000 tonnes during the same fortnight last year (2000). The off-take of sugar from factories during the first fortnight of December 2001 was 2,83,000 tonnes for internal consumption and 41,000 tonnes for export as against 1,54,000 tonnes for internal consumption and nil for exports during the same fortnight last season.’

  1. Present the data in tabular form.
  1. Suppose you were to present these data in diagrammatic form which of the diagrams would you use and why?
    1. Present these data diagrammatically.

Ans.

i. Data in the tabular form is as follows:

Sugar production and off-take in India
Year Production (in tones) Off-take for internal consumption

(in tones)

Off take for Exports

(in tones)

December 2000 3,78,000 1,54,000 0
December 2001 3,87,000 2,83,000 41,000

Source: The Indian Sugar Mills Association Report

ii. To present this data in diagrammatic form we will use multiple bar diagrams, because they are used for comparing two sets of data for different classes or years effectively.

iii. Diagrammatic presentation of the data:

Q.82 The following table shows the estimated sectoral real growth rates (percentage change over the previous year) in GDP at factor cost.

Year Agriculture and allied sectors Industry Services
1994-95 5.0 9.2 7.0
1995-96 -0.9 11.8 10.3
1996-97 9.6 6.0 7.1
1997-98 -1.9 5.9 9.0
1998-99 7.2 4.0 8.3
1999-2000 0.8 6.9 8.2

Represent the data as multiple time series graphs.

Ans.

Q.83 Collect the price of five vegetables from your local market every day for a week. Calculate their correlation coefficients. Interpret the result.

Ans.

(Hypothetical example, answer may vary)

Correlation between prices of potato and tomato

Likewise we can calculate correlation between different pairs of vegetables or we can use multivariate correlation to find the relation between the vegetables.

Q.84 Measure the height of your classmates. Ask them the height of their bench mate. Calculate the correlation coefficient of these two variables. Interpret the result.

Ans.

(Hypothetical example, results may vary)

Correlation coefficient between the height of classmate and height of his bench mate are negatively related.

Q.85 Calculate the correlation coefficient between the height of fathers in inches (X) and their sons (Y)

X 65 66 57 67 68 69 70 72
Y 67 56 65 68 72 72 69 71

Ans.

There is a positive correlation between height of fathers and their sons.

Q.86 Calculate the correlation coefficient between X and Y and comment on their relationship

X -3 -2 -1 1 2 3
Y 9 4 1 1 4 9

Ans.

Correlation coefficient between X and Y is zero. This means that X and Y have no linear relation between them.

(They a have a non–linear relationship Y = X2)

Q.87 Calculate the correlation coefficient between X and Y and Comment on their relationship

X 1 3 4 5 7 8
Y 2 6 8 10 14 16

Ans.

Correlation coefficient between X and Y is +1. There is a perfect positive correlation between X and Y. If there is a change in X, then there is equi–proportional change in Y.

Q.88 The monthly per capita expenditure incurred by workers for an industrial centre during 1980 and 2005 on the following items are 75, 10, 5, 6 and 4 respectively. Prepare a weighted index number for cost of living for 2005 with 1980 as the base.

Items Price in 1980 Price in 2005
Food 100 200
Clothing 20 25
Fuel & lighting 15 20
House rent 30 40
Misc 35 65

Ans.

Q.89 An enquiry into the budgets of the middle class families in a certain city gave the following information:

Expenses on items Food Fuel Clothing Rent Misc.
35% 10% 20% 15% 20%
Price (in ₹) in 2004 1500 250 750 300 400
Price (in ₹) in 1995 1400 200 500 200 250

What is the cost of living index during the year 2004 as compared with 1995?

Ans.

Q.90 In a town, 25% of the persons earned more than ₹ 45000 whereas 75% earned more than ₹ 18000. Calculate the absolute and relative values of dispersion.

Ans.

Q.91 The yield of wheat and rice per acre for 10 districts of state is as under:

District 1 2 3 4 5 6 7 8 9 10
Wheat 12 10 15 19 21 16 18 9 25 10
Rice 22 29 12 23 18 15 12 34 18 12

Calculate for each crop,
1. Range
2. Q.D.
3. Mean deviation about Mean
4. Mean deviation about Median
5. Standard Deviation
6. Which crop has greater variation?
7. Compare the values of different measures of for each crop.

Ans.

First, we arrange the values in an ascending order.

Wheat

Wheat |d|(mean) |D|(median) d d2
9 6.5 6.5 –6.5 42.25
10 5.5 5.5 –5.5 30.25
10 5.5 5.5 –5.5 30.25
12 3.5 3.5 –3.5 12.25
15 0.5 0.5 –0.5 0.25
16 0.5 0.5 0.5 0.25
18 2.5 2.5 2.5 6.25
19 3.5 3.5 3.5 12.25
21 5.5 5.5 5.5 30.25
25 9.5 9.5 9.5 90.25
∑X = 155 ∑|d| = 43 ∑|D| = 43 ∑d2 =254.50

Rice

Rice |d|(mean) |D|(median) d d2
12 7.5 6 –7.5 56.25
12 7.5 6 –7.5 56.25
12 7.5 6 –7.5 56.25
15 4.5 3 –4.5 20.25
18 1.5 0 –1.5 2.25
18 1.5 0 –1.5 2.25
22 2.5 4 2.5 6.25
23 3.5 5 3.5 12.25
29 9.5 11 9.5 90.25
34 14.5 16 14.5 210.25
∑X = 195 ∑|d| = 57.5 ∑|D| = 57 ∑d2 = 512.50

1. Range

Range for wheat = L – S

Largest value = 25 Smallest value = 9

= 25 – 9 = 16

Range for rice = L – S

Largest value = 34 Smallest value = 12

= 34 – 12 = 22

2. Quartile Deviation

3. Mean Deviation about mean

Mean for wheat = ∑X/N = 155/10 = 15.5

Mean for rice = 195/10 = 19.5

Mean Deviation from Mean = ∑|d|/n

Substituting values from above tables in the formula

Mean Deviation from Mean for Wheat = 43/10 = 4.3

Mean Deviation from Mean for Rice = 57.5/10 = 5.75

4. Mean Deviation about median

Position of Median = N+1th/2 item = 11/2 item = 5.5 item

Median= 5th + 6th /2 item

Median for wheat = 15 +16 / 2 = 15.5

Median for rice = 18 +18 / 2 = 18

Mean Deviation from Median= ∑|D|/n

here, D = X– median

Substituting values from above tables in the formula

Mean Deviation for wheat = 43/10 = 4.3

Mean Deviation for rice = 57/10 = 5.7

5. Standard Deviation

6. After carefully comparing the values for different measure of dispersion for each crop we reach to the conclusion that Rice has greater variation than Wheat.

7. Comparing values of different measures:

Wheat Rice
Range 16 22
Quartile Deviation 4.5 5.5
Mean Deviation about mean 4.3 5.75
Mean deviation about median 4.3 5.7
Standard Deviation 5.04 7.15

Q.92 In the previous question, calculate the relative measures of variation and indicate the value which, in your opinion, is more reliable.

Ans.

Q.93 A batsman is to be selected for a cricket team. The choice is between X and Y on the basis of their scores in five previous tests which are:

X: 25 85 40 80 120

Y: 50 70 65 45 80

Which bats man should be selected if we want.

    1. A higher run getter, or
    2. A more reliable in the team?

Ans.

To select batsmen first we need to calculate mean and standard deviation.

Batsmen X

d

d2

Batsmen Y

d

d2

25

–45

2025

50

–12

144

85

15

225

70

8

64

40

–30

900

65

3

9

80

10

100

45

–17

289

120

50

2500

80

18

324

∑x = 350

∑ d2 = 5750

∑y = 310

∑ d2 = 830

Now, calculate coefficient of variation to check for reliability of the batsmen

Coefficient of variation for Batsmen X = 33.91/70 × 100 = 48.44%

Coefficient of variation for Batsmen Y = 12.88/62 × 100 = 20.77%

  1. Batsmen X has a greater average (70 runs) as compared to Batsmen Y (62 runs), so he shall be selected if we want a high run getter.
  2. But, if we want reliable batsmen for the team then Batsmen Y shall be selected as he has a lower coefficient of variation (20.77) as compared to batsmen X (48.44). This means that batsmen Y are very consistent with his performance.

Q.94 To check the quality of two brands of light bulbs, their life in burning hours was estimated as under for 100 bulbs of each brand.

Life

(in hrs)

No. of bulbs

Brand A

Brand B

0–50

15

2

50–100

20

8

100–150

18

60

150–200

25

25

200–250

22

5

100

100

    1. Which brand gives higher life?
    2. Which brand is more dependable?

Ans.

To check for the quality of two brands of light bulbs first we calculate the mean and standard deviation of both the brand’s.

  1. Brand B gives a higher life as average life time of bulb from brand B(136.5 hrs) is greater than Brand A(134.5 hrs).
  2. Brand B is more dependable because variation in Brand B (27.34%) is lesser than Brand A (51.15%). This means that the average lifetime of samples of bulb from Brand B only differ by 27.34%, but average lifetime of samples of bulb from Brand B differ by 51.15%.

Q.95 Average daily wage of 50 workers of a factory was ₹200 with a standard Deviation of ₹40. Each worker is given a raise of ₹20. What is the new average daily wage and standard deviation? Have the wages become more or less uniform?

Ans.

Average daily wage (Mean) = ₹ 200

Standard deviation = ₹ 40

Since, wage of each worker is increased by ₹ 20 this is a situation of change in origin. With change in origin value of Mean changes but it has no impact on value of Standard Deviation.

i.e New Standard Deviation = Old Standard Deviation = ₹ 40

Sum of total wages of workers = 40 × 200 = ₹ 10000

Conclusion,

The coefficient of variation has declined with increase in wage rate; this means that wages have become more uniform than earlier.

Q.96 If in the previous question, each worker is given a hike of 10% in wages. How are the Mean and Standard deviation values affected?

Ans.

If each worker is given a hike of 10%

Then, total sum of wages (∑X) 10000 increases by 10%.

New total sum of wages = 10000 + 1000 = 11000

New mean = Rs. 220

New standard deviation:

With 10% increase in wage,

New Standard Deviation= Old Standard Deviation + 10% of Old Standard Deviation

= 40 + (40*10%)

= 40+4 = 44

With the hike of 10% in wages, mean increases to 220 and standard deviation increases to 44.

Q.97 Calculate the Mean deviation using mean and Standard deviation for the following distribution.

Classes Frequencies
20–40 3
40–80 6
80–100 20
100–120 12
120–140 9
50

Ans.

First we calculate all the relevant elements for mean deviation from mean and standard deviation

Classes f m fm d = m – 94.8 |d| f|d| fd fd2
20–40 3 30 90 –64.8 64.8 194.4 –194.4 12597.12
40–80 6 60 360 –34.8 34.8 208.8 –208.8 7266.24
80–100 20 90 1800 –4.8 4.8 96 –96 460.8
100–120 12 110 1320 15.2 15.2 182.4 182.4 2772.48
120–140 9 130 1170 35.2 35.2 316.8 316.8 11151.36
50 ∑fm = 4740 ∑f|d| = 998.4 ∑fd2 = 34248

Q.98 The sum of 10 values is 100 and the sum of their squares is 1090. Find out the coefficient of variation.

Ans.

Q.99 If the arithmetic mean of the data given below is 28, find (a) the missing frequency, and (b) the median of the series:

Profit per retail shop (in ₹) 0-10 10 -20 20-30 30-40 40-50 50-60
Number of retail shops 12 18 27 17 6

Ans.

1. Let us take the missing frequency as x

Profit per retail shop (in ₹) (X) Number of retail shops (f) Mid values (m) fm
0-10 12 5 60
10-20 18 15 270
20-30 27 25 675
30-40 X 35 35x
40-50 17 45 765
50-60 6 55 330
∑f = 80 + x ∑Fm = 2100 + 35x

Mean = ∑fm/∑f Mean = 28
Substituting the values in the formula we get,

Thus, the missing value frequency is 20.

(b)

Profit per retail shop(in ₹) (X) Number of retail shops (f) Cumulative frequency (cf)
0-10 12 12
10-20 18 30
20-30 27 57
30-40 20 77
40-50 17 94
50-60 6 100
∑f = 100

Q.100 The following table gives the daily income of ten workers in a factory.

Find the arithmetic mean.

Workers A B C D E F G H I J
Daily Income (in ₹.) 120 150 180 200 250 300 220 350 370 260

Ans.

Q.101

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Following information pertains to the daily income of 150 families

Calculate the arithmetic mean.

Income (in ₹) Number of families
More than 75 150
.. 85 140
.. 95 115
.. 105 95
.. 115 70
.. 125 60
.. 135 40
.. 145 25

Ans.

Income (in Rs.) Number of families(f) Mid values
(x)
fx
75-85 10 80 800
85-95 25 90 2250
95-105 20 100 2000
105-115 25 110 2750
115-125 10 120 1200
125-135 20 130 2600
135-145 15 140 2100
145-155 25 150 3750
∑f = 150 ∑fx = 17450

Q.102 The size of land holdings of 380 families in a village is given below. Find the median size of land holdings.

Size of Land Holdings (in acres) Less than 100 200-300 300-200 100-200 400 and above
Number of families 40 89 148 64 39

Ans.

To calculate the mean size of holding, first calculate the cumulative frequency.

Size of land holdings (in acres)(X) Number of families Cumulative frequency
0-100 40 40
100-200 89 129
200-300 148 277
300-400 64 341
400-500 39 380
∑f = 380

Q.103 The following series relates to the daily income of workers employed in affirm. Compute (a) highest income of lowest 50% workers (b) minimum income earned by the top 25% workers and (c) maximum income earned by 25% workers.

Daily Income (in ₹) 10-14 15-19 20-24 25-29 30-34 35-39
Numbers of workers 5 10 15 20 10 5

Ans.

Daily income(in ₹) Class interval
(X)
Number of workers (f) Cumulative frequency

(cf)

10-14 9.5-14.5 5 5
15-19 14.5-19.5 10 15
20-24 19.5-24.5 15 30
25-29 24.5-29.5 20 50
30-34 29.5-34.5 10 60
35-39 34.5-39.5 5 65
∑f = 65

Q.104 The following table gives production yield in kg. per hectare of wheat of 150 farms in a village. Calculate the mean, median and mode values.

Production yield (kg. per hectare) 50-53 53-56 56-59 59-62 62-65 65-68 68-71 71-74 74-77
Number of farms 3 8 14 30 36 28 16 10 5

Ans.

To calculate mean, median and mode values

Production yield (kg per hectare) (X) Number of farms
(f)
Mid values

(m)

Cumulative

Frequency

(cf)

fm
50-53 3 51.5 3 154.5
53-56 8 54.5 11 436
56-59 14 57.5 25 805
59-62 30 60.5 55 1815
62-65 36 63.5 91 2286
65-68 28 66.5 119 1862
68-71 16 69.5 135 1112
71-74 10 72.5 145 725
74-77 5 75.5 150 377.5
∑f = 150 ∑fm = 9573

Q.105 Width of bars in a bar diagram need not to be equal. (True/False)

Ans.

Width of bars in a bar diagram need not to be equal. – False

Reason: In bar diagrams all bars needs to be of equal width and also at equal distance from each other.

Q.106 Width of rectangles in a histogram should essentially be equal. (True/False)

Ans.

Width of rectangles in a histogram should essentially be equal – False

Reason: Width of rectangles in a histogram may or may not be equal it depends on the width of class intervals.

Q.107 Histogram can only be formed with continuous classification of data.(True/False)

Ans.

Histogram can only be formed with continuous classification of data. – True

Reason: A histogram is never drawn for a discrete variable/data. If the classes are not continuous they are first converted into continuous classes and then histogram is drawn.

Q.108 Histogram and column diagram are the same method of presentation of data.(True/False)

Ans.

Histogram and column diagram are the same method of presentation of data. – False

Reason: A histogram is a two dimensional diagram with set of rectangles with no spacing in between. On the other hand column (bar) diagram is a one-dimensional diagram with equal space between every Column (bar). Histogram is never drawn for a discrete variable/data whereas column diagram can be drawn for discrete variable/ data.

Q.109 Mode of a frequency distribution can be known graphically with the help of a histogram.(True/False)

Ans.

Mode of a frequency distribution can be known graphically with the help of a histogram. – True

Reason: The highest rectangle of the histogram gives the value of mode of the frequency distribution

Q.110 The monthly expenditure (Rs.) of a family on some important items and the Goods and Services Tax (GST) rates applicable to these items is as follows:

Item Monthly Expense(Rs) GST Rate %
Cereals 1500 0
Eggs 250 0
Fish, Meat 250 0
Medicines 50 5
Biogas 50 5
Transport 100 5
Butter 50 12
Babool 10 12
Tomato Ketchup 40 12
Biscuits 75 18
Cakes, Pastries 25 18
Branded Garments 100 18
Vacuum Cleaner, Car 1000 28

Calculate the average tax rate as far as this family is concerned.

Ans.

The average tax rate (Simple) for this family is = (0+0+0+5+5+5+12+12+12+18+18+18+28)/13 = 133/13= 10.23 %

The average tax rate (weighted) for this family is = (0×1500 +0×250 +0×250 +5×50 +5×50 +5×100 +12×50 +12×10+12×40+18×75+18×25+18×100+28×1000) / (1500+ 250+250 +50 +50+100+50+10+40+75+25+100+1000)= 33800/3500= 9.657 %

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