Construction of a multiple bar diagram

Step 1: Choose a scale. The largest value is 180 quintals, so take $1\text{ cm} = 40$ quintals on the Y-axis (180 quintals = 4.5 cm).

Step 2: Find the bar heights.

Step 3: Draw the diagram.

1. Draw the X-axis (years) and Y-axis (production in quintals) and mark the Y-axis as 0, 40, 80, 120, 160, 200.
2. For each year draw a group of three adjacent bars of equal width (Paddy, Maize, Wheat), using different shading for each crop. Leave a gap between one year's group and the next.
3. Give a title, e.g. "Production of paddy, maize and wheat, 2075-2077 B.S.", and an index showing the shading of each crop.

Rough sketch (each row = 20 quintals; bars in each group are Paddy, Maize, Wheat):

 180 |                                 ██
 160 |                                 ██
 140 |                   ██            ██
 120 |                   ██            ██  ██
 100 |     ██            ██  ██    ██  ██  ██
  80 |     ██  ██    ██  ██  ██    ██  ██  ██
  60 | ██  ██  ██    ██  ██  ██    ██  ██  ██
  40 | ██  ██  ██    ██  ██  ██    ██  ██  ██
  20 | ██  ██  ██    ██  ██  ██    ██  ██  ██
   0 +------------------------------------------
           2075          2076          2077

Interpretation: Production of all three crops rose every year, and maize had the highest production in each year.

Working table (base year 2020 = 100)

Laspeyre's price index (base-year quantities as weights)

$P_L = \frac{\sum p_1q_0}{\sum p_0q_0} \times 100 = \frac{1475}{1100} \times 100 = 134.09$

Paasche's price index (current-year quantities as weights)

$P_P = \frac{\sum p_1q_1}{\sum p_0q_1} \times 100 = \frac{2150}{1600} \times 100 = 134.38$

Conclusion: Laspeyre's index is 134.09 and Paasche's index is 134.38. The general price level in 2024 was about 34% higher than in 2020.

Working table (base year 2081 = 100)

Laspeyre's price index (base-year quantities as weights)

$P_L = \frac{\sum p_1q_0}{\sum p_0q_0} \times 100 = \frac{108}{82} \times 100 = 131.71$

Paasche's price index (current-year quantities as weights)

$P_P = \frac{\sum p_1q_1}{\sum p_0q_1} \times 100 = \frac{170}{129} \times 100 = 131.78$

Conclusion: Laspeyre's index is 131.71 and Paasche's index is 131.78. The general price level in 2083 was about 31.7% to 31.8% higher than in 2081.

Classification of data means arranging collected data into groups or classes according to their similarities and characteristics. The main methods are:

1. Geographical (spatial) classification: Data are classified according to place or region. Example: population of Nepal classified by province (Koshi, Madhesh, Bagmati, Gandaki, Lumbini, Karnali, Sudurpashchim).

2. Chronological (temporal) classification: Data are classified according to time such as year, month or period. Example: Nepal's GDP or export figures for the fiscal years 2078/79, 2079/80 and 2080/81.

3. Qualitative classification: Data are classified according to attributes or qualities that cannot be measured numerically, such as sex, religion, literacy or marital status. Example: population classified as male and female, or literate and illiterate.

4. Quantitative classification: Data are classified according to measurable numerical values such as age, income, height or marks. Example: families classified into income groups of below Rs 20,000, Rs 20,000-50,000 and above Rs 50,000.

5. Mixed classification: When data are classified by two or more of the above bases together, e.g. paddy production by district over several years (place and time).

Thus, data may be classified geographically, chronologically, qualitatively or quantitatively, depending on the nature of the data and the purpose of the study.

Chronological classification and spatial classification.

Construction of a multiple bar diagram

Step 1: Choose a scale. The largest value is 50 quintals, so take $1\text{ cm} = 10$ quintals on the Y-axis.

Step 2: Find the bar heights.

Step 3: Draw the diagram.

1. Draw the X-axis (years) and Y-axis (production in quintals), marking the Y-axis as 0, 10, 20, 30, 40, 50.
2. For each year draw three adjacent bars of equal width for Paddy, Maize and Wheat, shading each crop differently. Leave a gap between the groups of different years.
3. Write a title, e.g. "Production of paddy, maize and wheat, 2078-2080 B.S.", and an index showing the shading of each crop.

Rough sketch (each row = 10 quintals; bars in each group are Paddy, Maize, Wheat):

  50 | ██                              ██
  40 | ██  ██                ██    ██  ██
  30 | ██  ██  ██    ██      ██    ██  ██
  20 | ██  ██  ██    ██  ██  ██    ██  ██  ██
  10 | ██  ██  ██    ██  ██  ██    ██  ██  ██
   0 +------------------------------------------
           2078          2079          2080

Let me verify the calculations.

Given data:

Laspeyere's price index:

$P_{01}^{L}=\frac{\Sigma p_1q_0}{\Sigma p_0q_0}\times 100 = \frac{1200}{1900}\times 100 = 63.16$

Paasche's price index:

$P_{01}^{P}=\frac{\Sigma p_1q_1}{\Sigma p_0q_1}\times 100 = \frac{1575}{2300}\times 100 = 68.48$

All arithmetic checks out. The table values, formulas, and interpretation are all correct and complete.

OK

Given: Base year (2023) = 0, Current year (2024) = 1

Laspeyre’s Method:

$P_{01}=\frac{\Sigma P_1Q_0}{\Sigma P_0Q_0}\times 100$

$P_{01}=\frac{5700}{4700}\times 100 = 121.28$

Paasche’s Method:

$P_{01}=\frac{\Sigma P_1Q_1}{\Sigma P_0Q_1}\times 100$

$P_{01}=\frac{5900}{4900}\times 100 = 120.41$

Conclusion:

• Laspeyre’s price index = 121.28
• Paasche’s price index = 120.41

This shows that the general price level in 2024 increased by about 21.28% according to Laspeyre’s method and about 20.41% according to Paasche’s method compared to the base year 2023.

The two main methods of data collection are:

1. Primary method – data collected first-hand by the investigator through surveys, interviews, observation, etc.
2. Secondary method – data obtained from already published or recorded sources such as books, reports, and government publications.

The limitations of statistics are as follows:

1. Study of quantitative facts only: Statistics deals only with facts that can be expressed in numbers. Qualitative facts such as honesty, beauty, intelligence or happiness cannot be measured directly, so they fall outside its scope.

2. Does not study individuals: Statistics studies groups or aggregates, not individual units. For example, the average income of a country does not tell us the income of any particular person.

3. Results are true only on average: Statistical conclusions are true only in general, not in every individual case. The average rainfall of a place may not match the rainfall of any single year.

4. Lack of uniformity in data: Data collected by different persons, at different places or times, may not be uniform. Figures that are not uniform cannot be compared meaningfully.

5. Misuse is possible: Statistics can be manipulated by collecting biased samples, using wrong averages or presenting figures in a misleading way. As it is said, "Statistics can prove anything," so results may be wrong if used carelessly.

6. Requires expert knowledge: Collection, classification, tabulation and interpretation of data need special skill and experience. Without an expert, the conclusions drawn may be incorrect.

7. Does not reveal the whole truth: Statistics gives only one aspect of a problem at a time. It cannot explain the causes behind a fact, so it must be used along with other methods of study.

Given: Grouped frequency distribution of marks of 43 students.

Mean

$\bar{x}=\frac{\Sigma fm}{N}=\frac{1990}{43}=46.28$

Mean = 46.28 marks

Mode

Highest frequency = 16, so the modal class is 40 – 60.

Here, $L = 40$, $f_1 = 16$, $f_0 = 10$, $f_2 = 8$, $h = 20$.

$\text{Mode}=L+\frac{f_1-f_0}{2f_1-f_0-f_2}\times h$

$=40+\frac{16-10}{2(16)-10-8}\times 20$

$=40+\frac{6}{14}\times 20=40+8.57=48.57$

Mode = 48.57 marks

Answer: Mean = 46.28 marks, Mode = 48.57 marks.

The method of selecting a representative part of the population for study.

Statistics is the science that deals with the collection, organisation, presentation, analysis and interpretation of numerical facts or data. In the plural sense, statistics means numerical data themselves (such as figures of population, income or production); in the singular sense, it means the scientific method of handling such data. It is used in economics, business, government and research to draw meaningful conclusions from facts.

Four limitations of statistics:

1. Statistics deals only with quantitative facts: Statistics studies only those facts which can be expressed in numbers. Qualitative facts such as honesty, beauty, intelligence or loyalty cannot be measured directly, so they fall outside its scope unless converted into numerical form.

2. Statistics does not study individuals: Statistics studies groups or aggregates, not a single individual. For example, the average income of a group can be found, but statistics cannot describe the income or behaviour of one particular person. Conclusions apply to the mass, not to the individual.

3. Results are true only on average: Statistical conclusions are true only in general or on average. They may not hold true for every single case. For instance, the average height of Nepali men may be 5 feet 5 inches, but many men may be taller or shorter than this average.

4. Statistics can be misused: Statistical data and methods can be deliberately misused by dishonest persons to draw wrong conclusions. Figures can be manipulated, manipulated samples can be selected, or results can be presented in a misleading way, so statistics can prove anything if handled carelessly or with bad intention.

Other limitations include: results are affected by errors in data collection, statistics requires expert knowledge to handle properly, and it does not reveal the whole story behind the figures.

Data: Costs under five headings — Wages Rs 600, Cement Rs 800, Iron Rs 400, Wood Rs 300, Miscellaneous Rs 150.

Total cost $= 600 + 800 + 400 + 300 + 150 = \text{Rs } 2250$

Angle of each sector $= \dfrac{\text{Value of component}}{\text{Total value}} \times 360^\circ$

Construction of the pie-chart:

1. Draw a circle of convenient radius and mark its centre O.
2. Draw a horizontal radius as the base line and mark it as the starting line.
3. With the help of a protractor, mark the first sector of $96^\circ$ for Wages from the base line.
4. From the end of this sector, mark the next sector of $128^\circ$ for Cement.
5. Continue in the same way, marking $64^\circ$ for Iron and $48^\circ$ for Wood.
6. The remaining sector of $24^\circ$ is for Miscellaneous, which completes the circle ($96 + 128 + 64 + 48 + 24 = 360^\circ$).
7. Shade or label each sector clearly and give the chart a suitable title, e.g. "Pie-chart showing cost of production under different headings."

Note: The largest sector is Cement ($128^\circ$) and the smallest is Miscellaneous ($24^\circ$), so the chart shows that cement alone accounts for more than one-third of the total cost.

Given table:

1. Median

$\frac{N}{2} = \frac{160}{2} = 80$

The first cumulative frequency greater than 80 is 120, so the median class is 30–40, with $L = 30$, $i = 10$, $f = 50$, $cf = 70$.

$\text{Median} = L + \frac{\frac{N}{2} - cf}{f} \times i = 30 + \frac{80 - 70}{50} \times 10 = 30 + 2 = 32$

Median = Rs 32 per hour

2. Mode

The highest frequency is 50, so the modal class is 30–40, with $L = 30$, $i = 10$, $f_1 = 50$, $f_0 = 40$, $f_2 = 30$.

$\text{Mode} = L + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times i = 30 + \frac{50 - 40}{100 - 40 - 30} \times 10 = 30 + \frac{10}{30} \times 10 = 33.33$

Mode = Rs 33.33 per hour

Answer: Median wage = Rs 32 per hour; modal wage = Rs 33.33 per hour.

Primary and secondary sources.

Primary data is collected first-hand by the researcher (surveys, interviews, observation); secondary data is already collected and published by others (books, reports, government publications).

Construction of a pie-chart

Step 1: Total expenditure $= 1300 + 700 + 1000 + 900 + 600 = \text{Rs } 4500$

Step 2: Angle of each sector $= \dfrac{\text{Expenditure of the head}}{\text{Total expenditure}} \times 360^\circ$

Check: $104 + 56 + 80 + 72 + 48 = 360^\circ$.

Step 3: Draw the chart.

1. Draw a circle of convenient radius and mark its centre $O$; draw a horizontal radius as the base line.
2. With a protractor mark $104^\circ$ from the base line for Food, then $56^\circ$ for Health, $80^\circ$ for Cloth, $72^\circ$ for Education and the remaining $48^\circ$ for Others.
3. Shade or label each sector and give the chart a title, e.g. "Pie-chart showing household expenditure".

The largest sector is Food ($104^\circ$), so the family spends the biggest share of its budget on food.

Given data:

1. Calculation of Median

$N = 44,\quad \frac{N}{2}=22$

From the cumulative frequency column, the value 22 lies in the class 30–40.
So, the median class is 30–40.

For the median class:

• Lower limit, $L = 30$
• Frequency of median class, $f = 14$
• Cumulative frequency of preceding class, $cf = 15$
• Class size, $h = 10$

Formula:

$M = L + \frac{\frac{N}{2}-cf}{f}\times h$

Substituting the values:

$M = 30 + \frac{22-15}{14}\times 10$

$M = 30 + \frac{7}{14}\times 10$

$M = 30 + 5$

$\boxed{M = 35}$

So, the median value is 35.

2. Calculation of Mode

The highest frequency is 14, which lies in the class 30–40.
So, the modal class is 30–40.

For the modal class:

• Lower limit, $L = 30$
• Frequency of modal class, $f_1 = 14$
• Frequency of preceding class, $f_0 = 11$
• Frequency of succeeding class, $f_2 = 9$
• Class size, $h = 10$

Formula:

$Z = L + \frac{f_1-f_0}{2f_1-f_0-f_2}\times h$

Substituting the values:

$Z = 30 + \frac{14-11}{2(14)-11-9}\times 10$

$Z = 30 + \frac{3}{28-20}\times 10$

$Z = 30 + \frac{3}{8}\times 10$

$Z = 30 + 3.75$

$\boxed{Z = 33.75}$

So, the mode value is 33.75.

Answer:
Median = 35
Mode = 33.75

Working table (base year 2023 = 100)

Laspeyre's price index (base-year quantities as weights)

$P_L = \frac{\sum p_1q_0}{\sum p_0q_0} \times 100 = \frac{1180}{1010} \times 100 = 116.83$

Paasche's price index (current-year quantities as weights)

$P_P = \frac{\sum p_1q_1}{\sum p_0q_1} \times 100 = \frac{1330}{1140} \times 100 = 116.67$

Conclusion: Laspeyre's index is 116.83 and Paasche's index is 116.67. Prices in 2024 were about 16.8% (Laspeyre) and 16.7% (Paasche) higher than in 2023.

Statistics is the science of collecting, organising, presenting, analysing and interpreting numerical data. Its main functions are:

1. Collection and organisation of data: Statistics provides systematic methods of collecting facts and figures and arranging them in tables or classes so that they become easy to handle and understand.

2. Presentation of data: It presents data in condensed and attractive forms such as tables, graphs, diagrams and charts, which makes comparison and quick understanding possible.

3. Analysis and interpretation of data: It uses measures like averages, dispersion, correlation and index numbers to analyse data and draw meaningful conclusions from them.

4. Comparison of facts: It helps to compare data of different periods, places or groups (for example, this year's price level with last year's) so that trends and changes can be identified.

5. Formulation of plans and policies: Statistical data are the base for making economic plans, budgets and government policies, since targets and achievements can be measured only with statistics.

Thus, statistics is an indispensable tool in every field where numerical facts are used for understanding and action.

Pie-chart of prices of vegetables

Total price $= 90 + 40 + 80 + 60 + 30 = \text{Rs } 300$

Angle of each sector $= \dfrac{\text{Price}}{300} \times 360^\circ$

Steps to draw the pie-chart

1. Draw a circle of suitable radius, mark its centre $O$ and draw a horizontal radius as the base line.
2. With a protractor mark $108^\circ$ from the base line for Mushroom, then $48^\circ$ for Tomato, $96^\circ$ for Bean, $72^\circ$ for Brinjal and the remaining $36^\circ$ for Potato.
3. Shade or label each sector with its name and angle (or price).
4. Give the chart a title: "Prices of vegetables per kg".

Conclusion: Mushroom has the largest sector ($108^\circ$) and potato the smallest ($36^\circ$).

Mean

$\bar{x} = \frac{\Sigma fx}{N} = \frac{1230}{40} = 30.75$

Median

$\frac{N}{2} = \frac{40}{2} = 20$

The cumulative frequency first exceeds 20 at 30, so the median class is 30–40, with $L = 30$, $cf = 18$, $f = 12$, $h = 10$.

$\text{Median} = L + \frac{\frac{N}{2} - cf}{f} \times h = 30 + \frac{20 - 18}{12} \times 10 = 30 + 1.67 = 31.67$

Answer: Mean = 30.75; Median = 31.67.

Working table (base year 2022 = 100)

Laspeyre's price index (base-year quantities as weights)

$P_L = \frac{\sum p_1q_0}{\sum p_0q_0} \times 100 = \frac{229}{155} \times 100 = 147.74$

Paasche's price index (current-year quantities as weights)

$P_P = \frac{\sum p_1q_1}{\sum p_0q_1} \times 100 = \frac{596}{360} \times 100 = 165.56$

Conclusion: Laspeyre's index is 147.74 and Paasche's index is 165.56. Prices in 2024 were about 47.7% (Laspeyre) and 65.6% (Paasche) higher than in 2022. Paasche's index is larger because quantities shifted toward goods whose prices rose most.

Statistics helps in presenting economic data in a simple and comparable form, and it helps the government in formulating and evaluating economic policies.

Two importance of statistics in economics:

1. Presentation of economic data — Statistics collects, classifies and presents complex economic facts (production, price, income, employment) in a simple and understandable form.
2. Formulation of economic policies — Statistics provides the quantitative basis for planning, budgeting and evaluating the success of government economic policies.

Uses of Classification of Data

Classification is the process of arranging collected raw data into groups or classes according to their similarities, differences, and common characteristics. It is the first and most important step in the tabulation and analysis of data. Its main uses are described below:

1. Simplification of data: Raw data collected from the field is large, scattered and confusing. Classification arranges it into a small number of homogeneous groups, which makes the data simple, brief and easy to understand.

2. Facilitates comparison: When data is divided into similar groups, the values of one group can be easily compared with those of another group, or with the data of another time or place. For example, the income of farmers can be compared with that of workers only when the data is classified.

3. Basis for tabulation: Classification is the foundation of tabulation. Only after the data is classified into classes or groups can it be presented in tables, rows and columns in a systematic manner.

4. Helps in analysis and interpretation: Classified data makes it possible to calculate averages, percentages, ratios and measures of dispersion. It also helps to bring out the real nature, trend and relationship hidden in the raw figures.

5. Saves time and labour: Since the data is arranged in a systematic order, the researcher does not have to search through the whole mass of figures. This saves a lot of time, money and effort in research work.

6. Useful for policy making and planning: Government and other organisations use classified data to know the real situation of the country, such as the distribution of population, income, employment and production. On this basis, they formulate suitable plans and policies.

7. Makes presentation attractive and meaningful: Classified data can be easily presented through tables, graphs, diagrams and charts, which makes the information more attractive, clear and meaningful to the reader.

Conclusion: In short, classification converts disordered raw data into an orderly and meaningful form. Without classification, data remains a mere collection of figures and cannot be used for comparison, analysis or decision making. Hence, classification is the foundation of all statistical work.

Given:

Total frequency:

$N = 5+7+4+6+3 = 25$

Calculation of Mean

Formula:

$\bar{x}=\frac{\sum fm}{N}$

Substituting:

$\bar{x}=\frac{575}{25}=23$

So, the mean is 23.

Calculation of Median

Cumulative frequency table:

Position of median:

$\frac{N}{2}=\frac{25}{2}=12.5$

The cumulative frequency just greater than 12.5 is 16, so the median class is 20–30.

For the median class:

$l=20,\quad cf=12,\quad f=4,\quad h=10$

Formula:

$M=l+\frac{\frac{N}{2}-cf}{f}\times h$

Substituting:

$M=20+\frac{12.5-12}{4}\times 10$

$M=20+\frac{0.5}{4}\times 10$

$M=20+1.25=21.25$

So, the median is 21.25.

Answer: Mean = 23, Median = 21.25

Importance of Statistics in Economics (5 marks)

Statistics is the science of collecting, organising, presenting, analysing, and interpreting numerical data. In economics it is indispensable because economic problems are largely quantitative in nature. Its main importance can be clarified as follows:

1. Presentation of economic facts in numerical form: Statistics expresses economic facts such as production, consumption, income, price, and employment in exact numbers. This removes vagueness and makes the size and extent of an economic problem clear. For example, instead of saying "prices have risen very much," statistics tells us that the price index rose from 120 to 150, i.e. by 25 per cent.

2. Basis for formulating economic policies: The government needs reliable data to plan and frame policies. Budgets, five-year plans, tax rates, subsidy levels, and minimum wage decisions are all based on statistical information about national income, population, poverty, and unemployment. Without such data, policies would be guesswork.

3. Study of economic relationships: Statistics helps to establish and measure relationships between economic variables, such as the relation between price and demand, income and saving, or investment and employment. Tools like correlation and regression show how a change in one variable affects another.

4. Comparison over time and place: Through index numbers and time-series data, statistics allows comparison of economic conditions between different years (e.g. GDP of 2015 vs 2024) and between different countries or regions. This helps in judging whether the economy is progressing or declining.

5. Testing and verifying economic theories: Economic laws and theories are tested against real data using statistical methods. A theory is accepted only if it is supported by facts. For instance, the law of demand is verified by actual price–quantity data.

6. Forecasting and planning: On the basis of past and present data, statistics helps to forecast future trends in population, demand, production, and prices. Such forecasts are essential for development planning and for solving problems like inflation, unemployment, and poverty.

Conclusion: Thus, statistics is an indispensable tool of economic analysis. It supplies the raw material (data), the methods of analysis, and the basis for decision-making. As such, it is rightly regarded as the backbone of economics.

Data: Food = Rs 1000, Medicine = Rs 600, Education = Rs 800, Cloths = Rs 700, Saving = Rs 500.

Total expenditure $= 1000 + 600 + 800 + 700 + 500 = \text{Rs } 3600$

Calculation of angles: angle of each sector $= \dfrac{\text{amount}}{\text{total}} \times 360^\circ$

Steps to draw the pie-chart:

1. Draw a circle of suitable radius and mark its centre.
2. Draw a horizontal radius from the centre towards the right as the base line.
3. With a protractor, mark $100^\circ$ from the base line and draw the sector for Food.
4. From the new radius, mark in order $60^\circ$ (Medicine), $80^\circ$ (Education), $70^\circ$ (Cloths) and $50^\circ$ (Saving).
5. The last sector ends exactly at the base line, since $100^\circ + 60^\circ + 80^\circ + 70^\circ + 50^\circ = 360^\circ$.
6. Label each sector with its heading and angle (or percentage) and give the chart a title.

Food has the largest sector ($100^\circ$) and saving the smallest ($50^\circ$).

Given:

1. Mean

2. Median

Here,

The cumulative frequency just greater than 25 is 30, so the median class is 30–40.

For median class:

• Lower limit, $l = 30$
• Preceding cumulative frequency, $cf = 15$
• Frequency of median class, $f = 15$
• Class interval, $h = 10$

Therefore, the mean marks of 50 students is 37.6 and the median marks is 36.67.

Primary data is more reliable because the researcher collects it first-hand, directly from the source, for the specific purpose of the study. The researcher knows how it was collected and can control its accuracy, whereas secondary data was gathered earlier by others for other purposes and may be outdated or biased.

The class intervals are continuous and of equal width (5), so the histogram can be drawn directly.

Histogram

1. Take wages (Rs) on the X-axis with a scale such as 1 cm = Rs 5, and the number of workers on the Y-axis with 1 cm = 2 workers.
2. Mark the class limits 5, 10, 15, 20, 25, 30, 35 on the X-axis.
3. On each class, draw a rectangle with no gaps between them. The heights are:

The tallest rectangle is over 15-20 (10 workers).

Frequency polygon (drawn on the same histogram)

1. Find the mid-point of each class: $\dfrac{\text{lower} + \text{upper}}{2}$.

2. Mark a point at the top of each rectangle above its mid-point: (7.5, 3), (12.5, 5), (17.5, 10), (22.5, 8), (27.5, 4), (32.5, 2).
3. Add an empty class on each end so the polygon touches the X-axis: the points (2.5, 0) and (37.5, 0).
4. Join all the points with straight lines: (2.5, 0), (7.5, 3), (12.5, 5), (17.5, 10), (22.5, 8), (27.5, 4), (32.5, 2), (37.5, 0).

Conclusion: The graphs show that most workers earn between Rs 15 and Rs 20, and the number of workers falls as wages rise above that.

Primary data

Primary data is original information collected for the first time by the researcher directly from the field or source for a particular purpose. For example, a survey of the monthly income of households in a ward.

Methods of collecting primary data

1. Direct personal investigation (interview): The investigator meets the respondents face to face and asks questions. It gives accurate and detailed information but is costly and slow for a large area.
2. Indirect oral investigation: Information is collected from third persons who know the matter, such as local leaders or witnesses, when the people concerned cannot be reached directly.
3. Questionnaire (mailed or online): A list of questions is sent to respondents who fill in and return it. It covers a wide area cheaply, but response may be low and the respondents must be literate.
4. Schedule through enumerators: Trained enumerators visit respondents, ask the questions and fill in the form themselves. It suits illiterate respondents, as in a census.
5. Observation: The researcher watches and records events or behaviour directly, such as the traffic at a junction.
6. Telephone and online surveys: Quick and cheap, but limited to people with access to the phone or internet.
7. Local sources or correspondents: Local agents report information, which is quick but may not be accurate.

Conclusion: The method should be chosen according to the purpose, cost, time and the type of respondents.

Frequency distribution and median (the 45 values range from 3 to 68, class interval 10)

Step 1: Frequency distribution. Arrange the data in order, then tally. A value that lies on a class limit goes into the higher class (for example 10 is in 10-20).

Step 2: Median class. $\dfrac{N}{2} = \dfrac{45}{2} = 22.5$. The first cumulative frequency that is not less than 22.5 is 23, so the median class is 30-40.

Step 3: Median.

$Median = L + \frac{\frac{N}{2} - cf}{f} \times i$

Here $L = 30$, $cf = 14$ (cumulative frequency before the median class), $f = 9$, $i = 10$.

$Median = 30 + \frac{22.5 - 14}{9} \times 10 = 30 + \frac{8.5 \times 10}{9} = 30 + 9.44 = 39.44$

Check: The sorted data has 45 values, and the 23rd value is 39, which lies in the class 30-40, close to the result.

Median = 39.44

OR: Price index

Let $p_0, q_0$ be the base-year price and quantity and $p_1, q_1$ the current-year price and quantity.

Laspeyre's index (base-year quantities as weights):

$P_{01}^{L} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times 100 = \frac{1475}{1095} \times 100 = 134.70$

Paasche's index (current-year quantities as weights):

$P_{01}^{P} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times 100 = \frac{2260}{1665} \times 100 = 135.74$

Conclusion: Prices have risen by about 34.7% by Laspeyre's method and about 35.7% by Paasche's method compared with the base year.

Mode = 3 Median − 2 Mean

In an asymmetrical (skewed) series the mode cannot be located by inspection, so the empirical relation between the three averages is used — the value of mode is obtained by subtracting twice the mean from thrice the median.

To present statistical data in a simple, attractive and easily understandable visual form.

Construction of a pie-diagram

Step 1: Find the total expenditure

$\text{Total} = 1200 + 300 + 200 + 150 + 150 = \text{Rs. } 2000$

Step 2: Calculate the angle of each sector

Since a circle has $360^\circ$, the angle for each head is:

$\text{Angle} = \frac{\text{Expenditure of the head}}{\text{Total expenditure}} \times 360^\circ$

Step 3: Draw the pie-diagram

1. Draw a circle of suitable radius with a compass and mark its centre $O$.
2. Draw a horizontal radius as the base line and mark it as the starting line for "Food".
3. With the help of a protractor, mark an angle of $216^\circ$ at the centre from this base line. This sector represents Food.
4. From the end of the Food sector, mark the next angle of $54^\circ$ for Rent.
5. Continue in the same way, marking $36^\circ$ for Education, then $27^\circ$ for Miscellaneous, and the remaining $27^\circ$ for Clothing.
6. Shade or colour each sector differently and label it with the name of the head and its percentage or amount.
7. Write a suitable title, e.g. "Pie-diagram showing the distribution of expenditure of a family", and prepare an index showing the heads.

Verification: $216^\circ + 54^\circ + 36^\circ + 27^\circ + 27^\circ = 360^\circ$, so the construction is correct.

Given:

1. Mean

$\bar{x} = \frac{\Sigma fx}{N} = \frac{1360}{30} = 45.33$

Since income is in Rs. '000, mean income $= 45.33 \times 1000 = \text{Rs } 45{,}333$ (approx.).

2. Median

$\frac{N}{2} = \frac{30}{2} = 15$

The cumulative frequency first exceeds 15 at 24, so the median class is 40–60, with $l = 40$, $cf = 12$, $f = 12$, $h = 20$.

$\text{Median} = l + \frac{\frac{N}{2} - cf}{f} \times h = 40 + \frac{15 - 12}{12} \times 20 = 40 + 5 = 45$

Median income $= 45 \times 1000 = \text{Rs } 45{,}000$.

3. Comparison and conclusion

Mean (Rs 45,333) is slightly greater than median (Rs 45,000), a difference of only about Rs 333. The distribution is therefore almost symmetrical with a very slight positive skew, because the 4 high-income families (Rs 80,000–100,000) pull the mean up while the median is not affected by extreme values.

Conclusion: The typical family income is about Rs 45,000, and incomes are fairly evenly spread around this value.

1. Conversion into an exclusive continuous distribution

The given table is a less-than cumulative frequency distribution. The frequency of each class is found by subtracting the previous cumulative frequency.

2. Histogram

Take income (Rs. '000) on the X-axis and number of families on the Y-axis. All classes have width 10, so draw adjacent rectangles (no gaps) with heights equal to the frequencies: 5, 10, 15, 10 and 5.

No. of families
15 |              ██
10 |        ██    ██    ██
 5 |  ██    ██    ██    ██    ██
   +--+-----+-----+-----+-----+--
     0-10 10-20 20-30 30-40 40-50
          Income in Rs. '000

3. Modal class

The tallest rectangle stands on the class 20–30, so the modal class is 20–30 (Rs. 20,000–30,000).

Direct observation (personal investigation) and questionnaire/interview method.

Methods of Classifying Data

Classification of data means arranging collected raw data into groups or classes according to their similarities and characteristics. The main methods are as follows:

1. Geographical Classification

Data are classified according to place or region — for example, population by province, or production by district.

Practical application: Used by the government and planning bodies to compare development across regions and to allocate budget where it is needed most.

2. Chronological (Temporal) Classification

Data are arranged according to time — yearly, monthly or daily — in order of occurrence.

Practical application: Used in time-series analysis such as studying price trends, national income growth or rainfall over the years. It helps in forecasting the future.

3. Qualitative Classification

Data are classified according to attributes or qualities that cannot be measured numerically, such as sex, caste, religion, literacy or marital status. It may be simple (two classes, e.g., male–female) or manifold (more than two classes).

Practical application: Used in census reports and social surveys, for example classifying the population into male and female or literate and illiterate.

4. Quantitative Classification

Data are classified according to measurable numerical characteristics such as age, income, height or weight, by forming class intervals (e.g., 0–10, 10–20 years). It may be discrete or continuous.

Practical application: Used in frequency distribution tables, marks obtained by students, income distribution and in calculating averages and dispersion.

5. Spatial and Attribute Classification (Combined)

Often two methods are used together — for example, income groups (quantitative) within each province (geographical).

Practical application: Used in detailed government reports and research studies where comparison is needed on more than one basis.

Conclusion

Each method suits a particular type of data and purpose. Geographical classification is used for regional comparison, chronological for trend study, qualitative for social characteristics and quantitative for measurable variables. The choice of method depends on the nature of the data and the objective of the study.

Data: Total expenditure $= 1200 + 1600 + 600 + 800 + 400 + 200 = \text{Rs } 4800$

Formula: angle of each sector $= \dfrac{\text{value of item}}{\text{total value}} \times 360^\circ$

Steps to draw the pie-diagram:

1. Draw a circle of suitable radius and mark its centre.
2. Draw a horizontal radius as the base line.
3. With a protractor, mark angles at the centre in order: $90^\circ$ (Wages), $120^\circ$ (Cement), $45^\circ$ (Timber), $60^\circ$ (Iron), $30^\circ$ (Sand) and $15^\circ$ (Miscellaneous).
4. Join each marked point to the centre to get six sectors.
5. Shade or label each sector with its head and angle (or percentage).

Check: $90^\circ + 120^\circ + 45^\circ + 60^\circ + 30^\circ + 15^\circ = 360^\circ$. Cement has the biggest sector ($120^\circ$).

(a) Frequency distribution table (exclusive method; since full marks is 75, the last class is 70–80)

(b) Arithmetic mean

$\bar{X}=\frac{\sum fm}{\sum f}=\frac{750}{30}=25$

The simple arithmetic mean is 25 marks.

(c) Interpretation

The average mark is 25 out of 75, i.e. $\frac{25}{75}\times 100 = 33.33\%$ of the full marks. Most students scored in the lower intervals and none reached 60, so overall performance in Economics is weak and the class needs extra attention and practice.