Collection, organization and presentation of data | ISC Class 11 Economics Notes

This note covers the collection of data (primary and secondary sources, ways of collecting primary data, and the important sources of secondary data), the organisation of data (variables, frequency distributions, class limits and class marks), and the presentation of data (tabular presentation and diagrammatic presentation, including bar diagrams, pie diagrams, line graphs, histograms, frequency polygons and the ogive curve).

What are the sources of data?

Statistical data can come from two sources. When a researcher collects data directly by conducting an enquiry, the data are called Primary data, because they are based on first-hand information.

For example, if a student surveys classmates about the popularity of a film star, the information collected directly from them is primary data.

Definition: Primary data. Data collected directly by the investigator for the first time through an original enquiry, based on first-hand information.

If the data have already been collected and processed (scrutinised and tabulated) by some other agency, they are called Secondary data.

These can be obtained from published sources such as government reports, documents, newspapers, books written by economists, or a website.

The same data are primary to the source that first collects and processes them, and secondary for every source that later uses them. Using secondary data saves time and cost.

Definition: Secondary data. Data already collected and processed by some other agency, obtained from published or other existing sources rather than through a fresh enquiry.

How is primary data collected?

Primary data are usually collected through a Survey, a method of gathering information from individuals by asking questions about a product, candidate or issue.

The most common instrument used is a questionnaire or interview schedule, which may be self-administered by the respondent or administered by a researcher (enumerator) or trained investigator.

Designing a questionnaire

While preparing a questionnaire, several points should be kept in mind:

1. The questionnaire should not be too long; the number of questions should be as minimum as possible.
2. It should be easy to understand and avoid ambiguous or difficult words.
3. Questions should be arranged so the respondent feels comfortable, moving from general to specific.
4. Questions should be precise and clear, and not ask for information that is hard to calculate.
5. Questions should not be ambiguous, and should enable respondents to answer quickly, correctly and clearly.
6. Questions should not use double negatives, such as "Wouldn't you..." or "Don't you...", as these may lead to biased responses.
7. Questions should not be leading questions that hint at how the respondent should answer, and should not indicate alternatives within the question itself.

Questions may be closed-ended (structured), such as a two-way (yes/no) question or a multiple-choice question, or open-ended (unstructured), which allow individualised responses.

Closed-ended questions are easy to use, score and codify, but difficult to write well; the option "Any other (please specify)" is often provided so a true response that was not anticipated can still be recorded.

Open-ended questions allow more individualised answers but are difficult to interpret and hard to score.

Modes of collecting primary data

There are three basic ways of collecting data:

Once a questionnaire is ready, it is advisable to conduct a try-out with a small group, known as a Pilot Survey or pre-testing of the questionnaire.

This helps assess the suitability of the questions, the clarity of instructions, the performance of enumerators, and the likely cost and time of the actual survey.

What other ways of collecting primary data are used?

Besides the three basic modes, primary data may also be gathered by Indirect oral investigation and by Information from local sources or correspondents.

In an indirect oral investigation, the investigator does not question the person concerned directly. Instead, information is collected by interviewing other persons who are supposed to know the facts, such as neighbours, employers or local officials.

This method is useful where the persons concerned are unwilling to give information, cannot be contacted directly, or where the information sought is of a kind that others can report reliably.

In the method of information from local sources or correspondents, the investigator appoints local persons or correspondents in different places to collect and report information about the required facts.

This method is useful when the field of enquiry is wide, as in a survey spread over many districts or states, and it saves the time and cost of sending investigators everywhere.

Its main drawback is that the accuracy of the data depends on the honesty, ability and care of the local correspondents, whose reports may be biased or incomplete.

What is Census and what is a Sample Survey?

Census or complete enumeration

A survey, which includes every element of the population, is known as Census or the Method of Complete Enumeration.

The Census of India is carried out every ten years through a house-to-house enquiry covering all households in India, and the most recent completed Census of India was held in 2011.

Definition: Census. A survey that includes every element of the population, carried out as a complete enumeration rather than through a part of the population.

Population, sample and sampling methods

The Population or Universe in statistics means the totality of items under study, a group to which the results of the study are intended to apply.

When surveying the whole population is not possible, the researcher selects a Sample, a smaller group from which information is obtained, chosen so as to give reasonably accurate information about the population at lower cost and in less time.

Definition: Sample. A group or section of the population selected to represent the whole population from which information is obtained.

For example, in a study of the economic condition of agricultural labourers in Churachandpur district of Manipur, the population would be all agricultural labourers in that district, and the sample could be ten per cent of the agricultural labourers in Churachandpur district.

There are two main types of sampling. In Random sampling, every individual in the population has an equal chance of being selected; a common technique is the lottery method, where names are written on paper, mixed, and drawn one by one, and computer programmes may also be used.

In Non-random sampling, the units of the population do not all have an equal chance of selection, and the judgement, purpose, convenience or quota chosen by the investigator plays an important role in selecting the sample.

Definition: Random sampling. A method of sampling in which every individual unit of the population has an equal chance of being selected into the sample.

What are the important sources of secondary data?

Several agencies at the national level collect, process and tabulate statistical data. These include the Census of India, the National Sample Survey (NSS), the Central Statistics Office (CSO), the Registrar General of India (RGI), the Directorate General of Commercial Intelligence and Statistics (DGCIS) and the Labour Bureau, among others.

The Census of India provides the most complete and continuous demographic record of population, and has been regularly conducted every ten years since 1881; the first Census after Independence was conducted in 1951. It collects information on aspects such as size, density, sex ratio, literacy, migration and rural-urban distribution.

The National Sample Survey (NSS) was established by the Government of India to conduct nationwide surveys on socio-economic issues. It conducts continuous surveys in successive rounds, and releases data through reports and its quarterly journal, Sarvekshana.

The NSS provides periodic estimates of literacy, school enrolment, employment, unemployment and consumer expenditure, among other topics. The NSS 60th round survey (January-June 2004) covered morbidity and healthcare, and the NSS 68th round survey (2011-12) covered consumer expenditure.

The Central Statistics Office (CSO) compiles and publishes official statistics on the country's economy, including national income estimates and other economic data used for planning and analysis; in 2019 it was merged with the National Sample Survey Office to form the National Statistical Office (NSO).

The Registrar General of India (RGI) is the office responsible for conducting the Census of India and for maintaining the civil registration system of births and deaths, so its reports are a major published source of population data.

The Directorate General of Commercial Intelligence and Statistics (DGCIS) collects and publishes statistics of the country's foreign trade, that is, data on exports and imports.

The Labour Bureau collects and publishes statistics relating to labour, such as employment, wages and the cost of living of workers.

All these agencies publish their data in reports and documents, which other researchers can use as secondary data.

How is collected data organised?

Once data are collected, they are raw and unclassified, and are often large and cumbersome to handle. Classification brings order to raw data by arranging it into groups or classes based on some criterion, which makes it easier to locate information, compare values and draw conclusions.

A Variable is a quantity whose value varies, generally represented by letters such as X, Y or Z; each value of a variable is an observation. Variables are broadly of two types.

A Continuous variable can take any numerical value, including integral, fractional, or non-exact-fraction values (for example, height or weight).

A Discrete variable can take only certain values and changes by finite "jumps", for example the number of students in a class, which cannot take a fractional value.

Characteristics that cannot be expressed quantitatively, such as nationality, gender or marital status, are called qualities or attributes, and data classified on this basis is called Qualitative classification.

Characteristics such as height, weight, age, income or marks that are grouped into classes form a Quantitative classification.

Data can also be classified by time (Chronological classification, such as population by year) or by place (Spatial classification, such as wheat yield by country).

Frequency distribution

A Frequency distribution shows how different values of a variable are distributed across classes, along with the corresponding class frequencies. The Class frequency is the number of values in a particular class.

Definition: Frequency distribution. A comprehensive way of classifying raw data of a quantitative variable, showing how its values are distributed among classes along with the frequency of each class.

Each class is bounded by Class limits: the lowest value is the Lower class limit and the highest value the Upper class limit. The Class interval (or class width) is the difference between the upper and lower class limits.

The Class mark (or class mid-point) is the middle value of a class, used to represent it once data have been grouped.

Class Mark = (Upper Limit + Lower Limit)/2

Worked example 1. A class in a frequency distribution has a lower class limit of 60 and an upper class limit of 70.

Answer: Class interval = Upper limit − Lower limit = 70 − 60 = 10. Class mark = (70 + 60)/2 = 65.

In preparing a frequency distribution, questions to be addressed include whether to use equal or unequal class intervals, how many classes to have (usually between six and fifteen), the size of each class, how to fix the class limits, and how to obtain the frequency for each class using tally marks, counted in groups of five.

Class intervals may follow the Inclusive method, where values equal to the lower and upper limits are both included in that class, or the Exclusive method, where an item equal to the upper or lower limit is excluded from that class.

When using the inclusive method with a continuous variable, a gap appears between the upper limit of one class and the lower limit of the next; this is corrected by an adjustment: the difference between the lower limit of the next class and the upper limit of the present class is found, divided by two, then subtracted from all lower limits and added to all upper limits, giving the adjusted class limits and an Adjusted class mark.

Adjusted Class Mark = (Adj. Upper + Adj. Lower)/2

The Range is the difference between the largest and the smallest values of the variable, and is used along with the chosen class interval to decide the number of classes.

Range = Largest value − Smallest value

Grouping raw data into classes leads to a loss of information, since individual observations are replaced by the class mark for all further statistical calculations; this is why classes are formed so that the class mark lies as close as possible to the value around which observations in that class tend to concentrate.

Unequal class intervals may therefore be more appropriate where observations are unevenly spread. For a discrete variable, classification of data is called a Frequency array, since frequencies correspond to each integral value directly rather than to a range.

What is tabular presentation of data?

As data are usually voluminous, they need to be made usable and easily comprehended. There are three forms of presenting data: textual (described within the text, suitable when the quantity of data is not too large), tabular, and diagrammatic presentation.

In Tabulation, data are presented in rows (read horizontally) and columns (read vertically). Classification used in tabulation is of four kinds: qualitative (by attributes such as sex or location), quantitative (by measurable characteristics such as age or income), temporal (by time, such as years or months) and spatial (by place, such as state or country).

Definition: Tabulation. The systematic presentation of data in rows and columns so as to organise it for further statistical treatment and decision-making.

A good statistical table should have the following parts: a Table number for identification, a Title narrating its contents, Captions or column headings, Stubs or row headings, the Body containing the actual data, the unit of measurement, the Source of the data, and a Note explaining any feature of the data not otherwise self-explanatory.

This table, giving literacy rates in India by sex and location (per cent, Census of India 2011, for population aged 7 years and above), is a 3 × 3 table, giving nine items of information in nine cells, where each cell relates an attribute of gender with a literacy percentage.

What is diagrammatic presentation of data?

Diagrammatic presentation is the third method of presenting data, after textual and tabular presentation. It gives the quickest understanding of a situation compared with tabular or textual presentation, translating abstract numbers into a more concrete, easily comprehensible form; diagrams may be less accurate but are generally more effective than tables for presenting data.

The important kinds of diagram in common use are geometric diagrams (bar diagram and pie diagram), frequency diagrams (histogram, frequency polygon and ogive) and the arithmetic line graph.

Geometric diagrams include bar diagrams, which may be simple, multiple or component bar diagrams, and pie diagrams, where a circle's area is divided among the components it represents. Frequency diagrams are used specifically where data have already been organised as a grouped frequency distribution, and they include the histogram, the frequency polygon and the ogive or cumulative frequency curve, each built from the class limits, class marks or cumulative frequencies of the distribution.

Bar diagrams and pie diagrams

A Bar diagram comprises a group of equi-spaced, equi-width rectangular bars for each class or category of data, where the height of the bar reads the magnitude of that category and the bar's lower end touches the base line at zero.

Bar diagrams suit both frequency and non-frequency type data, and can represent discrete variables (such as family size) and attributes (such as gender or country).

Definition: Bar diagram. A diagram made up of a series of equi-spaced, equi-width rectangular bars whose heights are proportional to the values of the categories they represent.

Bar diagrams can take different forms. A Multiple bar diagram is used for comparing two or more sets of data, such as income and expenditure, or export and import figures over several years.

A Component bar diagram (or sub-diagram) shows a bar divided into two or more components, useful for comparing the sizes of different parts and their relationship, such as the components of a household's expenditure or the proportion of children enrolled and out of school within a given age group.

A Pie diagram (or pie chart) is also a component diagram, but here a circle's area is divided proportionally among the components it represents, by drawing straight lines from the centre to the circumference.

The values of each category are first expressed as a percentage of the total, and since a circle has 360°, each percentage figure is multiplied by 3.6° to find the angle each component should subtend at the centre.

Worked example 2. In the distribution of Indian population (2011) by working status, main workers were 29.8 per cent of the population.

Answer: Angle for main workers = 29.8 × 3.6° = 107.28°, which rounds to 107°.

Worked example 3. In the same distribution of Indian population (2011) by working status, the population was 121 crore in all, of which marginal workers numbered 12 crore and non-workers numbered 73 crore.

Answer: Marginal workers as a percentage of the total = (12 ÷ 121) × 100 = 9.9 per cent, giving an angle of 9.9 × 3.6° = 35.64°, rounded to 36°. Non-workers as a percentage of the total = (73 ÷ 121) × 100 = 60.3 per cent, giving an angle of 60.3 × 3.6° = 217.08°, rounded to 217°. These two angles, together with the 107° found for main workers, add up to 360°, completing the pie diagram.

What is a line graph?

The Arithmetic line graph, also called a time series graph or line diagram, is the third kind of diagram. It is used to show the trend of a variable over time, such as production, prices or population in successive years.

To draw it, time (years, months or days) is measured along the X-axis and the values of the variable along the Y-axis. Each pair of values is plotted as a point, and the points are joined in order by straight lines.

The line rises where the variable increases, falls where it decreases, and stays flat where it is unchanged, so the direction and steepness of the line show the trend at a glance.

Definition: Arithmetic line graph. A diagram in which time is plotted on the X-axis and the values of a variable on the Y-axis, with the plotted points joined by straight lines to show a trend.

The X-axis carries the years 1970-71, 1978-79, 1990-91, 1997-98, 2001-02, 2015-16 and 2016-17; the Y-axis carries production in million tonnes.

The plotted points are 108, 132, 176, 194, 212, 252 and 272 million tonnes for these years, joined by straight lines.

The line rises from 108 million tonnes in 1970-71 to 272 million tonnes in 2016-17, showing a rising long-term trend in food grain production over the years plotted.

Worked example 4. Food grain production in India was 108 million tonnes in 1970-71 and 272 million tonnes in 2016-17.

Answer: Absolute increase = 272 − 108 = 164 million tonnes. Percentage increase = (164 ÷ 108) × 100 = 151.85 per cent, so production in 2016-17 was about 2.5 times the level of 1970-71.

Histogram, frequency polygon and ogive

A Histogram is a two-dimensional diagram: a set of rectangles with class boundaries as the base (along the X-axis) and areas proportional to class frequency.

When class intervals are of equal width, as they generally are, the heights (not just areas) of the rectangles are directly proportional to the frequencies; when intervals are unequal, frequency density (class frequency divided by class width) must be used instead of absolute frequency.

Unlike a bar diagram, no space is left between the rectangles of a histogram, and a histogram can be drawn only for a continuous variable.

Definition: Histogram. A two-dimensional diagram of adjacent rectangles whose bases are the class intervals and whose areas are proportional to the class frequencies.

The X-axis carries the class boundaries of daily earnings (in rupees) in classes such as 45-49, 50-54, up to 115-119; the Y-axis carries the number of wage earners.

Adjacent rectangles with no gaps are drawn, each with a height equal to its class frequency, for example 2 for the 45-49 class rising to 13 for the 80-84 class and falling again towards the higher classes.

The histogram also shows the mode of the distribution, read as the x-coordinate corresponding to the highest rectangle.

A Frequency polygon is a plane figure bounded by straight lines, obtained from a histogram by joining the midpoints of the top of each rectangle; the two open ends are then joined to the base line at the midpoints of the classes with zero frequency immediately before and after the distribution, so that the total area under the polygon again represents the total frequency.

It is especially useful when comparing two or more distributions on the same axes, since lines are easier to compare than overlapping rectangles.

Class marks of the daily-earnings classes are plotted on the X-axis and the corresponding frequencies on the Y-axis; the plotted points are joined by short straight lines, with broken lines extending the curve down to the base line at the midpoints of the zero-frequency classes immediately before the first and after the last class, enclosing the full frequency within the figure.

An Ogive, or cumulative frequency curve, is drawn by plotting cumulative frequencies instead of simple frequencies. There are two kinds, corresponding to "less than" and "more than" type cumulative frequencies, giving two ogive curves for any grouped frequency distribution.

Plot upper class boundaries against "less than" cumulative frequency for the less-than ogive, and lower class boundaries against "more than" cumulative frequency for the more-than ogive, on the same axes; the point where the two curves intersect gives the median of the distribution.

Going beyond the syllabus

These ideas extend the unit and are not needed for the questions on it.

What are sampling and non-sampling errors?

The purpose of a sample is to estimate population parameters. Sampling error refers to the difference between the sample estimate and the corresponding population parameter (such as average income). It can be reduced by taking a larger sample.

Sampling Error = True value − Estimate

Worked example 5. The income of five farmers of Manipur is ₹500, ₹550, ₹600, ₹650 and ₹700. A sample of two farmers is selected, with incomes ₹500 and ₹600.

Answer: Population average = (500 + 550 + 600 + 650 + 700) ÷ 5 = 3000 ÷ 5 = ₹600. Sample average = (500 + 600) ÷ 2 = 1100 ÷ 2 = ₹550. Sampling error = ₹600 (true value) − ₹550 (estimate) = ₹50.

Non-sampling error is more serious than sampling error, because it is difficult to minimise even by taking a large sample; it can occur even in a Census.

Non-sampling errors include sampling bias, which occurs when some members of the target population could not possibly be included in the sample; non-response errors, when an interviewer cannot contact a listed person or that person refuses to respond; and errors in data acquisition, which arise from recording incorrect responses, for example writing 13 instead of 31.

What is a bivariate frequency distribution?

When more than one characteristic is recorded for each unit of a sample, such as sales and advertisement expenditure of several firms, the data can be summarised using a Bivariate frequency distribution, defined as the frequency distribution of two variables, with each cell showing the number of units falling into a particular row and column combination.

What is a frequency curve?

A Frequency curve is obtained by drawing a smooth freehand curve passing through the points of the frequency polygon as closely as possible, though it need not pass through every point exactly.

Glossary

• Primary data — data collected first-hand by the investigator through an original enquiry or survey.
• Secondary data — data already collected, processed and published by some other agency.
• Census — a survey covering every element of the population, also called complete enumeration.
• Sample — a smaller, representative section of the population from which information is obtained.
• Random sampling — a method in which every unit of the population has an equal chance of selection.
• Variable — a quantity whose value varies from one observation to another, usually denoted X, Y or Z.
• Frequency distribution — a table showing how values of a variable are distributed across classes with their frequencies.
• Class mark — the mid-point of a class interval, equal to half the sum of its upper and lower limits.
• Inclusive method — a classification in which both the upper and lower class limits are included in the frequency of that class.
• Exclusive method — a classification in which an item equal to the upper or lower limit is excluded from that class.
• Tabulation — the presentation of data in rows and columns, organised for further statistical use.
• Histogram — adjacent rectangles whose areas represent class frequencies of a continuous variable.
• Ogive — a cumulative frequency curve, of the "less than" or "more than" type.
• Arithmetic line graph — a diagram plotting a variable against time, with the points joined by straight lines to show a trend.

Common errors and misconceptions

• Misconception: Data collected by any investigator is always secondary data. Correct: Data are primary for the agency that first collects and processes them, and secondary only for later users of that same data.
• Misconception: Telephone interviews are the most suitable method regardless of the population. Correct: Telephone interviews work poorly where many people do not own telephones or the population is spread over a wide, inaccessible area.
• Misconception: A larger sample removes non-sampling error. Correct: A larger sample reduces sampling error, but non-sampling error (bias, non-response, recording mistakes) can persist even in a full Census.
• Misconception: A histogram and a bar diagram are the same thing. Correct: A histogram has no gaps between rectangles and is drawn only for a continuous variable, whereas a bar diagram leaves space between bars and can represent discrete variables or attributes.
• Misconception: The class mark can be calculated only after rounding the class limits. Correct: The class mark is simply half the sum of the actual upper and lower class limits, adjusted first only if there is a gap between classes.
• Misconception: In the exclusive method, a value equal to the class limit is always placed in the higher class. Correct: The investigator decides in advance whether the lower limit or the upper limit is excluded; the value equal to the boundary can be placed in either adjoining class.
• Misconception: A pie chart can be drawn directly from absolute values. Correct: Absolute values must first be converted to percentages of the total, then multiplied by 3.6° to obtain the angle for each component.
• Misconception: A line graph and a bar diagram show a trend equally well. Correct: Bars can also show values over time, but a line graph joins the plotted points in time order, so the direction and steepness of the trend are seen more directly.

Exam-style questions with model answers

Q1. State the meaning of primary data. [1 mark]

1. Primary data are data collected first-hand by the investigator through an original enquiry or survey.

Q2. Define the term variable. [1 mark]

1. A variable is a quantity whose value varies from one observation to another, generally represented by letters such as X, Y or Z.

Q3. The lower class limit of a class is 50 and the upper class limit is 60. Calculate the class interval and the class mark. [2 marks]

1. Class interval = Upper limit − Lower limit = 60 − 50 = 10.
2. Class mark = (Upper limit + Lower limit)/2 = (60 + 50)/2 = 55.

Q4. Distinguish between a discrete variable and a continuous variable, giving one example of each. [4 marks]

1. A continuous variable can take any numerical value, whole, fractional or otherwise, within its range.
2. For example, height can take values such as 90 cm, 100.85 cm or 149.99 cm.
3. A discrete variable can take only certain values and changes by finite jumps rather than passing through intermediate values.
4. For example, the number of students in a class can only be a whole number such as 25 or 26, and cannot take a value such as 25.5.

Q5. Explain the inclusive method and the exclusive method of forming class intervals. [4 marks]

1. In the inclusive method, values equal to both the lower limit and the upper limit of a class are included in the frequency of that same class, for example a class written as 0-10, 11-20, and so on.
2. In the exclusive method, an item equal to either the upper or the lower class limit is excluded from the frequency of that class, for example a class written as 0-10, 10-20, and so on.
3. With exclusive intervals, the investigator must decide in advance whether a value equal to a shared limit is placed in the lower class (upper limit excluded) or the higher class (lower limit excluded).
4. For continuous variables, such as height or weight, the inclusive form of class interval is used very often, with adjusted limits to maintain continuity between classes.

Q6. Distinguish between a bar diagram and a histogram. [4 marks]

1. A bar diagram leaves space between consecutive bars, while a histogram has no space between its rectangles because it represents a continuous variable.
2. In a bar diagram, it is only the height of the bar that matters, and a single vertical line could serve the same purpose; in a histogram, the width as well as the height of each rectangle is important, because area represents frequency.
3. A bar diagram can represent both discrete variables and attributes, whereas a histogram can be drawn only for a continuous variable.
4. A histogram also gives the mode of the distribution graphically, read from the highest rectangle, which a simple bar diagram does not show.

Q7. Explain the construction and use of an arithmetic line graph. [4 marks]

1. Time, such as years, months or days, is measured along the X-axis and the values of the variable along the Y-axis.
2. Each pair of values is plotted as a point, and the points are joined in order by straight lines.
3. The line rises where the variable increases, falls where it decreases, and stays flat where it is unchanged.
4. It is used to show the trend of a variable over time, such as production, prices or population in successive years.

Q8. Describe the parts of a good statistical table. [8 marks]

1. Table number: given for identification, usually at the top, distinguishing one table from another when several appear together.
2. Title: a clear, brief, carefully worded statement of the table's contents, placed at the head of the table after the table number.
3. Captions or column headings: the designation given at the top of each column to explain the figures contained in it.
4. Stubs or row headings: the designation given to each row, with the complete left column known as the stub column.
5. Body: the main part of the table containing the actual data, with each figure fixed by its row and column position.
6. Unit of measurement: the unit of the figures must be stated with the title.
7. Source: the source of the data is written at the bottom of the table.
8. Note: a note explains any feature of the data not otherwise self-explanatory.

Q9. Explain, with reference to a pie diagram, how the angle for each component of the circle is found. [4 marks]

1. The value of each category is first expressed as a percentage of the total value of all categories.
2. A circle, whatever its radius, is treated as having 360°, so each one per cent of the total corresponds to 3.6°.
3. Each component's percentage figure is multiplied by 3.6° to obtain the angle it should subtend at the centre of the circle.
4. The angles of all the components together add up to 360°, completing the circle.

Q10. Explain the construction of an ogive curve. [8 marks]

1. An ogive, also called a cumulative frequency curve, is drawn by plotting cumulative frequencies rather than simple class frequencies.
2. There are two types of cumulative frequency: "less than" type, where frequencies are accumulated from the lowest class upward, and "more than" type, where frequencies are accumulated from the highest class downward.
3. For the less-than ogive, the upper class boundaries are plotted on the X-axis against the corresponding less-than cumulative frequency on the Y-axis.
4. For the more-than ogive, the lower class boundaries are plotted against the corresponding more-than cumulative frequency.
5. Both curves can be drawn on the same axes for a single frequency distribution, and the two curves intersect at one point.
6. This point of intersection gives the median of the distribution, which is the main practical use of drawing both ogives together.
7. Each curve is drawn by joining the plotted points with a smooth curve or with straight lines.
8. Since cumulative frequencies rise or fall steadily, the less-than ogive slopes upward and the more-than ogive slopes downward from left to right.

Key takeaways

• Primary data are collected first-hand through a survey, while secondary data are already collected and published by another agency.
• A questionnaire should be short, clear, free of leading or ambiguous questions, and move from general to specific questions.
• Personal interviews, mailing surveys and telephone interviews each have distinct advantages and disadvantages for different situations.
• A census covers the entire population; a sample survey studies only a smaller, representative section of it, usually at lower cost.
• Random sampling gives every unit an equal chance of selection, while non-random sampling relies on the investigator's judgement or convenience.
• Classification organises raw data into groups; variables may be continuous (any value) or discrete (only specific values).
• A frequency distribution shows class limits, class intervals and class marks, formed using either the inclusive or exclusive method.
• Tabular presentation organises data in rows and columns with a table number, title, captions, stubs, body, units, source and note.
• Diagrammatic presentation includes bar diagrams, pie diagrams, line graphs, histograms, frequency polygons and ogive curves, each suited to different kinds of data.
• An arithmetic line graph plots a variable against time and joins the points by straight lines, so the direction of the trend is seen at a glance.

Test yourself

What is the key difference between primary and secondary data?

Primary data are collected first-hand by the investigator through an original enquiry; secondary data have already been collected and processed by some other agency.

Why might a leading question such as "Wouldn't you oppose a price rise?" be a poor question in a survey?

It hints at how the respondent should answer, which can bias the response rather than record the respondent's genuine, independent view.

Name the two main types of sampling.

Random sampling, where every unit has an equal chance of selection, and non-random sampling, where selection is based on the investigator's judgement or convenience.

Give one example each of a continuous variable and a discrete variable.

Height is a continuous variable, as it can take any value within a range; the number of students in a class is a discrete variable, as it can take only whole-number values.

What is the class mark of a class with lower limit 20 and upper limit 30?

Class mark = (20 + 30)/2 = 25.

What distinguishes a histogram from a frequency polygon?

A histogram is a set of adjacent rectangles with areas proportional to class frequency, while a frequency polygon is formed by joining the midpoints of the tops of these rectangles with straight lines.

What is an arithmetic line graph used for?

It shows the trend of a variable over time, such as production or prices in successive years, by plotting time on the X-axis and the variable on the Y-axis and joining the points by straight lines.

What does the point of intersection of the two ogive curves represent?

It represents the median of the frequency distribution, found graphically where the less-than and more-than cumulative frequency curves cross.