Presentation of Data | CBSE Class 11 Economics Notes

This note covers textual, tabular and diagrammatic presentation, classification of data, parts of a statistical table, bar diagrams, pie charts, histograms, frequency polygons, frequency curves, ogives and arithmetic line graphs.

Why do data need presentation, and when is text suitable?

Data are the collected facts or figures to be organised and presented. They are generally voluminous. Presentation puts them into a compact, understandable form so that the information can be used readily.

The three general forms are textual presentation, which describes data in sentences; tabular presentation, which arranges data in rows and columns; and diagrammatic presentation, which represents data visually. The choice depends on the quantity and nature of the information and the purpose of presenting it.

What are the advantages and limitations of text?

Textual presentation is more suitable when the quantity of data is not too large. It often enables the writer to emphasise particular points. Its serious drawback is that the reader must go through the complete text to understand the information.

Consider the bandh, or shutdown, on 08 September 2005 against higher petrol and diesel prices in a town of Bihar. Five petrol pumps were open and 17 were closed. Two schools were closed and the remaining nine schools were open.

This small set of observations can be understood through a short description. A larger description, combining population, gender, residence and working status, requires the reader to keep track of several categories and numbers at once.

For example, India's population in 2001 was described as 102 crore, including 49 crore females and 53 crore males. A crore is ten million. Rural residence accounted for 74 crore people and urban residence for 28 crore.

The same description also distinguished 62 crore non-workers from 40 crore workers. Here, workers and non-workers identify the working-status groups in the population. These different classifications can be brought together in a table so their relationships are easier to locate.

Note: A presentation method should make the information easier to understand. A short paragraph can emphasise a small set of facts, while a table can accommodate a large quantity of data concerning one or more characteristics.

How are data classified for tabular presentation?

A table arranges data in rows, read horizontally, and columns, read vertically. A cell is the box where a row and column meet. Its value must be read together with both headings.

Tabulation's most important advantage is that it organises data for further statistical treatment and decision-making. The classification, or grouping, used in a table can be qualitative, quantitative, temporal or spatial.

What distinguishes the four kinds of classification?

A variable is a characteristic that can take different values. Quantitative classification groups its values using class limits, the limits assigned to each group. In an age distribution, age in years is the measurable characteristic.

Temporal classification could arrange a tea shop's sales by year. Spatial classification could arrange India's exports by destination. The presence of numbers does not by itself make the classification quantitative: identify what is being used to group the observations.

How should a literacy table be read?

Literacy rate expresses the literate population as a percentage of the relevant population. The following rates relate to people aged seven years and above. A percentage expresses a quantity per hundred.

Table: Literacy in India by sex and location, 2011, in per cent

The data body has three rows and three columns, making nine cells. The female row and rural column give 59 per cent. The male row and urban column give 90 per cent. Sex and location are qualitative attributes used together to organise the information.

What are the essential parts of a statistical table?

A good statistical table combines its figures with the information needed to interpret them. One-way, two-way and three-way classification use one, two and three characteristics respectively. The number of characteristics involved determines the form of tabulation.

The reader needs to identify what the table concerns, what each row and column represents, and which units apply. A number without those details does not communicate its intended meaning.

How does each part help the reader?

Table numbers generally follow ascending order. A number such as 4.5 identifies the fifth table in the fourth chapter. A clear title prevents ambiguity before the reader begins examining individual figures.

Units should always accompany the title when they apply to the table as a whole. If different rows or columns use different units, state them with their stubs or captions. When large figures are rounded, indicate the rounding method.

For example, population figures in crores require the unit to be visible. A note that figures are rounded to the nearest crore explains the degree of precision. If several sources supply the data, all should appear in the source statement.

How can a table be read systematically?

1. Read the title to identify the population or subject and the relevant period.
2. Check the units and any note explaining rounding or other features.
3. Find the required row and column, then read their intersection.
4. Use the source statement to identify the origin of the data.

In the population table for 2001, the rural female non-worker entry is 25 crore. Its meaning depends on combining location, gender and working status, rather than reading the figure in isolation.

How do simple and multiple bar diagrams compare data?

Geometric diagrams use forms such as bars and circles to represent data. Bar diagrams and pie diagrams belong to this group. The other important groups are frequency diagrams and arithmetic line graphs, discussed below.

Diagrams translate numerical information into a concrete visual form and provide the quickest understanding compared with textual or tabular presentation. They may be less accurate but are much more effective than tables in presenting data.

What does a simple bar show?

A simple bar diagram uses rectangular bars of equal width and equal spacing for the categories. The bar's height or length represents the magnitude. Its lower end touches the baseline, the line from which measurement begins, at zero.

A bar diagram is one-dimensional in its representation of magnitude: the length or height matters. Its width is not the quantity being compared. Longer bars represent more of the measured or counted characteristic.

Bars can represent frequency, meaning the number of observations in a category, or non-frequency values such as production, income or exports. They can also represent attributes such as gender, religion or country.

Discrete variables take separate values, as with family size or spots on a die. Bar diagrams can represent these variables and are more convenient for non-frequency data such as income and expenditure or exports and imports over time.

Separate vertical bars show male literacy rates in 2011. The horizontal labels identify major states and India; the vertical scale measures literacy in per cent. Kerala's bar is labelled 96 and West Bengal's 82.7.

See Fig. 4.1 in your NCERT textbook

When are multiple bars useful?

A multiple bar diagram compares two or more sets of data. Examples include income and expenditure, imports and exports in different years, and marks obtained in different subjects or classes.

For female literacy, separate bars for 2001 and 2011 allow comparison within each state. The viewer can compare the years for one state and compare the same year's values across states.

Paired bars show female literacy percentages for 2001 and 2011 across major states. The later bar is higher in each pair, showing the increase between the two years.

See Fig. 4.2 in your NCERT textbook

How do component bars show the parts of a total?

A component bar diagram, also called a sub-diagram, divides each bar into its constituent parts. A component is an element making up a whole. Such diagrams compare component sizes and show the relationships among the parts.

Examples include sales from different products, family expenditure on food, rent, medicine, education and power, and components of a population. Components are usually shaded or coloured to make them distinguishable.

Table: School enrolment of children aged 6 to 14 years in a district of Bihar

How is a component bar constructed?

1. Place the bars on the x-axis, the horizontal reference axis.
2. Make each complete bar represent its total. For percentage data, its height is 100 units.
3. Divide the bar into proportional component heights, giving smaller components priority in parting the bar.
4. Shade or colour the components so that their identities can be recognised.

For data expressed as amounts rather than percentages, the height corresponds to the total value. Work out each component's proportional height using the unitary method: find the height corresponding to one unit, then multiply by the component's value.

Worked example 1. Represent the girls' school-enrolment figures: 58.6 per cent enrolled and 41.4 per cent out of school.

Answer: Draw a bar of height 100 units. Divide it into 58.6 units for enrolled girls and 41.4 units for girls out of school. Both parts together represent the whole group of girls.

Three bars labelled Boys, Girls and All reach 100. Each is divided into enrolled and out-of-school portions, with the corresponding percentages written inside. A legend identifies the two shadings.

See Fig. 4.3 in your NCERT textbook

A legend is the key identifying the meaning of shading or colour. Read the complete bar and its parts together: equal total heights in this percentage diagram do not mean equal numbers of boys and girls.

How are pie-chart sectors calculated?

A pie diagram, or pie chart, is a component diagram using a circle. Its area is divided proportionally among the components. Straight lines from the centre to the circumference divide the circle into parts called sectors.

The circumference is the circle's boundary, and its radius is the distance from centre to boundary. Irrespective of the radius, the whole circle represents the total. The relative size of a sector represents the component's share.

How are values converted into angles?

Pie charts usually are not drawn with absolute values, meaning the original amounts rather than percentages. First express each component as a percentage of the total. The full circle has 360 degrees, written 360°. Each percentage point corresponds to 3.6°.

Let C mean the component value, T the total value of all components, and P the numerical percentage share. The multiplication sign is × and the division sign is /. Then:

P = 100 × C / T

Let A mean the sector's angle at the centre, measured in degrees. Convert the percentage into an angle as follows:

A = 3.6 × P

1. Identify all the components of the total being represented.
2. Express each component as a percentage of the total.
3. Multiply each percentage by 3.6° to obtain its central angle.
4. Draw the circle and divide it using the calculated angles.

Worked example 2. For boys aged 6 to 14 years in the Bihar district, 91.5 per cent are enrolled and 8.5 per cent are out of school. Find the angles for a pie chart.

Answer: The enrolled sector is 91.5 × 3.6° = 329.4°. The out-of-school sector is 8.5 × 3.6° = 30.6°. Together the angles make 360°, representing all boys in the group.

Data shown through a component bar diagram can also be represented equally well by a pie chart. The required conversion is from absolute component values into percentages before using them for the pie diagram.

Both methods show the composition of a total. The bar divides a length into component heights, while the pie divides a circle into proportional areas through the central angles.

How does a histogram represent a frequency distribution?

A grouped frequency distribution lists class intervals and the number of observations in each. A class interval is a range used to group values. Frequency diagrams generally represent such distributions.

A histogram consists of rectangles whose bases lie between class boundaries and whose areas are proportional to class frequencies. Class boundaries are the dividing values between continuous classes. The y-axis is the vertical reference axis.

Why do class widths matter?

Histograms represent continuous variables, which can take values throughout a range. For continuous classes, one class's upper boundary meets the next class's lower boundary. The rectangles therefore touch, leaving no open space between consecutive rectangles.

If the original classes are not continuous, convert them into continuous classes before drawing the histogram. With equal class widths, which generally occur, rectangle heights can represent frequencies directly because all bases have the same width.

With unequal widths, heights must be adjusted. Use frequency density, defined as class frequency divided by class width, so that rectangle areas remain proportional to frequencies. Raw frequency heights would not make areas comparable when bases differ.

Let U and L mean the upper and lower class boundaries, and w the class width. The minus sign, −, means subtraction.

w = U − L

Let f mean the class frequency and D the frequency density.

D = f / w

Worked example 3. The daily-earnings class of 80 to 84 rupees contains 13 wage earners. Its continuous boundaries are 79.5 and 84.5 rupees. Calculate its width and frequency density.

Answer: Width = 84.5 − 79.5 = 5 rupees. Frequency density = 13 / 5 = 2.6 wage earners per rupee of class width. When every class has this same width, the frequency height can instead be 13.

Adjacent rectangles represent 85 wage earners. Continuous wage boundaries appear on the horizontal axis and frequency on the vertical axis. Lines near the tallest rectangle locate the mode, labelled 80.6.

See Fig. 4.5 in your NCERT textbook

The mode is the value occurring most frequently. A histogram can locate it graphically; the horizontal coordinate, meaning the value read on the horizontal axis, of the dotted vertical line gives the mode.

How do histograms, frequency polygons and frequency curves differ?

A histogram and a bar diagram can look similar, but the quantities encoded by their dimensions differ. In a bar diagram, compare heights or lengths. In a histogram, both class widths and rectangle heights matter because area represents frequency.

How is a frequency polygon drawn?

A frequency polygon joins plotted frequencies with straight line segments. A class mark is the midpoint of a class interval. Whether boundaries or class marks label the horizontal axis, plot each frequency against its class midpoint.

1. Draw the histogram or locate the class midpoints and their frequencies directly.
2. Mark the midpoint of the top of each histogram rectangle.
3. Join consecutive points using short straight lines.
4. Join the two ends to the baseline at the midpoints of adjoining classes with zero frequency.

Closing the ends makes the total area under the polygon represent total frequency, as in the histogram. These additional zero-frequency classes are immediately before and after the observed distribution.

Straight segments join points above successive daily-wage classes, overlaid on the histogram. The line reaches the baseline at both ends. The axes identify daily wages in rupees and the number of wage earners.

See Fig. 4.6 in your NCERT textbook

When two or more distributions share the same axes, a polygon is likely to be more useful because the horizontal and vertical lines of histograms may coincide. This is a reason for choosing it when comparing distributions.

How is a frequency curve different?

A frequency curve is a smooth freehand curve drawn as close as possible to the polygon's points. It may not necessarily pass through every point. Its smooth shape replaces the polygon's connected straight segments.

The graph shows a solid frequency line and a dotted smooth curve, identified in a legend. Both rise towards the central wage values and fall towards the ends.

See Fig. 4.7 in your NCERT textbook

How are cumulative frequencies and ogives constructed?

Cumulative frequency is the accumulated number of observations up to, or beyond, a specified class limit. An ogive is a cumulative frequency curve. The two forms are the less-than ogive and the more-than ogive.

For the less-than form, plot cumulative frequencies against upper class limits. For the more-than form, plot them against lower class limits. In both forms, cumulative frequency belongs on the vertical axis.

Table: Mathematics marks and cumulative frequencies

The total number of students is 64. For the less-than series, successive addition gives the accumulated number below each upper limit. For the more-than series, begin with the total and remove frequencies below successive lower limits.

Worked example 4. The marks classes 0 to 20, 20 to 40, 40 to 60, 60 to 80 and 80 to 100 have frequencies 6, 5, 33, 14 and 6 respectively. Find the cumulative frequencies for less than 60 and more than 60.

Answer: Less than 60 = 6 + 5 + 33 = 44 students. More than 60 = 14 + 6 = 20 students. These values use different class limits in the two cumulative series.

What do the shapes and intersection show?

The less-than ogive is never decreasing; the more-than ogive is never increasing. These descriptions allow a flat portion when a class contributes no observations. They do not require every segment to change strictly.

The median is the central value dividing an ordered distribution into two equal halves. The intersection of the two ogives locates the median graphically. Read its value on the horizontal marks axis, not on the vertical frequency axis.

Separate plots show the rising less-than curve and falling more-than curve. A combined plot shows their intersection and a dotted vertical line down to the marks axis, labelled Median.

See Fig. 4.8 in your NCERT textbook

How do time-series graphs and worked tables support interpretation?

An arithmetic line graph is also called a time-series graph. A time series records a variable at successive times. Plot time on the horizontal axis and the variable's value on the vertical axis, then join the plotted points.

Time may be measured in hours, days, weeks, months or years. Such graphs help reveal a trend, the general direction of change, and periodicity, repeated patterns over time. They can help in understanding long-term movement, cyclicity and seasonality.

Cyclicity refers to recurring rises and falls, while seasonality refers to patterns linked to seasons. A line joining observations in time order shows how the measured quantity develops across the period.

What does the trade graph show?

Exports are goods or services sold abroad; imports are goods or services bought from abroad. India's trade graph for 1993-94 to 2013-14 shows imports above exports throughout that period.

Two lines identified as Exports and Imports run across annual labels from 1993-94 to 2013-14. The imports line is higher. Both lines rise rapidly after 2001-02, and the gap between them widens after that year.

See Fig. 4.9 in your NCERT textbook

The values are measured in units of 100 crores of rupees. The graph presents both series on common axes, so the viewer can compare their levels and movement over the same years.

How can a missing table entry be recovered?

A table also makes totals and component values easier to relate. In an election study in Bihar, an age distribution covers 542 respondents. A respondent is a person supplying information in a study.

Worked example 5. There are 542 respondents. Counts for age groups 20 to 30, 30 to 40, 40 to 50, 50 to 60, 70 to 80 and 80 to 90 years are 3, 61, 132, 153, 51 and 2. Find the missing count for 60 to 70 years and its percentage.

Answer: The known counts total 3 + 61 + 132 + 153 + 51 + 2 = 402. The missing count is 542 − 402 = 140 respondents. Its percentage is 100 × 140 / 542, approximately 25.83 per cent.

Choose the display to match the relationship: tables for organised figures, bars for comparisons, component bars or pies for composition, frequency diagrams for distributions, and arithmetic line graphs for observations over time.

Glossary

• Textual presentation — Presentation in which data are described within sentences, especially suitable when their quantity is not too large.
• Tabulation — Arrangement of data in rows and columns to support interpretation, statistical treatment and decision-making.
• Qualitative classification — Grouping according to attributes such as social status, nationality, sex or location.
• Quantitative classification — Grouping according to numerically measurable characteristics such as age, height, production or income.
• Temporal classification — Classification using time, such as hours, months or years, as the grouping variable.
• Spatial classification — Classification based on place, such as a village, district, state or country.
• Caption — A column heading that explains the figures appearing in that column of a table.
• Stub — A row heading that identifies the category represented by that row in a table.
• Component bar diagram — A bar diagram divided into parts to compare components and their relationships within a whole.
• Pie diagram — A circle divided into sectors whose areas proportionally represent the components of a total.
• Histogram — Adjacent rectangles representing continuous classes, with areas proportional to their respective class frequencies.
• Frequency density — Class frequency divided by class width, used to adjust histogram heights when class widths differ.
• Frequency polygon — A diagram joining frequencies plotted at class midpoints through a series of straight lines.
• Ogive — A cumulative frequency curve plotting accumulated frequencies against the appropriate class limits.
• Arithmetic line graph — A graph joining successive observations, with time horizontally and values of the variable vertically.

Common errors and misconceptions

• Misconception: Any table containing numbers uses quantitative classification. Correct: The basis of grouping determines the classification; sex and location are qualitative attributes even when the cells contain percentages.
• Misconception: A table's figures explain themselves. Correct: Read each figure with its title, row, column and unit, and check notes explaining rounding or other features.
• Misconception: The width of a bar represents the magnitude being compared. Correct: Height or length represents magnitude; width is unimportant for comparison in a bar diagram.
• Misconception: Histogram rectangles must have equal widths. Correct: They follow class widths. Unequal classes require frequency-density heights so that areas remain proportional to frequencies.
• Misconception: A frequency curve must pass through every polygon point. Correct: It may not necessarily pass through all points; it follows them as closely as possible.
• Misconception: Ogives plot ordinary frequencies against class midpoints. Correct: They plot cumulative frequencies against upper limits for less-than curves and lower limits for more-than curves.
• Misconception: An ogive intersection gives the mode. Correct: The intersection locates the median; a histogram can locate the mode graphically.
• Misconception: Every segment of a less-than ogive must rise. Correct: It is never decreasing, which allows flat portions; the more-than ogive is never increasing.

Exam-style questions with model answers

Q1. State one advantage and one drawback of textual presentation. [2 marks]

1. Textual presentation often enables particular points to be emphasised and is more suitable when the quantity of data is not too large.
2. Its serious drawback is that the reader must go through the complete text to comprehend the information presented.

Q2. Distinguish qualitative, quantitative, temporal and spatial classification, giving an example of each. [4 marks]

1. Qualitative classification groups observations by attributes such as nationality or sex, rather than by the numerical measurement of a characteristic.
2. Quantitative classification uses measurable characteristics, such as age or income, whose values can be arranged within class limits.
3. Temporal classification uses time as the classifying variable, as when a tea shop's sales are arranged by year.
4. Spatial classification uses place as the basis of grouping, as when exports from India are arranged by destination.

Q3. For boys aged 6 to 14 years in a district of Bihar, 91.5 per cent are enrolled in school and 8.5 per cent are out of school. Calculate the two pie-chart angles and check the total. Use 3.6° per percentage point. [3 marks]

1. The enrolled share is converted to a central angle by multiplying its percentage by 3.6°: 91.5 × 3.6° = 329.4°.
2. The out-of-school share uses the same conversion: 8.5 × 3.6° = 30.6°. This sector represents the smaller component of the group.
3. The two angles sum to 329.4° + 30.6° = 360°. They therefore make a complete circle representing all boys in this age group.

Q4. Explain five differences between a simple bar diagram and a histogram, covering magnitude, width, spacing, variables and graphical use. [5 marks]

1. In a simple bar diagram, magnitude is represented by the height or length of each bar. In a histogram, the area of each rectangle is proportional to its class frequency.
2. Bar width is unimportant for comparison, although equal widths are used. Histogram widths represent class intervals and may be unequal.
3. Simple bars have spaces between them. Histogram rectangles for continuous classes touch because each upper boundary meets the next lower boundary.
4. Bar diagrams can represent discrete as well as continuous variables. A histogram is drawn only for a continuous variable with continuous classes.
5. A bar diagram supports visual comparison between categories. A histogram represents a frequency distribution and also allows its mode to be located graphically.

Q5. Daily earnings of 80 to 84 rupees have frequency 13. The continuous boundaries are 79.5 and 84.5 rupees. Calculate the class width and frequency density, then explain which heights to use for equal and unequal class widths. [4 marks]

1. Class width is the difference between the upper and lower boundaries: 84.5 − 79.5 = 5 rupees.
2. Frequency density is class frequency divided by class width: 13 / 5 = 2.6 wage earners per rupee of width.
3. If all class widths are equal, use frequencies as rectangle heights. This class can then have a height of 13.
4. If widths differ, use frequency densities as heights. Areas, rather than unadjusted heights, must remain proportional to the class frequencies.

Q6. Mathematics marks classes 0 to 20, 20 to 40, 40 to 60, 60 to 80 and 80 to 100 have frequencies 6, 5, 33, 14 and 6. Give both cumulative series, identify the limits used for plotting each, state their directions and explain what their intersection locates. [6 marks]

1. The total is 6 + 5 + 33 + 14 + 6 = 64 students. The less-than cumulative frequencies are 6, 11, 44, 58 and 64.
2. Plot those cumulative values against the upper class limits 20, 40, 60, 80 and 100 respectively to construct the less-than ogive.
3. The more-than cumulative frequencies are 64, 58, 53, 20 and 6, obtained by removing the frequencies below successive lower limits.
4. Plot those values against the lower limits 0, 20, 40, 60 and 80 respectively to construct the more-than ogive.
5. The less-than ogive is never decreasing and the more-than ogive is never increasing. Their vertical axes show accumulated numbers of students.
6. The intersection of the two ogives locates the median graphically. Read the corresponding marks value on the horizontal axis.

Q7. Explain four steps for drawing a frequency polygon from a histogram, including how to close its ends. [4 marks]

1. Identify the midpoint of the top of every histogram rectangle. Each point represents a class midpoint and its associated frequency.
2. Join successive points with short straight lines, proceeding across the classes in order to form the polygon.
3. Take an adjoining class with zero frequency immediately before the first observed class and immediately after the last observed class.
4. Join the two polygon ends to the baseline at these additional class midpoints. The closed area represents total frequency, as in the histogram.

Q8. State how an arithmetic line graph is constructed and explain its purpose. [2 marks]

1. Plot time along the horizontal axis and the variable's values along the vertical axis, then join the successive plotted points.
2. The graph helps explain the long-term trend and periodic patterns in time-series data, including cyclicity and seasonality.

Key takeaways

• Text suits smaller quantities of data, while tables organise larger quantities for interpretation, statistical treatment and decision-making.
• Identify whether the basis of classification is an attribute, a measurable characteristic, time or place.
• A clear table needs identification, headings, actual data, units, sources and notes explaining relevant features.
• Simple and multiple bars compare magnitudes; component bars and pie diagrams show the parts of a whole.
• Convert each pie-chart component to a percentage, then multiply by 3.6° to calculate its central angle.
• Histogram area represents frequency; unequal class widths require frequency-density heights to keep areas comparable.
• Frequency polygons join class-midpoint frequencies; smooth frequency curves may not necessarily pass through every plotted point.
• Ogives use cumulative frequencies and locate the median; arithmetic line graphs show values changing through time.

Test yourself

What does a cell in a statistical table represent?

It gives the information at the intersection of a particular row and column, interpreted using both headings.

What is the difference between a caption and a stub?

A caption is a column heading; a stub is a row heading identifying the category in that row.

Where should a table's source statement appear, and what should it include if several sources supply the data?

A source statement generally appears at the bottom; when several sources supply the data, identify all of them.

What total height is used for a percentage component bar?

The complete bar has a height of 100 units, divided into proportional component heights.

Why is frequency density needed for unequal histogram intervals?

Different widths require adjusted heights so that rectangle areas, rather than raw heights, remain proportional to frequencies.

Where are frequencies plotted in a frequency polygon?

Each frequency is plotted against its class midpoint, even if class boundaries label the horizontal axis.

Which limits are used for less-than and more-than ogives?

Less-than cumulative frequencies use upper class limits; more-than cumulative frequencies use lower class limits.

Where does time appear on an arithmetic line graph?

Time appears on the horizontal axis, while values of the variable appear on the vertical axis.