This MCQ module is based on: Diagrammatic Presentation — Bar, Pie, Histogram, Ogive & Exercises
TOPIC 7 OF 15
Diagrammatic Presentation — Bar, Pie, Histogram, Ogive & Exercises
🎓 Class 11
Social Science
CBSE
Theory
Ch 4 — Presentation of Data
⏱ ~28 min
🌐 Language: [gtranslate]
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Diagrammatic Presentation — Bars, Pies, Histograms, Ogives and Line Graphs
Tables put data in order. Diagrams put data on display. NCERT lists three families of diagrams — geometric (bar and pie), frequency (histogram, frequency polygon, frequency curve, ogive) and the arithmetic line graph. This part rebuilds every NCERT figure as an interactive Chart.js render, walks through each kind of bar diagram, the four-way distinction inside frequency diagrams, and finishes with the full set of end-of-chapter Exercises with model answers — plus a clean recap and key-term sheet.
4.7 Why Diagrams? The Third Form of Presentation
Diagrammatic presentation? translates the abstract numerical content of a table into shapes that the eye can compare in one glance. Diagrams are usually less precise than tables — you can read 23.6 % off a table but only "about a quarter" off a pie chart — yet diagrams are far more effective at communicating the overall pattern. NCERT groups diagrams into three families.
Geometric Diagram
Bar diagrams (simple, multiple, sub-divided/component, percentage) and the pie diagram. Used for non-frequency data, attributes and time series alike.
Frequency Diagram
Histogram, frequency polygon, frequency curve and ogive. Used only for grouped frequency distributions of a continuous variable.
Arithmetic Line Graph
Also called a time series graph. Time on the X-axis, value on the Y-axis. Reveals trend, periodicity and seasonality in long time series.
4.8 Geometric Diagrams — Bar Diagrams
4.8.1 Simple Bar Diagram
A bar diagram? uses a row of equispaced, equiwidth rectangular bars; the height of each bar reads the value of the variable. Bars rise from a common base line at zero. Bars can be drawn vertically or horizontally — vertical bars are also called columns, especially in time-series data.
NCERT's Figure 4.1 plots male literacy rates of major Indian states in 2011 (Table 4.6). Each state gets one bar; the height equals its male literacy percentage. The chart shows Kerala (96.0) at the top and Bihar (73.4) near the bottom.
| State | Male 2001 | Female 2001 | Male 2011 | Female 2011 |
|---|---|---|---|---|
| Andhra Pradesh (AP) | 70.3 | 50.4 | 75.6 | 59.7 |
| Assam (AS) | 71.3 | 54.6 | 78.8 | 67.3 |
| Bihar (BR) | 59.7 | 33.1 | 73.4 | 53.3 |
| Jharkhand (JH) | 67.3 | 38.9 | 78.4 | 56.2 |
| Gujarat (GJ) | 79.7 | 57.8 | 87.2 | 70.7 |
| Haryana (HR) | 78.5 | 55.7 | 85.3 | 66.8 |
| Karnataka (KA) | 76.1 | 56.9 | 82.9 | 68.1 |
| Kerala (KE) | 94.2 | 87.7 | 96.0 | 92.0 |
| Madhya Pradesh (MP) | 76.1 | 50.3 | 80.5 | 60.0 |
| Chhattisgarh (CH) | 77.4 | 51.9 | 81.5 | 60.6 |
| Maharashtra (MR) | 86.0 | 67.0 | 89.8 | 75.5 |
| Odisha (OD) | 75.3 | 50.5 | 82.4 | 64.4 |
| Punjab (PB) | 75.2 | 63.4 | 81.5 | 71.3 |
| Rajasthan (RJ) | 75.7 | 43.9 | 80.5 | 52.7 |
| Tamil Nadu (TN) | 82.4 | 64.4 | 86.8 | 73.9 |
| Uttar Pradesh (UP) | 68.8 | 42.2 | 79.2 | 59.3 |
| Uttarakhand (UK) | 83.3 | 59.6 | 88.3 | 70.7 |
| West Bengal (WB) | 77.0 | 59.6 | 82.7 | 71.2 |
| India | 75.3 | 53.7 | 82.1 | 65.5 |
Fig 4.1 — Simple bar diagram showing male literacy rates of major states of India, 2011 (population aged 7+).
4.8.2 Multiple Bar Diagram
A multiple bar diagram? compares two or more sets of related data — for example imports versus exports, or female literacy in 2001 versus 2011. The bars for each category sit side by side, separated from the next category by a small gap. Different colours or shading distinguish the data sets.
Fig 4.2 — Multiple bar diagram showing female literacy rates over the two census years 2001 and 2011 across major Indian states. (Source data: Table 4.6.)
🔍 Reading Figure 4.2
Female literacy rose throughout the country between 2001 and 2011. The sharpest rise can be read off bars for Bihar, Jharkhand and Uttar Pradesh — three of the most lagging states made the biggest gains. Kerala stays at the top in both years.
4.8.3 Component / Sub-Divided Bar Diagram
A component bar diagram? (also called a sub-divided bar diagram or stacked bar) shows the total of each category as one bar, then divides that bar into the proportions of its components. It is excellent for comparing both the total and the composition across categories.
NCERT's Figure 4.3 uses Table 4.7 — enrolment by gender in a Bihar district for children aged 6–14. Each total bar (Boys, Girls, All) reaches 100 % and is sub-divided into "Enrolled" and "Out of school".
| Gender | Enrolled (%) | Out of School (%) |
|---|---|---|
| Boy | 91.5 | 8.5 |
| Girl | 58.6 | 41.4 |
| All | 78.0 | 22.0 |
Fig 4.3 — Component bar diagram of enrolment at primary level, Bihar district. The girl bar shows nearly 41% out-of-school — much higher than for boys.
4.8.4 Percentage Bar Diagram
When every component bar is drawn to a uniform total of 100 %, the result is a percentage bar diagram. The Bihar enrolment chart above is already a percentage bar — Boys, Girls and All bars all reach exactly 100 %, only the internal division differs. This format isolates the compositional shift across categories.
📖 Construction Steps for a Component Bar
Step 1 — Draw a base bar on the X-axis with height equal to the total value (or 100 units for percentage data).Step 2 — Use the unitary method to convert each component into a proportional height.
Step 3 — Stack the smaller components first (priority bottom). Shade or colour each component differently.
Step 4 — Label every component clearly so the legend explains the shading.
4.8.5 Pie Diagram
A pie diagram? (or pie chart) is a circle whose area is proportionally divided among the categories it represents. Like a component bar, a pie chart shows composition — but the components are wedges of a circle instead of stacked rectangles.
To construct a pie:
- Convert every component value to a percentage of the total.
- Multiply each percentage by 3.6° (since 360°/100 = 3.6°) to get the angle subtended at the centre.
- Draw the circle and cut it into wedges using straight lines from the centre to the circumference.
NCERT's Table 4.8 shows the angle calculation for the working status of the Indian population (2011).
| Status | Population (Crore) | Per cent | Angular Component |
|---|---|---|---|
| Marginal Worker | 12 | 9.9 | 36° |
| Main Worker | 30 | 29.8 | 107° |
| Non-worker | 60 | 60.3 | 217° |
| All | 102 | 100.0 | 360° |
Fig 4.4 — Pie diagram for the working-status distribution of the Indian population (2011). Non-workers occupy 60% of the circle.
⚖️ Pie vs Component Bar — When to Use Which
Both diagrams show the same composition information. A pie's circular geometry makes the relative shares pop out (the eye reads angles well), but the area of the pie has no bearing on the absolute total. A component bar can be drawn at heights proportional to the absolute totals, so the eye can compare both composition and total magnitude at the same time. NCERT's activity asks you exactly this — to redraw Fig 4.4 as a component bar.
4.9 Frequency Diagrams
When the data are a grouped frequency distribution of a continuous variable, NCERT lists four standard diagrams: histogram, frequency polygon, frequency curve, and ogive (less-than and more-than).
4.9.1 Histogram
A histogram? is a two-dimensional diagram. The X-axis carries the class boundaries, the Y-axis carries the frequencies. Each class becomes a rectangle whose area is proportional to its frequency. When all classes have the same width, the heights themselves can serve as the frequency.
If the class widths are unequal, the height must be replaced by the frequency density — the frequency divided by the class width — so that the rectangle's area (not its height) measures the frequency.
Frequency Density = Frequency ÷ Class Width
NCERT's Table 4.9 records 85 daily wage earners' incomes in 15 classes of width Rs 5. The histogram (Fig. 4.5) is drawn from this distribution.
| Daily earning (Rs) | No. of wage earners (f) |
|---|---|
| 45–49 | 2 |
| 50–54 | 3 |
| 55–59 | 5 |
| 60–64 | 3 |
| 65–69 | 6 |
| 70–74 | 7 |
| 75–79 | 12 |
| 80–84 | 13 |
| 85–89 | 9 |
| 90–94 | 7 |
| 95–99 | 6 |
| 100–104 | 4 |
| 105–109 | 2 |
| 110–114 | 3 |
| 115–119 | 3 |
| Total | 85 |
Source: Unpublished data.
Fig 4.5 — Histogram for the distribution of 85 daily wage earners. The mode lies in the class with the tallest bar (80–84, frequency 13).
⚖️ Histogram vs Bar Diagram — Five Key Differences
1. Bar widths are arbitrary; histogram widths are not — they equal the class widths.2. Bars have gaps between them; histogram rectangles share boundaries (no gaps).
3. A bar diagram works for both discrete and continuous variables; a histogram is only for continuous variables.
4. In a bar diagram only the height matters; in a histogram height and width together form the area, which is the meaningful quantity.
5. A histogram graphically shows the mode; a bar diagram does not.
4.9.2 Frequency Polygon
A frequency polygon? is a many-sided plane figure derived from the histogram. The simplest method: join the midpoints of the tops of consecutive rectangles with straight lines. Add an extra empty class at each end (frequency = 0) and join down to the X-axis so that the area enclosed equals the total frequency.
Frequencies are always plotted against the mid-point (class mark) of each class on the X-axis. NCERT's Fig. 4.6 draws the frequency polygon for Table 4.9.
Fig 4.6 — Frequency polygon for the wage data of Table 4.9. The two ends touch the base line at the mid-points of empty adjacent classes.
💡 Why a Frequency Polygon is Useful
Two or more distributions plotted as histograms tend to overlap and become unreadable. The same distributions drawn as frequency polygons sit on top of one another cleanly — you can directly compare their shapes.
4.9.3 Frequency Curve
A frequency curve? is obtained by drawing a smooth free-hand curve through the points of the frequency polygon — passing as close to each point as possible (it does not need to pass through every point). Smoothing reveals the underlying shape of the distribution: bell-shaped, skewed, U-shaped and so on.
Fig 4.7 — Frequency curve fitted to the wage data of Table 4.9. The smoothing reveals a slightly right-skewed peak around the Rs 80–85 class.
4.9.4 Ogive — The Cumulative Frequency Curve
An ogive? plots cumulative frequency on the Y-axis against class limits on the X-axis. There are two kinds:
- Less-than ogive: Cumulative frequencies are plotted against the upper class limits. The curve is non-decreasing — it rises from 0 to N (the total).
- More-than ogive: Cumulative frequencies are plotted against the lower class limits. The curve is non-increasing — it falls from N down to 0.
The two ogives together intersect at the median of the distribution.
| Marks | Number of students (f) | Less than CF | More than CF |
|---|---|---|---|
| 0–20 | 6 | 6 (less than 20) | 64 (more than 0) |
| 20–40 | 5 | 11 (less than 40) | 58 (more than 20) |
| 40–60 | 33 | 44 (less than 60) | 53 (more than 40) |
| 60–80 | 14 | 58 (less than 80) | 20 (more than 60) |
| 80–100 | 6 | 64 (less than 100) | 6 (more than 80) |
| Total | 64 |
Fig 4.8 — Less-than and More-than ogives for Table 4.10. The intersection X-coordinate is the median (close to mark 56).
4.10 The Arithmetic Line Graph (Time-Series Graph)
An arithmetic line graph? plots time on the X-axis and the value of a variable on the Y-axis, joining successive points by straight line segments. It reveals long-term trend, periodicity (cycles) and seasonality. NCERT's Table 4.11 lists India's exports and imports from 1993–94 to 2013–14 (Rs in 100 crores).
| Year | Exports | Imports |
|---|---|---|
| 1993–94 | 698 | 731 |
| 1994–95 | 827 | 900 |
| 1995–96 | 1,064 | 1,227 |
| 1996–97 | 1,188 | 1,389 |
| 1997–98 | 1,301 | 1,542 |
| 1998–99 | 1,398 | 1,783 |
| 1999–2000 | 1,591 | 2,155 |
| 2000–01 | 2,036 | 2,309 |
| 2001–02 | 2,090 | 2,452 |
| 2002–03 | 2,549 | 2,964 |
| 2003–04 | 2,934 | 3,591 |
| 2004–05 | 3,753 | 5,011 |
| 2005–06 | 4,564 | 6,604 |
| 2006–07 | 5,718 | 8,815 |
| 2007–08 | 6,559 | 10,123 |
| 2008–09 | 8,408 | 13,744 |
| 2009–10 | 8,455 | 13,637 |
| 2010–11 | 11,370 | 16,835 |
| 2011–12 | 14,660 | 23,455 |
| 2012–13 | 16,343 | 26,692 |
| 2013–14 | 19,050 | 27,154 |
Source: DGCI&S, Kolkata.
Fig 4.9 — Arithmetic line graph for India's exports and imports, 1993–94 to 2013–14. Imports stayed above exports throughout. Both rose sharply after 2001–02 and the gap (trade deficit) widened thereafter.
🔍 Reading Figure 4.9
Three quick observations: (i) imports stay above exports in every year — a persistent trade deficit; (ii) the rise after 2001–02 is steeper than the rise before it; (iii) the export–import gap widens sharply post-2001, reflecting India's rising oil and gold imports during the boom.
4.11 Choropleth Maps and Cartograms — Brief Mention
Beyond the bar, pie, histogram and line graph, two more diagrammatic tools appear in modern statistical work, mentioned here because economic data are often spatial.
Choropleth Map
A geographical map shaded or coloured in proportion to a statistical variable — for example district-wise literacy intensity. Darker shades = higher values. Used widely by Census of India.
Cartogram
A map in which the geographical area of each region is distorted in proportion to the variable being shown — e.g. countries scaled by population. Highlights statistical magnitude over physical size.
4.12 Conclusion
By now you can choose between three forms of presentation. Textual works for small datasets where emphasis is needed. Tabular accommodates any volume across one or more variables. Diagrams — bar, pie, histogram, frequency polygon/curve, ogive, time-series line graph — translate the table into shapes that make the pattern visible at a glance. The right choice of form depends entirely on the message and the audience.
EXPLORE — Bar Diagram for Your School
Bloom: L3 Apply
NCERT's activity asks you to collect the number of students in each class of your school in the current year and draw a bar diagram from that table.
✅ Sample
Suppose Class VI = 80, Class VII = 78, Class VIII = 75, Class IX = 90, Class X = 95, Class XI = 60, Class XII = 55. Plot equal-width bars on the X-axis (one per class) with heights proportional to the enrolment numbers. The two highest bars (IX and X) and the dip at XI–XII immediately reveal a pattern that's invisible in the raw row of numbers.
DISCUSS — Reading Figure 4.2
Bloom: L4 Analyse
NCERT poses two follow-up questions on Figure 4.2 (female literacy 2001 vs 2011 across major states). Answer both.
- How many of the major states had female literacy higher than the national average (65.5 %) in 2011?
- Did the gap between maximum and minimum female literacy across states narrow between 2001 and 2011?
✅ Sample
Q1. States above 65.5 % female literacy in 2011 (from Table 4.6): Assam 67.3, Gujarat 70.7, Haryana 66.8, Karnataka 68.1, Kerala 92.0, Maharashtra 75.5, Punjab 71.3, Tamil Nadu 73.9, Uttarakhand 70.7, West Bengal 71.2 — that is 10 states above the national average.Q2. In 2001 the range was Kerala (87.7) − Bihar (33.1) = 54.6. In 2011 the range was Kerala (92.0) − Bihar (53.3) = 38.7. The gap shrank by about 16 percentage points — yes, the gap declined.
THINK — Pie Area and Total Value
Bloom: L5 Evaluate
NCERT asks: does the area of a pie chart have any bearing on the total value of the data being represented? Justify in 2–3 lines.
✅ Sample
No. The radius of a pie is chosen by the diagram-maker for visual convenience — the same data can be drawn as a small pie or a large pie and the message is identical. A pie communicates the relative share of each component (the angles), not the absolute total. To convey the total magnitude, a component bar drawn at a height equal to the absolute total is preferable.
4.13 Worked CBQ — Choosing the Right Diagram
📊 Case-Based Question — Picking a Diagram for the Job
A research student has the following datasets and must decide which kind of diagram suits each best: (i) annual rainfall in Mumbai for 12 months of last year, (ii) the religious composition of Delhi's population, (iii) the cost components of running a textile factory, (iv) a frequency distribution of monthly household income.
Q1. Which diagram best represents the monthly rainfall over 12 months and why?
L2 UnderstandAnswer: A simple bar (column) diagram or an arithmetic line graph. Time on the X-axis (months), rainfall (mm) on the Y-axis. A bar diagram makes month-to-month comparison easy; a line graph emphasises the rising-falling monsoon trend.
Q2. Which diagram should represent the religious composition of Delhi?
L2 UnderstandAnswer: A pie diagram (or, equivalently, a percentage component bar). Composition data — where the parts must sum to 100 % — are most naturally shown as wedges of a circle. The reader instantly sees which religion forms the majority and how big the minority shares are.
Q3. The textile factory's cost components are: raw material 45%, wages 30%, overheads 15%, marketing 10%. Which diagram and why?
L3 ApplyAnswer: A component bar diagram (sub-divided bar) with one bar of 100 % height divided into four shaded segments — 45, 30, 15, 10. A pie chart works equally well. If the analyst wants to compare the same composition across multiple years, several component bars side-by-side beat a single pie.
Q4. The frequency distribution of household income (10 classes) needs a graph that also shows the median visually. Suggest a diagram and explain.
L5 EvaluateAnswer: Draw both the less-than ogive and the more-than ogive on the same axes. The X-coordinate at which they intersect is the median of the income distribution — exactly the diagram NCERT shows in Fig. 4.8(b). A histogram alone cannot show the median visually but does reveal the mode.
⚖️ Assertion–Reason Questions (Class 11)
Choose: (A) Both A and R are true and R is the correct explanation of A. (B) Both A and R are true but R is not the correct explanation of A. (C) A is true, R is false. (D) A is false, R is true.
Assertion (A): A histogram cannot have any space between two consecutive rectangles, but a bar diagram must have space between consecutive bars.
Reason (R): Histograms are drawn for continuous variables where the upper boundary of one class fuses with the lower boundary of the next, while bar diagrams represent discrete categories that have no such mathematical adjacency.
Correct: (A) — Both statements are true and R correctly explains A. Continuity of class boundaries forces histogram rectangles to be adjacent; discrete categories in a bar diagram have arbitrary spacing.
Assertion (A): The intersection point of the less-than and more-than ogives gives the median of the frequency distribution.
Reason (R): At the median, exactly half the observations lie below it and half above it — which is precisely the point at which the cumulative-up curve and the cumulative-down curve meet.
Correct: (A) — Both A and R are true and R correctly explains A. NCERT illustrates this property in Fig. 4.8(b).
Assertion (A): An arithmetic line graph is the most suitable diagram for representing the cyclical and seasonal pattern of long time-series data.
Reason (R): The arithmetic line graph plots time along the X-axis and joins successive data points with straight-line segments, allowing trend, periodicity and seasonality to be read at a glance.
Correct: (A) — Both A and R are true and R correctly explains A. NCERT's Fig. 4.9 shows exactly this — the rising trend, the post-2001 acceleration and the widening export-import gap.
4.14 NCERT Exercises — Full Set with Model Answers
EXERCISES — Chapter 4: Presentation of Data
Reproduce of NCERT end-of-chapter questions (Q1–Q15) with model answers. Answer the multiple-choice questions 1–4, the True/False questions 5–10, and the conceptual questions 11–15.
1Bar diagram is a — (i) one-dimensional diagram, (ii) two-dimensional diagram, (iii) diagram with no dimension, (iv) none of the above.
Answer: (i) one-dimensional diagram. In a bar diagram only the height (length) of each bar carries information — the width is uniform and arbitrary. So the bar conveys magnitude in only one dimension, height. Histograms by contrast are two-dimensional because both height and width matter.
2Data represented through a histogram can help in finding graphically the — (i) mean, (ii) mode, (iii) median, (iv) all the above.
Answer: (ii) mode. A histogram graphically locates the mode — by joining the upper corners of the modal rectangle with diagonal lines to the corresponding corners of the adjacent rectangles, the X-coordinate of their intersection gives the modal value. Mean and median cannot be read straight from a histogram.
3Ogives can be helpful in locating graphically the — (i) mode, (ii) mean, (iii) median, (iv) none of the above.
Answer: (iii) median. The X-coordinate of the intersection of the less-than ogive and the more-than ogive is the median of the distribution. The mode is read from a histogram, not an ogive.
4Data represented through arithmetic line graph help in understanding — (i) long term trend, (ii) cyclicity in data, (iii) seasonality in data, (iv) all the above.
Answer: (iv) all the above. A time-series line graph reveals all three at once — the long-term trend (overall rise/fall), cyclical fluctuations (multi-year swings) and seasonality (within-year repetitive patterns).
5Width of bars in a bar diagram need not be equal. (True/False)
False. By convention, all bars in a bar diagram are drawn with the same width to ensure that visual comparison is fair — only the height differs. Unequal widths would mislead the eye, since width is otherwise arbitrary.
6Width of rectangles in a histogram should essentially be equal. (True/False)
False. Histogram rectangles can have unequal widths if the underlying class intervals are unequal. In that case the height is replaced by the frequency density (frequency ÷ class width) so that the rectangle's area remains proportional to the frequency.
7Histogram can only be formed with continuous classification of data. (True/False)
True. A histogram requires a continuous variable so that the rectangles share class boundaries and lie adjacent without gaps. Discrete data with inclusive classes must first be converted to continuous form (the ±0.5 adjustment) before a histogram can be drawn.
8Histogram and column diagram are the same method of presentation of data. (True/False)
False. A column diagram is a vertical bar diagram and shows discrete or non-frequency data with gaps between bars. A histogram is a frequency diagram for continuous data with no gaps, where area (not just height) measures frequency.
9Mode of a frequency distribution can be known graphically with the help of a histogram. (True/False)
True. The modal class is the rectangle of greatest height. By joining the upper-left corner of the modal rectangle to the upper-left corner of the next rectangle, and the upper-right corner of the modal rectangle to the upper-right corner of the previous rectangle, the X-coordinate of their intersection gives the mode.
10Median of a frequency distribution cannot be known from the ogives. (True/False)
False. The median can be read off the ogives — it is the X-coordinate at which the less-than and more-than ogives intersect. NCERT's Fig. 4.8(b) demonstrates exactly this.
11What kind of diagrams are more effective in representing the following? (i) Monthly rainfall in a year. (ii) Composition of the population of Delhi by religion. (iii) Components of cost in a factory.
(i) Monthly rainfall — a simple bar (column) diagram or an arithmetic line graph. Time on X-axis, rainfall (mm) on Y-axis.
(ii) Composition of Delhi's population by religion — a pie diagram (or percentage component bar). Composition data sum to 100% and translate naturally into wedges/segments.
(iii) Components of cost in a factory — a component bar diagram (sub-divided bar) or a pie chart, since the parts add up to total cost.
(ii) Composition of Delhi's population by religion — a pie diagram (or percentage component bar). Composition data sum to 100% and translate naturally into wedges/segments.
(iii) Components of cost in a factory — a component bar diagram (sub-divided bar) or a pie chart, since the parts add up to total cost.
12Suppose you want to emphasise the increase in the share of urban non-workers and lower level of urbanisation in India as shown in Example 4.2. How would you do it in tabular form?
Answer: Use a two-way table classified by location (rural/urban/all) along the stub and by status (workers/non-workers/total) along the captions, with figures both in absolute crore values and as percentages of the row total. Add a head note "(Crore; per cent in parentheses)".
The percentages in parentheses make the urban non-worker share (68%) jump out compared to the rural share (58%); the row totals (74 vs 28 crore) show the lower urbanisation.
| Location | Workers (% of total) | Non-workers (% of total) | Total |
|---|---|---|---|
| Rural | 31 (42%) | 43 (58%) | 74 |
| Urban | 9 (32%) | 19 (68%) | 28 |
| All India | 40 (39%) | 62 (61%) | 102 |
13How does the procedure of drawing a histogram differ when class intervals are unequal in comparison to equal class intervals in a frequency table?
Answer: When class intervals are equal, the height of every rectangle equals the class frequency directly — area is automatically proportional to frequency. When intervals are unequal, the height is replaced by the frequency density (frequency ÷ class width) so that the area of each rectangle (height × width) remains proportional to its frequency. Without this adjustment the eye would mistakenly read frequency from height alone, producing a distorted picture of the distribution.
14The Indian Sugar Mills Association reported that sugar production in the first fortnight of December 2001 was 3,87,000 tonnes against 3,78,000 tonnes a year earlier (2000). The off-take of sugar from factories during the first fortnight of December 2001 was 2,83,000 tonnes for internal consumption and 41,000 tonnes for exports, against 1,54,000 tonnes (consumption) and nil for exports a year earlier. (i) Present the data in tabular form. (ii) Which diagrammatic form is suitable and why? (iii) Present these data diagrammatically.
(i) Tabular form (000 tonnes):
Source: Indian Sugar Mills Association.
(ii) Diagram choice: a multiple bar diagram (two bars per item, one per year) is most suitable. The data has three items × two years = six values, perfectly suited for paired comparison. Alternatively a sub-divided/component bar diagram showing total off-take split into "consumption + exports" makes the doubling of off-take starkly visible.
(iii) See the chart below — multiple bar diagram comparing 2000 and 2001 figures for production, internal consumption and exports.
| Item | 1st fortnight Dec 2000 | 1st fortnight Dec 2001 |
|---|---|---|
| Production | 378 | 387 |
| Off-take — Internal Consumption | 154 | 283 |
| Off-take — Exports | 0 | 41 |
| Total Off-take | 154 | 324 |
(ii) Diagram choice: a multiple bar diagram (two bars per item, one per year) is most suitable. The data has three items × two years = six values, perfectly suited for paired comparison. Alternatively a sub-divided/component bar diagram showing total off-take split into "consumption + exports" makes the doubling of off-take starkly visible.
(iii) See the chart below — multiple bar diagram comparing 2000 and 2001 figures for production, internal consumption and exports.
15The following table shows the estimated sectoral real growth rates (percentage change over the previous year) in GDP at factor cost. Represent the data as multiple time-series graphs.
Answer:
Plot all three series on a single set of axes — year on the X-axis, growth-rate on the Y-axis. Three differently-coloured lines reveal the volatility of agriculture (range −1.9 % to 9.6 %), the steady-but-declining industry, and the consistently strong services sector.
| Year | Agriculture & allied (%) | Industry (%) | Services (%) |
|---|---|---|---|
| 1994–95 | 5.0 | 9.2 | 7.0 |
| 1995–96 | −0.9 | 11.8 | 10.3 |
| 1996–97 | 9.6 | 6.0 | 7.1 |
| 1997–98 | −1.9 | 5.9 | 9.0 |
| 1998–99 | 7.2 | 4.0 | 8.3 |
| 1999–2000 | 0.8 | 6.9 | 8.2 |
📋 Summary — Recap of Chapter 4
- Data — even voluminous data — speak meaningfully only through presentation.
- For small data, textual presentation serves the purpose better, allowing emphasis on particular figures.
- For large data, tabular presentation accommodates any volume across one or more variables. Tables have eight standard parts: number, title, captions, stubs, body, unit, source, footnote.
- Tabulation classifies data four ways: qualitative (attribute), quantitative (measurement), temporal (time), and spatial (place).
- Tabulated data can be presented through diagrams for quicker comprehension. NCERT identifies three families: geometric (bar/pie), frequency (histogram, polygon, curve, ogive) and arithmetic line graph.
- Bar diagrams come in four flavours — simple, multiple, component (sub-divided), and percentage. They are one-dimensional (only height matters).
- Histograms are two-dimensional and apply only to continuous variables; the area of each rectangle is proportional to the frequency. The mode can be read graphically from a histogram.
- Frequency polygons (joined midpoints), frequency curves (smoothed polygons), and ogives (cumulative — less-than and more-than) all derive from grouped frequency data. The intersection of the two ogives gives the median.
- The arithmetic line graph (time-series graph) places time on the X-axis and the variable on the Y-axis — best for showing long-term trend, cyclicity and seasonality.
- Choropleth maps and cartograms add a geographical dimension to statistical presentation.
🔑 Key Terms
Textual presentation — Data described inside the running text of a paragraph; suitable for very small datasets.
Tabular presentation — Data arranged in rows and columns according to a logical classification.
Caption / Column heading — Designation of each column at the top of a table.
Stub / Row heading — Designation of each row, written in the leftmost stub column.
Head note / Unit — Brief statement of the unit of measurement, placed near the title.
Source note — Brief statement at the bottom of the table indicating the origin of the data.
Footnote — Final note explaining any feature of the data not self-explanatory.
Bar diagram — A row of equispaced rectangular bars; height proportional to value.
Multiple bar diagram — Side-by-side bars used to compare two or more sets of related data.
Sub-divided / Component bar diagram — A single bar split into proportional segments showing composition.
Pie diagram — A circle with wedges proportional (in angle) to the percentage shares of components.
Histogram — Two-dimensional rectangles for continuous frequency data; area = frequency.
Frequency polygon — Lines joining the mid-points of histogram tops; many-sided figure.
Frequency curve — Smooth free-hand curve passing through frequency-polygon points.
Ogive — Cumulative frequency curve; less-than (rising) or more-than (falling). Their intersection is the median.
Arithmetic line graph — Time-series graph with time on X-axis and value on Y-axis; reveals trend, cyclicity and seasonality.
Frequently Asked Questions — Diagrammatic Presentation — Bars, Pies, Histograms, Ogives and Line Graphs
What is the difference between a histogram and a bar diagram in NCERT Class 11?
A bar diagram has visible gaps between bars and is used for discrete or categorical data such as production by industry or marks by subject; the height of each bar represents the value or frequency. A histogram has no gaps between rectangles and is used for continuous grouped data; the rectangles are drawn over class intervals and the area is proportional to the frequency. NCERT Class 11 Statistics Chapter 4 Part 2 stresses that histograms must use exclusive class intervals and equal class widths (or frequency density for unequal widths), while bar diagrams allow any uniform spacing.
How do you construct a pie chart in NCERT Class 11 Statistics Chapter 4?
To construct a pie chart in NCERT Class 11 Statistics Chapter 4 Part 2, first calculate the percentage that each component contributes to the total, then multiply each percentage by 3.6 (since 100% equals 360 degrees) to get the angle of each sector. Draw a circle, mark the angles from the centre using a protractor starting at the top (12 o'clock position), and shade or label each sector with its category name and percentage. Pie charts are best when there are between three and seven categories that together sum to a meaningful whole, such as a household budget.
What is a multiple bar diagram and when is it used?
A multiple bar diagram (also called compound bar diagram) groups two or more related bars side by side for each category to allow direct comparison between sub-categories. NCERT Class 11 Statistics Chapter 4 Part 2 gives the example of male and female literacy rates plotted side by side for several years, where each year has two adjacent bars in different colours. The multiple bar diagram is used when comparing the same set of components across categories, but if you want to show how parts make up a whole, a component (stacked) bar diagram is used instead.
How do you find the median graphically using ogives in Class 11 Statistics?
To find the median graphically using ogives in NCERT Class 11 Statistics Chapter 4 Part 2, draw both the less-than ogive and the more-than ogive on the same axes; the X-coordinate of the point where the two curves intersect gives the median. Alternatively, using just the less-than ogive, mark N/2 on the Y-axis (where N is total frequency), draw a horizontal line to the curve, then drop a perpendicular to the X-axis and read off the median value. Both methods give the same answer and are commonly tested in CBSE board exams.
What is a time-series line graph in NCERT Class 11 Statistics?
A time-series line graph plots time on the X-axis (years, months, quarters) and the value of a variable on the Y-axis, joining successive points with straight line segments to show the variable's movement over time. NCERT Class 11 Statistics Chapter 4 Part 2 explains that line graphs are ideal for displaying trends, fluctuations and turning points in economic data — for example yearly GDP, monthly inflation or daily Sensex closes. Multiple variables can be plotted on the same graph using different line styles or colours, allowing direct comparison of growth rates or cyclical patterns across series.
What is the difference between a frequency polygon and a frequency curve?
A frequency polygon is a line graph that joins the midpoints of the tops of histogram rectangles using straight line segments, with the lines closing on the X-axis at imaginary classes just before and after the data. A frequency curve is the smoothed version of the polygon, drawn freehand to remove sharp angles and represent the underlying continuous distribution. NCERT Class 11 Statistics Chapter 4 Part 2 explains that the frequency curve is preferred for visualising the general shape of a distribution (symmetrical, skewed, U-shaped, J-shaped) once large amounts of data make the polygon's segments look almost continuous.
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