Presentation of Data Class 11: Tables, Bar and Pie Diagrams, Histogram and Ogive

What Presentation of Data is about

Weightage: This chapter is part of Unit 2, Collection, Organisation and Presentation of Data. The syllabus groups Unit 2 with Unit 1, Introduction: together they carry 15 of the 80 theory marks in the school annual exam.Source: CBSE Economics (030) syllabus 2026-27, Class XI course structure, Unit 2 Collection, Organisation and Presentation of Data

Key concepts

Textual and tabular presentation

NCERT section 2 and 3

Textual (descriptive) presentation: data are written in sentences. Example: "Of the 40 students, 15 scored between 30 and 40 marks, the largest group." Suitable only when there are few figures; with many figures the reader gets lost.

Tabular presentation: data are arranged in rows (horizontal) and columns (vertical). Advantages: many figures in little space; easy comparison; ready for analysis; no repetition of words.

Classification of tables (by the basis used to classify the data):

• Qualitative: by attributes (literacy by gender).
• Quantitative: by measurable characteristics (families by income class).
• Temporal: by time (production by year).
• Spatial: by place (rainfall by state).

Tables may also be simple (one characteristic) or complex (two or more characteristics, with sub-columns).

Worked example

Suppose data show the number of male and female students in Classes 11 and 12 of a school. Which kind of table is this, and how many cells does the body need?

Try it first, then show the solutionSolution

1. Classified by attribute (gender) and by class: a qualitative, two-way (complex) table
2. Rows: Class 11, Class 12 (plus total); columns: Male, Female (plus total)
3. Body: 2 × 2 = 4 cells, or 3 × 3 = 9 with totals

Answer: A qualitative two-way table; 4 data cells (9 with totals).

Parts of a table

NCERT section 4

1. Table number: for reference (Table 4.1).
2. Title: says what, where, when and how classified; short and clear, placed at the top.
3. Captions or column headings: describe each column; may have sub-heads.
4. Stubs or row headings: describe each row.
5. Body: the numerical data, the main part of the table.
6. Unit of measurement: (₹ crore, kg, per cent) stated with the title or headings.
7. Source: where the data come from (for secondary data), placed below the table.
8. Note: explains anything special (figures rounded, provisional, excludes a region), placed at the end.

A good table is simple, has a clear title, avoids repetition, shows totals, and arranges items in a logical order (alphabetical, by size, by time).

Worked example

Write a suitable title for a table showing the number of two-wheelers sold by three dealers in Pune in each quarter of 2025-26.

Try it first, then show the solutionSolution

1. Title must say what (two-wheeler sales), where (Pune), when (2025-26) and how classified (by dealer and quarter)

Answer: "Table 1: Quarterly sales of two-wheelers by three dealers in Pune, 2025-26 (number of vehicles)".

Bar diagrams: simple, multiple and component

NCERT section 5

A bar diagram uses bars of equal width with equal gaps; only the height (length) matters. Bars can be vertical or horizontal. It suits both discrete and continuous variables, and attributes.

• Simple bar diagram: one bar per item (production in different years; population of states). Bars may be arranged in order of size.
• Multiple bar diagram: two or more bars side by side for each item, to compare related series (wheat and rice output each year; male and female literacy by state). A legend shows which bar is which.
• Component (sub-divided) bar diagram: one bar per item divided into parts, showing the composition of a total (sales by region in each year). Parts can be shown in actual values or in percentages (all bars of equal height = 100).

Year	Wheat (thousand tonnes)	Rice (thousand tonnes)
2022-23	120	80
2023-24	135	85
2024-25	150	95

Component bars by percentage for the same data:

Year	Wheat share	Rice share	Total (thousand tonnes)
2022-23	60%	40%	200
2023-24	61.4%	38.6%	220
2024-25	61.2%	38.8%	245

Worked example

Which bar diagram suits each: (a) exports of India in five years; (b) boys' and girls' enrolment in three schools; (c) the share of rent, food and fees in a family's spending in three years?

Try it first, then show the solutionSolution

1. (a) One series over years: simple bar
2. (b) Two related series compared side by side: multiple bar
3. (c) Parts of a total: component bar (percentage)

Answer: (a) Simple; (b) multiple; (c) component bar diagram.

Pie diagram

NCERT section 5

A pie diagram is a circle divided into sectors whose angles are proportional to the parts of a total. It shows the composition of a whole.

Angle of a sector = (Value of the part ÷ Total) × 360°, or = Percentage × 3.6°.

Steps: convert values to percentages or angles; draw a circle; draw the largest sector first from the 12 o'clock position, going clockwise; label each sector (or use a legend).

Pie diagrams are not drawn for absolute values of different totals; they show shares. With too many small parts, they become hard to read.

Item	Expenditure (₹)	Percentage	Angle at centre
Food	10,800	45%	162°
Housing	6,000	25%	90°
Education	3,600	15%	54°
Transport	2,400	10%	36°
Others	1,200	5%	18°
Total	24,000	100%	360°

Worked example

Find the angle for "Housing" in the family budget above (₹6,000 out of ₹24,000).

Try it first, then show the solutionSolution

1. Angle = 6,000 ÷ 24,000 × 360°
2. = 0.25 × 360° = 90°

Answer: 90°

Histogram, including unequal classes

NCERT section 5

A histogram is a set of adjacent rectangles drawn for a frequency distribution of a continuous variable: the class intervals are on the x-axis and the area of each rectangle is proportional to the class frequency.

• With equal classes, heights = frequencies.
• With unequal classes, heights must be adjusted: use frequency density = frequency ÷ class width, or the adjusted frequency = (frequency × standard width) ÷ class width.
• Inclusive classes must be made exclusive first so that the rectangles touch.
• The mode can be located from a histogram (the tallest rectangle).

Histogram vs bar diagram: in a histogram there is no gap between rectangles, and width matters (area represents frequency); in a bar diagram there are gaps and only height matters. Bar diagrams suit discrete and qualitative data too; histograms only continuous data.

Age (years)	Customers (f)	Class width	Frequency density (f ÷ width)	Adjusted frequency (width 10)
10-20	10	10	1	10
20-30	25	10	2.5	25
30-40	30	10	3	30
40-60	24	20	1.2	12
60-80	11	20	0.55	5.5

Worked example

In the table above, why is the height for 40-60 taken as 12 and not 24?

Try it first, then show the solutionSolution

1. The class 40-60 is twice as wide as the standard width of 10
2. Adjusted frequency = 24 × 10 ÷ 20 = 12
3. So the area (12 × 20) still equals 24 × 10, keeping area proportional to frequency

Answer: Because unequal width must be corrected; the height is the frequency density scaled to width 10.

Frequency polygon and frequency curve

NCERT section 5

A frequency polygon is drawn by plotting class marks (x-axis) against frequencies (y-axis) and joining the points with straight lines. It can be drawn from a histogram by joining the mid-points of the tops of the rectangles. To close it, the line is extended to the mid-points of an imaginary class with zero frequency at each end.

For the marks distribution (classes 0-10 ... 40-50) the points are (5, 1), (15, 6), (25, 12), (35, 15), (45, 6).

A polygon is better than a histogram for comparing two or more distributions on the same graph.

A frequency curve is a smooth freehand curve drawn through (or close to) the points of the frequency polygon. It shows the general shape of the distribution (symmetric, skewed).

Worked example

Give the points to plot for a frequency polygon of: 10-20: 4; 20-30: 9; 30-40: 6.

Try it first, then show the solutionSolution

1. Class marks 15, 25, 35
2. Points (15, 4), (25, 9), (35, 6); close at (5, 0) and (45, 0)

Answer: (5, 0), (15, 4), (25, 9), (35, 6), (45, 0).

Ogive (cumulative frequency curve)

NCERT section 5

An ogive is a curve of cumulative frequencies. There are two:

• Less than ogive: less than cumulative frequencies plotted against upper limits; it rises (never decreases).
• More than ogive: more than cumulative frequencies plotted against lower limits; it falls (never increases).

The two ogives intersect at the median: drop a perpendicular from the point of intersection to the x-axis.

For the marks of 40 students:

Upper limit	Less than c.f.	Lower limit	More than c.f.
10	1	0	40
20	7	10	39
30	19	20	33
40	34	30	21
50	40	40	6

The ogives cross where both cumulative frequencies equal 20 (half of 40), at about 30.67 marks, which matches the median found by formula: 30 + (20 − 19) ÷ 15 × 10 = 30.67.

Worked example

Against which values are the points of a less than ogive and a more than ogive plotted?

Try it first, then show the solutionSolution

1. Less than ogive: upper class limits
2. More than ogive: lower class limits

Answer: Upper limits (less than) and lower limits (more than).

Arithmetic line graph (time series graph)

NCERT section 5

An arithmetic line graph shows the value of a variable over time: time (years, months) on the x-axis and the variable on the y-axis; points are joined by straight lines. It shows trends, fluctuations and turning points at a glance. Two or more series can be drawn on the same graph to compare them (e.g. exports and imports).

Year	Sales of a cycle firm (₹ lakh)
2019-20	42
2020-21	31
2021-22	38
2022-23	47
2023-24	53
2024-25	58

Plotted, this shows a fall in 2020-21 and a steady rise afterwards. A false base line (a break in the y-axis) is used when values are large and far from zero, so that changes are visible.

Worked example

From the sales table, in which year did sales fall, and by what percentage?

Try it first, then show the solutionSolution

1. Sales fell from 42 to 31 lakh in 2020-21
2. Fall = 11 ÷ 42 × 100 ≈ 26.2%

Answer: 2020-21, a fall of about 26.2%.

Presentation of Data in real life

Each concept, met in an everyday situation. Read the situation, answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Topics in this chapter

• Histogram, Frequency Polygon and Ogive
• Tabulation, Bar Diagrams and Pie Diagrams

Presentation of Data: key facts

Angle of a pie sector

Angle = (Part ÷ Total) × 360° = Percentage × 3.6°

All angles add up to 360°.

Frequency density

Frequency density = Class frequency ÷ Class width

Used for histograms with unequal classes.

Adjusted frequency

Adjusted frequency = Frequency × (Standard width ÷ Class width)

Standard width: usually the smallest or most common width.

Class mark (for polygon)

Class mark = (Lower limit + Upper limit) ÷ 2

x-coordinate of each polygon point.

Ogive points

Less than: (upper limit, less than c.f.); More than: (lower limit, more than c.f.)

Intersection gives the median.

Percentage share (component bar)

Share = Part ÷ Total × 100

All percentage bars have height 100.

How the school exam asks Presentation of Data

The question patterns to expect, and the method that works for each.

Tricks to remember Presentation of Data

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Parts of a table: "N-T-C-S-B-U-S-N"

  "Nine Tigers Can Swim But Usually Stay Near": Number, Title, Captions, Stubs, Body, Unit, Source, Note.

  Use it when: Listing parts of a table.

• Ogive rule

  Less than → Upper limits; More than → Lower limits ("LU, ML").

  Use it when: Plotting ogives.

• Pie formula

  Percent × 3.6 = degrees.

  Use it when: Pie angles.

Presentation of Data: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 2 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Presentation of Data

The slips examiners see most often, and what to write instead.

• ✗Mistake: Leaving gaps between histogram rectangles

  Habit from bar diagrams.

  ✓Do this: Histogram rectangles are adjacent because the variable is continuous.

  Usually costs: 1 mark.

• ✗Mistake: Not adjusting heights for unequal classes

  Using frequencies directly.

  Wrong working

  Class 40-60 drawn with height 24

  Right working

  Height 24 × 10 ÷ 20 = 12

  ✓Do this: Use frequency density or adjusted frequency.

  Usually costs: 1-2 marks.

• ✗Mistake: Plotting the less than ogive against lower limits

  Mixing up the two ogives.

  ✓Do this: Less than: upper limits. More than: lower limits.

  Usually costs: 1-2 marks.

• ✗Mistake: Pie angles not adding to 360°

  Rounding each angle.

  ✓Do this: Keep one decimal place or adjust the largest sector so the total is exactly 360°.

  Usually costs: ½-1 mark.

• ✗Mistake: No title, scale or legend on a diagram

  Students focus on the drawing.

  ✓Do this: Every diagram needs a title, a scale for each axis, axis labels and a legend for multiple or component bars.

  Usually costs: 1 mark.

NCERT exercise guide: Presentation of Data

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Presentation of Data

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   In a histogram, the area of each rectangle is proportional to:
   (a) class mark
   (b) class frequency
   (c) cumulative frequency
   (d) class width only

   Hint

   Area, not height.

   Answer

   (b) class frequency

2. Q2 · 1 mark · MCQExam level

   The part of a table that gives the column headings is called:
   (a) stub
   (b) caption
   (c) body
   (d) note

   Hint

   Columns.

   Answer

   (b) caption

3. Q3 · 1 mark · Very short answerExam level

   What is an arithmetic line graph also called?

   Hint

   Time.

   Answer

   A time series graph.

4. Q4 · 4 marks · Long answerExam level

   Explain any four parts of a table.

   Hint

   Number, title, captions, stubs, body, unit, source, note.

   Answer

   Table number for reference; title stating what, where, when; captions (column headings) and stubs (row headings) describing columns and rows; body containing the data (also unit, source, note).

5. Q5 · 3 marks · Short answerHarder exam level

   Find the pie angles: exports of a firm by region: North ₹45 lakh, South ₹30 lakh, East ₹15 lakh, West ₹30 lakh.

   Hint

   Total 120.

   Answer

   North 135°, South 90°, East 45°, West 90° (total 360°).

6. Q6 · 3 marks · Short answerHarder exam level

   Find adjusted heights for a histogram: 0-10: 5; 10-20: 12; 20-40: 16; 40-70: 18.

   Hint

   Standard width 10.

   Answer

   5, 12, 16 × 10/20 = 8, 18 × 10/30 = 6.

7. Q7 · 3 marks · Short answerHarder exam level

   Give the points to plot for a more than ogive: 0-20: 8; 20-40: 15; 40-60: 20; 60-80: 7.

   Hint

   Lower limits.

   Answer

   Total 50. Points: (0, 50), (20, 42), (40, 27), (60, 7).

8. Q8 · 3 marks · Short answerHarder exam level

   Distinguish between a histogram and a bar diagram on three bases.

   Hint

   Gaps, width, data type.

   Answer

   Gaps: none in a histogram vs equal gaps between bars. Width: matters (area = frequency) vs only height matters. Data: continuous frequency distributions vs discrete, continuous or qualitative data.

9. Q9 · 4 marks · Long answerHarder exam level

   Explain simple, multiple and component bar diagrams with an example each.

   Hint

   One series, many series, parts.

   Answer

   Simple: one bar per item, e.g. rice output by year. Multiple: bars side by side for related series, e.g. male and female literacy by state. Component: one bar divided into parts, e.g. spending on food, rent and others each year.

10. Q10 · 4 marks · Long answerHarder exam level

    What are the advantages and limitations of diagrammatic presentation?

    Hint

    Attractive vs approximate.

    Answer

    Advantages: attractive and easy to understand; quick comparison; shows trends and patterns; remembered longer. Limitations: less precise than tables; can mislead with poor scales; only approximate values can be read; cannot show many details.

11. Q11 · 3 marks · Short answerHarder exam level

    Suppose sales of a firm (₹ crore) were 52, 54, 55, 57 and 60 in five years. A graph starting the y-axis at 0 makes the line look almost flat. What can be done, and what care is needed?

    Hint

    False base line.

    Answer

    Use a false base line: break the y-axis and start, say, at 50, so changes are visible. Care: show the break clearly so readers do not think the growth is larger than it is.

12. Q12 · 4 marks · Long answerHOTS / competency

    Suppose a newspaper shows two pie diagrams of the same size for a state's budget in 2015 and 2025, though the 2025 budget is three times larger. What does the reader learn and what may be misunderstood?

    Hint

    Shares vs totals.

    Answer

    Equal-sized pies correctly compare shares (percentages) of each head in the two years. But a reader may think spending on a head fell if its share fell, though in rupees it may have grown because the total tripled. To show totals, use areas proportional to totals or add a bar diagram of absolute values.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.Name the three kinds of diagrams in the syllabus.Show answer

   Geometric (bar and pie), frequency diagrams (histogram, polygon, curve, ogive) and arithmetic line graphs.

2. 2.What are stubs?Show answer

   Row headings of a table.

3. 3.Formula for a pie angle?Show answer

   Part ÷ total × 360°.

4. 4.What is frequency density?Show answer

   Class frequency ÷ class width.

5. 5.How is the median found from ogives?Show answer

   From the x-value of their intersection.

6. 6.Which diagram can locate the mode?Show answer

   Histogram.

7. 7.Which ogive never decreases?Show answer

   The less than ogive.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Can you list the eight parts of a table?
• ☐Can you classify tables as qualitative, quantitative, temporal or spatial?
• ☐Can you choose between simple, multiple and component bars?
• ☐Can you compute pie angles that add to 360°?
• ☐Can you adjust histogram heights for unequal classes?
• ☐Can you plot a frequency polygon from class marks?
• ☐Can you draw both ogives and read the median?
• ☐Can you draw a time series graph with a false base line?

Presentation of Data: frequently asked questions