Collection, Classification and Presentation of Statistical Data

Every statistical method in this paper starts with a pile of raw figures that someone collected, sorted and laid out. This chapter covers that first stage: primary and secondary data, the ways primary data is collected, how raw figures are classified into a frequency distribution, how a table is built, and the diagrams and graphs that present a distribution — bar diagram, pie chart, histogram, frequency polygon and ogive. Paper-II asks these as quick definition and construction questions, and the traps are almost always about class boundaries and unequal class widths.

Primary and secondary data

Think of data as having a first owner. Primary data are collected for the first time, by the investigator, for the purpose in hand — a survey of household spending designed for your own study. Secondary data were collected by someone else for some other purpose and are now being reused — census tables, an RBI bulletin, a company's annual report. The same figures can be primary for the agency that collected them and secondary for everyone who later quotes them.

Primary against secondary
PointPrimarySecondary
Collected bythe investigator, first timeanother agency, earlier
Fit to purposeexactmust be checked
Cost and timehighlow
Sourcessurveys, interviews, schedulespublished reports, records, journals
⚠️ Exam trap
Before using secondary data, check three things: reliability (who collected it and how), suitability (same definitions and units as your enquiry) and adequacy (enough coverage). A question that asks what must be checked before reusing data is asking for these three.

Methods of collecting primary data

Primary data can be gathered in five standard ways. They differ in who asks the questions, who writes the answers, and how large an area can be covered.

The five methods
MethodHow it worksBest when
Direct personal investigationinvestigator meets every informantsmall area, high accuracy needed
Indirect oral investigationthird parties who know the facts are questionedinformants unwilling or unreachable
Information through local correspondentsagents in each area send reportsregular, approximate figures
Mailed questionnaireinformants fill and return the formwide area, literate informants
Schedules filled by enumeratorstrained enumerators ask and recordlarge surveys such as a census
🧠 Memory trick
Questionnaire — the informant writes the answers. Schedule — the enumerator writes them. The census uses schedules because many informants cannot be relied on to fill a form themselves.

Coverage is a separate choice. A census (complete enumeration) studies every unit of the population; a sample survey studies a part and infers the rest. A census avoids sampling error but is slow and costly, and it is impossible when observing a unit destroys it — testing the life of bulbs, for example.

Classification, class intervals and tabulation

Classification groups raw data by a common character: geographical (by place), chronological (by time), qualitative (by an attribute such as literacy) and quantitative (by a measured variable such as income). A quantitative variable is discrete if it jumps between values (number of children) and continuous if it can take any value in a range (height). Continuous data are grouped into class intervals.

The vocabulary of a class interval
TermMeaningExample: inclusive class 10–19
Class limitsthe two numbers written10 and 19
Class boundarieslimits pushed half the gap outward9.5 and 19.5
Class widthupper boundary − lower boundary10
Mid-value(lower + upper limit) ÷ 214.5

In the exclusive method (10–20, 20–30, …) the upper limit of one class is the lower limit of the next and a value of exactly 20 goes into 20–30. In the inclusive method (10–19, 20–29, …) both limits belong to the class, and before drawing a histogram or using an interpolation formula you must convert to boundaries by subtracting and adding half the gap — here 0.5. Sturges' rule suggests the number of classes: k = 1 + 3.322 log₁₀N; for N = 100 it gives 7.644, so about 8 classes.

A statistical table has fixed parts: the table number and title; the caption (column headings); the stub (row headings); the body (the figures); a head-note (units, placed under the title); and a footnote and source note below. Tabulation follows classification — you cannot tabulate what you have not classified.

Diagrams and graphs of a distribution

A bar diagram compares separate categories with bars of equal width whose heights carry the values; the gaps between bars are deliberate. A pie chart divides 360° in proportion to each component: angle = (component ÷ total) × 360°.

Worked pie chart: a monthly budget of ₹12,000
ItemSpending (₹)Angle
Food4800144°
Rent300090°
Education180054°
Others240072°

A histogram is for a continuous frequency distribution: rectangles stand on the class boundaries with no gaps, and the area of each rectangle is proportional to its frequency. When widths are equal, height = frequency. When they are not, the height must be the frequency density (frequency ÷ width), or the frequency rescaled to a standard width.

Unequal widths: heights rescaled to a width of 10
ClassWidthFrequencyHeight to draw
0–101055
10–201088
20–4020126
40–501066

A frequency polygon joins the points (mid-value, frequency) — equivalently the mid-points of the tops of the histogram's rectangles — and is closed by extending to the mid-values of an empty class at each end. An ogive is the graph of a cumulative frequency: the less-than ogive plots cumulative frequency against upper boundaries and rises; the more-than ogive plots against lower boundaries and falls. The two ogives intersect at the median.

⚠️ Exam trap
Drawing a histogram with frequency as the height when the widths differ is the single most common construction error. A class twice as wide with the same frequency must be drawn half as tall.

Exam traps and a quick revision

  • Census data quoted in your study are secondary for you, primary for the census office.
  • Inclusive classes must be turned into boundaries before a histogram, a median or a mode is computed.
  • Bar diagram: gaps, heights matter. Histogram: no gaps, areas matter.
  • Frequency polygon uses mid-values; ogive uses class boundaries and cumulative frequency.
  • Pie angle = share × 360°; a quarter of the total is always 90°.
🧠 Memory trick
Caption is on columns, stub is on the side: 'caption — columns, stub — side rows'.

Key takeaways

  • Primary data are first-hand and purpose-built; secondary data are reused and must be checked for reliability, suitability and adequacy.
  • A questionnaire is filled by the informant, a schedule by the enumerator.
  • Convert inclusive classes to boundaries by moving each limit half the gap outward; mid-value = (lower + upper limit) ÷ 2.
  • In a histogram area, not height, represents frequency; with unequal widths plot frequency density.
  • The less-than and more-than ogives meet at the median; a pie angle is the component's share of 360°.

Practice questions (10)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

  1. A researcher conducts her own survey of 400 households to study their spending on transport. For her study, these figures are:

    1. secondary data
    2. published data
    3. derived data
    4. primary data
    Show answer

    Answer: D — primary data

    She collected them herself, for the first time and for this purpose, so they are primary. They would become secondary for anyone who later reuses her figures.

  2. In which method of primary data collection does a trained enumerator ask the questions and record the answers himself?

    1. Indirect oral investigation
    2. Information through local correspondents
    3. Schedules filled by enumerators
    4. Mailed questionnaire
    Show answer

    Answer: C — Schedules filled by enumerators

    A schedule is filled by the enumerator, a questionnaire by the informant. That is why large surveys such as the census rely on schedules.

  3. For the inclusive class 30–39 (data recorded to whole numbers), the upper class boundary is:

    1. 34.5
    2. 39
    3. 40
    4. 39.5
    Show answer

    Answer: D — 39.5

    The gap between 39 and the next class's lower limit 40 is 1, so each limit moves 0.5 outward: boundaries 29.5 and 39.5.

  4. A frequency polygon is drawn by plotting frequencies against the:

    1. lower class limits
    2. upper class boundaries
    3. cumulative frequencies
    4. mid-values of the classes
    Show answer

    Answer: D — mid-values of the classes

    The polygon joins the points (mid-value, frequency). Upper boundaries with cumulative frequencies give the less-than ogive instead.

  5. In a pie chart of a family's monthly spending of ₹3,600, rent is ₹900. The angle of the rent sector is:

    1. 60°
    2. 90°
    3. 120°
    4. 25°
    Show answer

    Answer: B — 90°

    Angle = (900 ÷ 3600) × 360° = 0.25 × 360° = 90°. 25 is the percentage share, not the angle.

  6. In a histogram where most classes have width 10, the class 20–40 has frequency 18. To keep area proportional to frequency, its rectangle should be drawn to a height of:

    1. 9
    2. 0.9
    3. 18
    4. 36
    Show answer

    Answer: A — 9

    The class is twice the standard width, so the height is 18 × 10 ÷ 20 = 9. Drawing it at 18 doubles its area and overstates the class.

  7. The less-than ogive and the more-than ogive of the same distribution intersect at a point whose abscissa (x-value) is the:

    1. range
    2. median
    3. mean
    4. mode
    Show answer

    Answer: B — median

    At the crossing, the number of items below equals the number above, which is N/2 on each side — the definition of the median.

  8. By Sturges' rule, k = 1 + 3.322 log₁₀N, the suggested number of classes for N = 64 observations is about (log₁₀64 = 1.806):

    1. 64
    2. 7
    3. 8
    4. 6
    Show answer

    Answer: B — 7

    k = 1 + 3.322 × 1.806 = 1 + 6.0 = 7.0, so 7 classes. Sturges gives exactly 1 + log₂N, and log₂64 = 6.

  9. In a statistical table, the headings of the columns are called the:

    1. head-note
    2. stub
    3. body
    4. caption
    Show answer

    Answer: D — caption

    Caption = column headings; stub = row headings at the side; head-note = the unit line under the title; body = the figures.

  10. Testing every bulb produced by a factory for its burning life is impractical mainly because:

    1. the test destroys the unit, so only a sample can be tested
    2. secondary data on bulbs are always available
    3. bulb life is a qualitative attribute
    4. a census always has a larger sampling error
    Show answer

    Answer: A — the test destroys the unit, so only a sample can be tested

    A destructive test leaves nothing to sell, so a census is impossible here. A census has no sampling error at all, and burning life is a measured, quantitative variable.