6 Statistics: Data, Sampling & Presentation · 6.1 Data, sampling, probability and presentation
6.1.4 Classification, frequency distribution and tabulation; diagrams and graphs (bar, pie, histogram, frequency polygon, ogive)
Raw data are classified into groups, condensed into a frequency distribution, set out in tables, and shown by diagrams (bars, pies, pictograms) for comparison or graphs (histogram, frequency polygon, ogive) for frequency distributions.
- APPSC Paper I
- Advanced level
- 8 min read
- Free
Core facts
- Classification: arranging data into groups by common characteristics.
- Four bases: geographical (by place), chronological (by time), qualitative (by attribute: sex, literacy), quantitative (by size of a variable: marks, income).
- Variable: discrete (whole values: number of children) or continuous (any value in a range: height, weight).
- Frequency = number of times a value or class occurs.
- Class limits: lower and upper limit; class interval (width) = upper − lower (true limits).
- Class mark (mid-point) = (lower limit + upper limit) ÷ 2.
- Exclusive method: 10–20, 20–30 (upper limit excluded; 20 goes in 20–30). Inclusive method: 10–19, 20–29 (both limits included).
- Sturges' rule: number of classes k = 1 + 3.322 log₁₀ N.
- Cumulative frequency: "less than" (adds from the top) and "more than" (adds from the bottom).
- Parts of a table: table number, title, head note, captions (column headings), stubs (row headings), body, foot note, source note.
- One-dimensional diagrams: bar diagrams (simple, multiple, sub-divided/component, percentage). Two-dimensional: rectangles, squares, circles, pie diagram. Three-dimensional: cubes, cylinders. Also pictograms and cartograms (maps).
- Pie diagram: angle of a component = (component ÷ total) × 360°.
- Graphs of frequency distributions: histogram, frequency polygon, frequency curve, ogive (cumulative frequency curve).
- Histogram = adjacent rectangles on class intervals; area ∝ frequency.
- Ogives: "less than" and "more than" ogives intersect at the median.
Explained
1. Building a frequency distribution.
Marks of 20 students: 12, 25, 37, 8, 41, 19, 33, 27, 15, 46, 22, 30, 38, 11, 29, 24, 35, 17, 43, 26.
| Marks | Frequency | Less-than c.f. | More-than c.f. |
|---|---|---|---|
| 0–10 | 1 | 1 (below 10) | 20 (0 and above) |
| 10–20 | 5 | 6 (below 20) | 19 (10 and above) |
| 20–30 | 6 | 12 (below 30) | 14 (20 and above) |
| 30–40 | 5 | 17 (below 40) | 8 (30 and above) |
| 40–50 | 3 | 20 (below 50) | 3 (40 and above) |
Class marks: 5, 15, 25, 35, 45. Width: 10. (Exclusive classes: 30 goes in 30–40.)
2. Inclusive to exclusive. For 10–19, 20–29, …: correction = (20 − 19)/2 = 0.5. Subtract 0.5 from lower limits and add 0.5 to upper limits: 9.5–19.5, 19.5–29.5. Do this before drawing a histogram or finding the median.
Example 1. Find the number of classes by Sturges' rule for N = 100. Solution. log₁₀ 100 = 2; k = 1 + 3.322 × 2 = 7.644. Answer: about 8 classes
3. Good classification: classes exhaustive (every item fits), mutually exclusive (no item in two classes), equal widths where possible, open-end classes ("below 10", "60 and above") avoided where possible, about 5–15 classes.
4. Kinds of tables. Simple (one characteristic), double (two), complex / manifold (three or more); general-purpose (reference) tables and special-purpose (summary) tables.
5. Pie diagram.
Example 2. A family's monthly spending: food ₹4,800, rent ₹2,400, education ₹1,800, others ₹3,000. Find the pie angles. Solution. Total ₹12,000. Food 4,800/12,000 × 360 = 144°; rent 72°; education 54°; others 90°. Answer: 144°, 72°, 54°, 90° (sum 360°)
6. Which diagram when.
| Need | Diagram |
|---|---|
| Compare totals of one item | simple bar |
| Compare two or more items side by side | multiple bar |
| Show parts of a total | sub-divided bar or pie |
| Compare shares across years | percentage bar |
Bars have equal width and equal gaps; only the length carries meaning. Bars may be vertical or horizontal.
7. Graphs of a frequency distribution.
- Histogram: class intervals on the x-axis, rectangles touching each other; for equal classes, height = frequency.
- Frequency polygon: join the mid-points of the tops of histogram rectangles (or plot class marks against frequencies), and close it on the x-axis at the mid-points of the empty classes at both ends; with equal class widths its area equals the histogram's.
- Frequency curve: a smoothed frequency polygon.
- Ogive: plot "less than" c.f. against upper limits, "more than" c.f. against lower limits.
Example 3. In the marks table above, find the median from the ogives (as the textbooks interpolate). Solution. N/2 = 10 lies in 20–30 (c.f. before 6, f = 6). Median = 20 + (10 − 6)/6 × 10 = 26.67. Answer: about 26.67 (the x-value where the two ogives cross)
Deeper
Unequal class widths. In a histogram, area, not height, shows frequency. Use frequency density = frequency ÷ width (or adjust to the standard width).
Example 4. Classes 0–10 (f 5), 10–20 (f 8), 20–40 (f 20), 40–60 (f 6). Find the heights on a base of width 10. Solution. Widths 10, 10, 20, 20. Heights 5, 8, 20/2 = 10, 6/2 = 3. Answer: 5, 8, 10, 3 (Plotting 20 and 6 directly would double the area of the wide classes.)
Uses of graphs.
- Histogram: locate the mode (from the tallest rectangle, by joining its corners to the adjacent rectangles' corners).
- Ogive: find the median, quartiles, deciles and percentiles graphically (read x at N/4, N/2, 3N/4).
- Lorenz curve: cumulative % of population against cumulative % of income; the farther from the line of equal distribution (45° line), the greater the inequality (basis of the Gini coefficient).
- Histograms and polygons cannot be drawn for open-end classes without assuming a width.
Diagrams versus graphs.
| Point | Diagram | Graph |
|---|---|---|
| Drawn on | plain paper | graph paper |
| Used for | comparison, attractive display | frequency distributions, time series, relations |
| Accuracy | approximate | can be read for values (median, mode) |
Time series graph (historigram). Time on the x-axis, values on the y-axis; a false base line (a broken y-axis) is used when values are far from zero; a ratio (semi-log) scale shows rates of change.
Example 5. Which pair is correctly matched? (a) Ogive – mode (b) Histogram – median (c) Ogive – median (d) Pie diagram – frequency distribution of a continuous variable Solution. Ogives give the median (and quartiles); histogram gives the mode. Answer: (c)
Common traps
- ✗ "In a histogram, height always shows frequency." ✓ Area shows frequency; with unequal widths use frequency density.
- ✗ "Bars in a bar diagram touch each other." ✓ Bar diagrams have gaps; histogram rectangles touch.
- ✗ "Ogives intersect at the mode." ✓ They intersect at the median.
- ✗ "Less-than ogive is plotted at lower limits." ✓ Less-than at upper limits; more-than at lower limits.
- ✗ "Pie angle = component/total × 100." ✓ × 360° (× 100 gives the percentage).
- ✗ "Stubs are column headings." ✓ Captions head columns; stubs head rows.
- ✗ "In 10–20, 20–30, the value 20 goes in 10–20." ✓ In the exclusive method it goes in 20–30.
- ✗ "Sturges: k = 1 + 3.322 log N with natural log." ✓ The log is to base 10.
Quick revision
- Classification bases: geographical, chronological, qualitative, quantitative.
- Class mark = (lower + upper)/2; width = upper − lower.
- Exclusive 10–20, 20–30; inclusive 10–19, 20–29 (correct by ±0.5).
- Sturges: k = 1 + 3.322 log₁₀ N.
- Table parts: number, title, head note, captions (columns), stubs (rows), body, foot note, source.
- Pie angle = share × 360°.
- Histogram: touching rectangles, area ∝ frequency; gives the mode.
- Frequency polygon joins class-mark points, closed at both ends.
- Less-than ogive at upper limits; ogives cross at the median.
- Lorenz curve shows inequality against the 45° line.
Practise this topic. Questions on Frequency distribution, tabulation & diagrams, with answers explained.
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