S. S. Stevens (1946) classified measurements into four scales arranged in increasing order of mathematical strength.
Numbers / labels are used only as names or categories. There is no order, no equal distance, and no true zero. Only operations valid: = and ≠.
Gender coded as 1 = Male, 2 = Female. The numbers are just labels — saying "2 > 1" makes no sense.
Jersey numbers of cricket players (Dhoni-7, Kohli-18). Number 18 is not "bigger" than 7 in any meaningful way.
Categories can be ordered or ranked, but the differences between ranks are not necessarily equal. Operations valid: =, ≠, <, >.
Customer satisfaction: Very Poor < Poor < Average < Good < Excellent. We can say "Excellent > Good" but cannot say the gap from Good to Excellent equals the gap from Poor to Average.
Position in a 100 m race — 1st, 2nd, 3rd. The time gap between 1st and 2nd may be 0.1 s while between 2nd and 3rd may be 1.5 s — ranks don't preserve distance.
Differences between values are meaningful and equal, but the zero is arbitrary (not a true zero). Ratios are not meaningful. Valid operations: =, ≠, <, >, +, –.
Temperature in Celsius. 30°C – 20°C = 10°C and 20°C – 10°C = 10°C (equal differences). But 20°C is not "twice as hot as" 10°C, because 0°C does not mean "no temperature".
Calendar years (1990, 2000, 2026). 2026 – 1990 = 36 years (meaningful), but 2026 is not "twice 1013".
Has all properties of interval scale plus a true zero. Ratios are meaningful. All arithmetic operations are valid: =, ≠, <, >, +, –, ×, ÷.
Weight in kg. 0 kg = no weight. 80 kg is twice 40 kg. All operations valid.
Monthly income (₹0, ₹15 000, ₹30 000). ₹0 means no income; ₹30 000 is twice ₹15 000.
| Scale | Order? | Equal differences? | True zero? | Examples |
|---|---|---|---|---|
| Nominal | No | No | No | Gender, religion, blood group |
| Ordinal | Yes | No | No | Ranks, grades, satisfaction |
| Interval | Yes | Yes | No | Temperature, IQ, calendar years |
| Ratio | Yes | Yes | Yes | Weight, height, income, age |
A frequency distribution is a tabular arrangement of data showing each possible value (or class) and the number of times (frequency) it occurs.
Number of children per family in 20 households:
2, 1, 0, 3, 2, 4, 1, 2, 1, 3, 0, 2, 1, 2, 3, 1, 2, 0, 1, 2.
| No. of children (x) | Tally | Frequency (f) |
|---|---|---|
| 0 | ||| | 3 |
| 1 | |||| | | 6 |
| 2 | |||| || | 7 |
| 3 | ||| | 3 |
| 4 | | | 1 |
| Total | 20 | |
Marks of 30 students:
45, 52, 38, 71, 64, 49, 58, 33, 67, 72, 55, 48, 60, 39, 51,
66, 74, 57, 41, 50, 63, 47, 36, 69, 56, 44, 53, 61, 46, 59.
Range = 74 – 33 = 41. Take 5 classes of width 10 starting from 30 (exclusive classes 30–40, 40–50, …, where a value equal to the upper limit is counted in the next class, so 50 falls in 50–60).
| Class | Tally | Frequency (f) |
|---|---|---|
| 30 – 40 | |||| | 4 |
| 40 – 50 | |||| || | 7 |
| 50 – 60 | |||| |||| | 9 |
| 60 – 70 | |||| || | 7 |
| 70 – 80 | ||| | 3 |
| Total | 30 | |
Cumulative "less than" frequencies: 4, 11, 20, 27, 30.
Pictures of data — easier and more attractive than tables.
A line diagram showing variation of a variable over time. Time on X-axis, value on Y-axis; points joined by straight lines.
Note: Don't confuse Historiogram (time line) with Histogram (frequency bars).
Plotting India's GDP from 2015 to 2025 year-by-year as connected dots gives a historiogram.
Daily temperature of Kadapa city for one week shown as a line graph.
Rectangular bars of equal width drawn at equal gaps; height proportional to value. Used for categorical data.
Production (in tonnes) of a factory in 5 years: 200, 250, 300, 280, 350.
Number of students enrolled in B.A., B.Sc., B.Com., B.B.A. — one bar per stream.
Two or more sets of bars drawn side-by-side for the same category, distinguished by different colours / shades. Used to compare related variables.
Boys-vs-Girls strength in 4 colleges. For each college, two bars (one for boys, one for girls).
Imports and Exports of India for 5 years — two bars per year side by side.
A circle divided into sectors whose angles are proportional to component values.
Total ₹20 000 spent as: Food 8 000, Rent 5 000, Education 4 000, Savings 3 000.
| Item | Amount (₹) | Angle |
|---|---|---|
| Food | 8 000 | (8 000/20 000)×360 = 144° |
| Rent | 5 000 | 90° |
| Education | 4 000 | 72° |
| Savings | 3 000 | 54° |
| Total | 20 000 | 360° |
Sleep 8 h, College 6 h, Self-study 4 h, Sports/Recreation 3 h, Others 3 h.
Each angle = (hours / 24) × 360°. So Sleep = 120°, College = 90°, Self-study = 60°, Sports = 45°, Others = 45°. (Check sum = 360°.)
Used mainly for frequency distributions.
Adjacent rectangles erected on class boundaries with no gap; area of each rectangle is proportional to its frequency. For equal class widths, the height itself = frequency.
If classes are unequal, height is proportional to frequency density = \(f / h\).
Using marks data of Example 2 (Section 2): draw bars of width 10 on intervals 30–40, 40–50, 50–60, 60–70, 70–80 with heights 4, 7, 9, 7, 3 respectively. Bars touch each other.
| Class | Frequency f | Width h | f / h (height) |
|---|---|---|---|
| 0–10 | 5 | 10 | 0.5 |
| 10–20 | 15 | 10 | 1.5 |
| 20–40 | 20 | 20 | 1.0 |
| 40–50 | 10 | 10 | 1.0 |
Plot the mid-values of classes against frequencies and join the points by straight lines. Closed at both ends by joining to the X-axis at one extra class on each side.
Marks data: mid-values 35, 45, 55, 65, 75 with frequencies 4, 7, 9, 7, 3. Plot the five points and join by lines; close the polygon at the adjacent mid-values 25 and 85 on the X-axis.
To compare distributions of two classes, two frequency polygons can be drawn on the same axes — a histogram cannot do this clearly.
An ogive is a graph of cumulative frequencies plotted against class boundaries. Two types:
The two ogives intersect at the median of the distribution.
| Class | f | Less-than CF | More-than CF |
|---|---|---|---|
| 30–40 | 4 | 4 (<40) | 30 (≥30) |
| 40–50 | 7 | 11 (<50) | 26 (≥40) |
| 50–60 | 9 | 20 (<60) | 19 (≥50) |
| 60–70 | 7 | 27 (<70) | 10 (≥60) |
| 70–80 | 3 | 30 (<80) | 3 (≥70) |
Plotting (40,4), (50,11), (60,20), (70,27), (80,30) and joining gives the less-than ogive. The more-than ogive uses (30,30), (40,26), (50,19), (60,10), (70,3).
Total \(N = 30\); \(N/2 = 15\). On the less-than ogive, read off the X-coordinate where the cumulative frequency = 15 — this gives the median ≈ 54.4 marks.
The graphical reading is exactly the linear-interpolation formula for the median of a grouped distribution. The median class is the one containing the \((N/2)\)-th observation:
\[ \text{Median} \;=\; L + \dfrac{\tfrac{N}{2} - CF}{f}\times h, \]
where \(L\) = lower boundary of the median class, \(CF\) = cumulative frequency before it, \(f\) = its frequency and \(h\) = its width. Here \(N/2 = 15\) first exceeds the cumulative total in the class 50–60, so \(L = 50,\; CF = 11,\; f = 9,\; h = 10\):
\[ \text{Median} \;=\; 50 + \dfrac{15 - 11}{9}\times 10 \;=\; 50 + \dfrac{40}{9} \;=\; 54.44. \]
This confirms the ≈ 54.4 read from the ogive intersection — the graph and the formula are the same calculation, one geometric and one algebraic. (The full treatment of the median is in Unit 3.)
Additional worked problems with step-by-step procedures to support self-study, matching this unit's topics.
Inclusive → Exclusive conversion: correction factor \(= \dfrac{\text{lower limit of next class} - \text{upper limit of current class}}{2}\); add it to upper limits and subtract from lower limits.
Prepare a discrete frequency distribution from the following data (number of letters in each word):
5, 5, 2, 6, 1, 5, 2, 9, 5, 4, 3, 4, 11, 7, 2, 5, 12, 6
Solution. Arrange in ascending order, then tally each distinct value:
| Variable (X) | Tally | Frequency (f) |
|---|---|---|
| 1 | | | 1 |
| 2 | ||| | 3 |
| 3 | | | 1 |
| 4 | || | 2 |
| 5 | |||| | 5 |
| 6 | || | 2 |
| 7 | | | 1 |
| 9 | | | 1 |
| 11 | | | 1 |
| 12 | | | 1 |
| Total | 18 |
20 students appear in an examination (max 50 marks). Prepare a frequency distribution taking class width 10. Marks: 5, 16, 17, 17, 20, 21, 22, 22, 22, 25, 25, 26, 26, 30, 31, 31, 34, 35, 42, 48.
Inclusive method:
| Marks | No. of students |
|---|---|
| 1–10 | 1 |
| 11–20 | 4 |
| 21–30 | 9 |
| 31–40 | 4 |
| 41–50 | 2 |
| Total | 20 |
Exclusive method:
| Marks | No. of students |
|---|---|
| 0–10 | 1 |
| 10–20 | 3 |
| 20–30 | 9 |
| 30–40 | 5 |
| 40–50 | 2 |
| Total | 20 |
Number of classes by Sturges' rule \( k = 1 + 3.322\log_{10} N \). To convert an inclusive series to exclusive, the correction factor is \(\frac{\text{lower limit of 2nd class} - \text{upper limit of 1st class}}{2}\); here \(\frac{11-10}{2}=0.5\), giving exclusive classes 0.5–10.5, 10.5–20.5, 20.5–30.5, …
Prepare a simple bar diagram for India's merchandise exports (₹ million):
| Year | 1971 | 1972 | 1973 | 1974 | 1975 | 1976 | 1977 | 1978 |
|---|---|---|---|---|---|---|---|---|
| Exports | 1962 | 2174 | 2419 | 3024 | 3852 | 4688 | 5555 | 5112 |
Solution. Take year on the X-axis, exports on the Y-axis (scale: 1000), and draw equal-width vertical bars of heights equal to the export figures.
50 part-time college students bought books as follows: 11 bought 1 book, 10 bought 2, 16 bought 3, 6 bought 4, 5 bought 5, 2 bought 6. Determine the bin size and draw the histogram.
Solution. Smallest value 1, largest 6. Using 0.5 and 6.5 as boundaries the range is \(6.5-0.5 = 6\); with 6 bins the bin size = 1. Bars of heights 11, 10, 16, 6, 5, 2 are drawn over the intervals 0.5–1.5, 1.5–2.5, …, 5.5–6.5 with the number of books on the X-axis and the frequency on the Y-axis.
A family's weekly expenditure: Mortgage ₹300, Food ₹225, Fuel ₹75. Draw a pie chart.
Solution. Total = ₹600. Angle for each segment \(= \dfrac{\text{value}}{600}\times 360^\circ\):
| Expense | Amount (₹) | Percentage | Angle |
|---|---|---|---|
| Mortgage | 300 | 50.0% | 180° |
| Food | 225 | 37.5% | 135° |
| Fuel | 75 | 12.5% | 45° |
| Total | 600 | 100% | 360° |
Construct a frequency polygon for the Calculus final-test scores:
| Lower | Upper | Mid-value | Frequency |
|---|---|---|---|
| 49.5 | 59.5 | 54.5 | 5 |
| 59.5 | 69.5 | 64.5 | 10 |
| 69.5 | 79.5 | 74.5 | 30 |
| 79.5 | 89.5 | 84.5 | 40 |
| 89.5 | 99.5 | 94.5 | 15 |
Solution. Plot the frequency against each mid-value and join the points by straight line segments (closing to the X-axis at the mid-values 44.5 and 104.5).