Arithmetic mean: direct, short-cut and step-deviation methods
Business Statistics · Unit II · Topic 4 of 15 · 5 min read
On this page
- 1. Central tendency and the qualities of a good average
- 2. Formulas
- 3. Worked: grouped data (January 2026)
- 4. Worked: cumulative tables (January 2025, August 2024)
- 5. Worked: missing frequency (January 2025)
- 6. Worked: combined mean (August 2024)
- 7. Weighted mean and properties
- 8. Merits and demerits
- How this is examined
Cases checked to exist. Read the judgment before quoting a holding.
Every case cited here has been checked to exist — the citation resolves and names the right parties. The propositions attributed to each case have not been verified line by line against the judgment. Case names link through to the full judgment: read it before quoting a holding in an examination answer, and never carry a proposition from notes into court on trust.
1. Central tendency and the qualities of a good average
An average is a single value that represents a whole series; since it lies somewhere in the middle of the values, it is called a measure of central tendency. The measures are the arithmetic mean, median, mode (positional), geometric mean and harmonic mean.
A good average (Yule’s requisites) should be: rigidly defined; easy to understand and calculate; based on all the observations; capable of further algebraic treatment; little affected by sampling fluctuations; and not unduly affected by extreme values.
2. Formulas
- Individual series: x̄ = Σx / n.
- Discrete series: x̄ = Σfx / N, where N = Σf.
- Continuous series (x = mid-point of class):
- Direct: x̄ = Σfm / N.
- Short-cut: x̄ = A + Σfd / N, where d = m − A and A is an assumed mean.
- Step-deviation: x̄ = A + (Σfd′ / N) × h, where d′ = (m − A) / h and h is the class width.
3. Worked: grouped data (January 2026)
Classes 0–25, 25–50 … 125–150 with frequencies 12, 25, 40, 37, 28, 18. Take A = 87.5, h = 25.
| Class | f | m | d′ | fd′ | c.f. |
|---|---|---|---|---|---|
| 0–25 | 12 | 12.5 | −3 | −36 | 12 |
| 25–50 | 25 | 37.5 | −2 | −50 | 37 |
| 50–75 | 40 | 62.5 | −1 | −40 | 77 |
| 75–100 | 37 | 87.5 | 0 | 0 | 114 |
| 100–125 | 28 | 112.5 | 1 | 28 | 142 |
| 125–150 | 18 | 137.5 | 2 | 36 | 160 |
| Total | 160 | −62 |
x̄ = 87.5 + (−62 / 160) × 25 = 87.5 − 9.6875 = 77.81. (Median 77.03 and mode 70.83 from the same table are worked in the next note.) Check by the direct method: Σfm = 150 + 937.5 + 2,500 + 3,237.5 + 3,150 + 2,475 = 12,450; 12,450 / 160 = 77.81.
4. Worked: cumulative tables (January 2025, August 2024)
When the paper gives “below 10, below 20 …” or “above 0, above 10 …”, first convert to ordinary class frequencies by subtracting successive cumulative figures.
January 2025 (ages of persons): below 10: 2, below 20: 5, below 30: 9, below 40: 12, below 50: 14, below 60: 15, below 70: 19. Class frequencies: 0–10: 2, 10–20: 3, 20–30: 4, 30–40: 3, 40–50: 2, 50–60: 1, 60–70: 4; N = 19.
| Class | f | m | d′ (A = 35, h = 10) | fd′ |
|---|---|---|---|---|
| 0–10 | 2 | 5 | −3 | −6 |
| 10–20 | 3 | 15 | −2 | −6 |
| 20–30 | 4 | 25 | −1 | −4 |
| 30–40 | 3 | 35 | 0 | 0 |
| 40–50 | 2 | 45 | 1 | 2 |
| 50–60 | 1 | 55 | 2 | 2 |
| 60–70 | 4 | 65 | 3 | 12 |
| Total | 19 | 0 |
x̄ = 35 + (0 / 19) × 10 = 35 years.
August 2024 (wages, “above” table): above 0: 80, above 10: 77, above 20: 72, above 30: 65, above 40: 55, above 50: 43, above 60: 28, above 70: 16, above 80: 10, above 90: 8, above 100: 0. Frequencies for 0–10 to 90–100: 3, 5, 7, 10, 12, 15, 12, 6, 2, 8 (N = 80). With A = 55, h = 10, d′ runs −5 to +4 and Σfd′ = (−15 − 20 − 21 − 20 − 12) + (0 + 12 + 12 + 6 + 32) = −88 + 62 = −26. x̄ = 55 − 26/80 × 10 = 55 − 3.25 = Rs. 51.75.
5. Worked: missing frequency (January 2025)
Marks 0–10: 12, 10–20: 18, 20–30: 27, 30–40: f, 40–50: 17, 50–60: 6; mean = 28. Find f.
Mid-points 5, 15, 25, 35, 45, 55. Σfm = 60 + 270 + 675 + 35f + 765 + 330 = 2,100 + 35f. N = 80 + f.
28 = (2,100 + 35f) / (80 + f) ⇒ 2,240 + 28f = 2,100 + 35f ⇒ 7f = 140 ⇒ f = 20.
Check: N = 100, Σfm = 2,100 + 700 = 2,800, mean = 28.
6. Worked: combined mean (August 2024)
Combined mean x̄12 = (n1x̄1 + n2x̄2) / (n1 + n2).
Mean weight of 100 students is 60 kg; boys average 65 kg and girls 55 kg. Find the number of boys and girls. Let boys = b, girls = 100 − b. 65b + 55(100 − b) = 60 × 100 ⇒ 65b + 5,500 − 55b = 6,000 ⇒ 10b = 500 ⇒ b = 50 boys, 50 girls.
7. Weighted mean and properties
Weighted mean x̄w = Σwx / Σw, used when items are of unequal importance (an index of prices weighted by expenditure, a grade average weighted by credits).
Mathematical properties: (1) the sum of deviations from the mean is zero, Σ(x − x̄) = 0; (2) the sum of squared deviations is least when taken from the mean; (3) means of groups can be combined (section 6); (4) if each value is increased by a constant, the mean increases by that constant.
8. Merits and demerits
Merits: rigidly defined; simple to understand and compute; based on all observations; capable of algebraic treatment (combined mean, SD, correlation all use it); least affected by sampling fluctuations; gives a good basis of comparison.
Demerits: unduly affected by extreme values (the mean of 2, 3, 4 and 91 is 25, which represents none of them); cannot be found from open-end classes without assuming limits; cannot be located graphically; may be a value not in the series (2.4 children); unsuitable for qualitative data and for averaging ratios and rates (use GM or HM).
How this is examined
Mean, median and mode together: January 2026 (grouped, and a discrete weights table), January 2025 (below table), August 2024 (above table), and the 100-mark January 2026 paper (discrete marks). Missing frequency: January 2025. Combined mean: August 2024. Theory: “Explain the merits and demerits of mean” (January 2025), “merits and demerits of mean, median and mode” (June 2025), and “Explain the various measures of central tendency” (January 2025, August 2024).
Now try the real questions: Business Statistics past question papers.