Measures of Central Tendency and Dispersion for CA Foundation
Averages (mean, median, mode, GM, HM) and measures of spread (range, quartile deviation, mean deviation, standard deviation), with their properties.
Quantitative AptitudePaper 3, Statistics13 MCQs2 lessons2 flashcards3 memory tricks
Practise Measures of Central Tendency and Dispersion free
What you'll learn
- Measures of central tendency
Calculate mean, median, mode, GM and HM and use their relations. - Measures of dispersion
Calculate range, QD, SD and CV and know their properties.
Key formulas
| Empirical relation | Mode = 3 Median − 2 Mean |
|---|---|
| Order | AM ≥ GM ≥ HM |
| Two numbers | GM² = AM × HM |
| Combined mean | (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂) |
| Range | Largest − smallest |
| Quartile deviation | (Q₃ − Q₁) ÷ 2 |
| Standard deviation | σ = √[Σ(x − x̄)² ÷ n] |
| Coefficient of variation | CV = σ ÷ x̄ × 100 |
Concepts explained
Mean, median, mode and their relation
The mean adds and divides, the median takes the middle value after arranging, and the mode is the most frequent value. In moderately skewed data, Mode = 3 Median − 2 Mean.
What it means
- Mean uses every value, so extreme values pull it.
- Median is positional; arrange the data first. For n items it is the (n + 1)/2 th item.
- Mode is the most frequent value; a distribution can be bimodal.
- Combined mean = (n₁x̄₁ + n₂x̄₂) ÷ (n₁ + n₂).
- 3, 7, 1, 9, 5 → arranged 1, 3, 5, 7, 9
Answer: Median = 5
- Mean 30, median 28
- Mode = 84 − 60
Answer: 24
- 10 items mean 20; 15 items mean 30
- (200 + 450) ÷ 25
Answer: 26
Related terms
- Weighted mean: Σwx ÷ Σw
- Symmetric distribution: mean = median = mode
Exam tip: For income data with a few very high earners, the median represents a typical person better than the mean.
Arithmetic, geometric and harmonic means
For positive numbers, AM ≥ GM ≥ HM, with equality only if all values are equal. For two numbers, GM² = AM × HM.
What it means
- GM = n-th root of the product; used for growth rates and ratios.
- HM = n ÷ Σ(1/x); used for average speeds over equal distances.
- For two numbers a and b: AM = (a + b)/2, GM = √ab, HM = 2ab/(a + b).
- AM = 10, GM = 8
- HM = GM² ÷ AM = 64 ÷ 10
Answer: 6.4
- Equal distances at 30 and 60 km/h
- HM = 2 × 30 × 60 ÷ 90
Answer: 40 km/h
Related terms
- Weighted HM: average speed over unequal distances
Exam tip: Average speed over equal distances is the harmonic mean, never the simple mean.
Standard deviation, variance and coefficient of variation
Standard deviation measures how far values typically lie from the mean; CV = SD ÷ mean × 100 compares consistency across series.
What it means
- σ = √[Σ(x − x̄)² ÷ n]; variance = σ².
- Change of origin (adding k): SD unchanged. Change of scale (multiplying by k): SD × |k|, variance × k².
- Quartile deviation = (Q₃ − Q₁) ÷ 2; range = largest − smallest.
- Lower CV means greater consistency.
- 2, 4, 4, 4, 5, 5, 7, 9: mean 5
- Σ(x − 5)² = 32; variance = 4
Answer: SD = 2
- Mean 50, SD 10
Answer: CV = 20%
Related terms
- Mean deviation: average absolute deviation from mean or median
- SD of first n natural numbers: √[(n² − 1)/12]
Exam tip: Variance is not SD: take the square root.
Memory tricks
AM ≥ GM ≥ HM for positive numbers.
Mode = 3 Median − 2 Mean (moderately skewed data).
SD is independent of origin but not of scale.
Try 3 CA Foundation-style questions
Q1. The median of 3, 7, 1, 9 and 5 is
- 7
- 5
- 1
- 25
Show answer
Answer: (2) 5
Arrange in order: 1, 3, 5, 7, 9. The middle (3rd) value is 5.
Q2. If every observation is multiplied by 3, the standard deviation
- is multiplied by 3
- stays the same
- is multiplied by 9
- increases by 3
Show answer
Answer: (1) is multiplied by 3
SD is affected by a change of scale: it is multiplied by the same factor (variance by its square, 9).
Q3. The mode of 2, 3, 3, 5, 7, 3, 5 is
- 5
- 4
- 3
- 7
Show answer
Answer: (3) 3
3 occurs three times, more than any other value.
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