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Measures of Central Tendency & Dispersion

Mean / Median / Mode + Range / SD / CV — typically 4–5 marks per attempt. Memorise the empirical relation and the formulas; the rest is plug-and-chug.

Last reviewed: 25 April 2026

Measures of central tendency

  • •Arithmetic Mean (AM): sum of values divided by count. Distorted by outliers.
  • •Median: middle value when data is sorted. Outlier-resistant.
  • •Mode: most frequently occurring value. Useful for categorical data.
  • •Geometric Mean (GM): nth root of product. Used for ratios, growth rates.
  • •Harmonic Mean (HM): n / Σ(1/x). Used for averaging speeds.
  • •Relation: AM ≥ GM ≥ HM (equality when all values are equal).

Quartiles, deciles, percentiles

  • •Q1 (lower quartile): 25th percentile.
  • •Q2 = Median = 50th percentile.
  • •Q3 (upper quartile): 75th percentile.
  • •Inter-Quartile Range (IQR) = Q3 − Q1.

Measures of dispersion

  • •Range: Max − Min. Crude.
  • •Mean Deviation: average of |xi − mean|.
  • •Variance: average of (xi − mean)². Squared units.
  • •Standard Deviation (SD): square root of variance. Same units as the data.
  • •Coefficient of Variation (CV): (SD / Mean) × 100. Unit-free, used to compare two distributions.

Empirical relation

  • •For moderately skewed distributions: Mode ≈ 3 × Median − 2 × Mean.
  • •Useful when one of the three is hard to compute directly.

Formulas

Mean (ungrouped)
x̄ = Σxi / n
Mean (grouped)
x̄ = Σ(fi · xi) / Σfi
Median (continuous)
Median = L + ((n/2 − cf) / f) × h

L = lower limit of median class, cf = cumulative freq before, f = freq of median class, h = class width.

Mode (continuous)
Mode = L + ((f1 − f0) / (2f1 − f0 − f2)) × h

f1 = freq of modal class, f0/f2 = freq of preceding/succeeding classes.

Variance
σ² = Σ(xi − x̄)² / n
Standard Deviation
σ = √[Σ(xi − x̄)² / n]
Coefficient of Variation
CV = (σ / x̄) × 100%

Must know before the exam

  • ★Median is preferred for skewed data; AM is preferred for symmetric data.
  • ★SD is unaffected by adding a constant to every value (shift invariant) BUT is multiplied by k if every value is multiplied by k (scale linear).
  • ★CV is unitless — use it to compare variability of two different scales (e.g., heights in cm vs weights in kg).
  • ★For combined mean: x̄_c = (n1·x̄1 + n2·x̄2) / (n1 + n2).

Common mistakes & fixes

✗ Forgetting to take the square root of variance to get SD.
✓ SD = √variance. They have different units (variance is squared).
✗ Confusing population SD (divide by n) with sample SD (divide by n−1).
✓ ICAI Foundation typically uses n. If question says 'sample', use n−1.

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