Grades 7–10 Question Generator Weakness Buddy Practice Quiz Dashboard CBSE CBSE Class 10 Doubts Important QsPricing
All Class 10 Maths notes

Class 10 Maths Notes — Chapter 13: Statistics

Mean, median, and mode of grouped data.

Detailed NCERT notes

  • For grouped frequency distributions, class marks xᵢ = (upper + lower)/2 represent each class.
  • Direct method for mean: x̄ = Σfᵢxᵢ / Σfᵢ.
  • Assumed mean method: choose a (usually a middle xᵢ); compute dᵢ = xᵢ − a; then x̄ = a + Σfᵢdᵢ / Σfᵢ.
  • Step-deviation method: choose class width h; uᵢ = (xᵢ − a)/h; x̄ = a + h · Σfᵢuᵢ / Σfᵢ.
  • Median formula: Median = l + [(n/2 − cf)/f] × h, where l = lower boundary of median class, n = Σfᵢ, cf = cumulative frequency before median class, f = frequency of median class, h = class width.
  • Median class = the class in which the cumulative frequency first reaches or exceeds n/2.
  • Mode formula: Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h, where f₁ = frequency of modal class, f₀ = previous, f₂ = next.
  • Modal class = class with the highest frequency.
  • Empirical relation: 3 Median = Mode + 2 Mean (approximate).
  • For continuous data, ensure classes are 'exclusive' (like 10–20, 20–30) or convert 'inclusive' classes (like 10–19, 20–29) by adjusting boundaries.
  • Note: ogive/less-than and more-than curves were rationalised; focus on the three central-tendency formulas.

Formulas & key results

  • Mean (direct): x̄ = Σfᵢxᵢ / Σfᵢ
  • Mean (assumed): x̄ = a + Σfᵢdᵢ / Σfᵢ
  • Mean (step-deviation): x̄ = a + h · Σfᵢuᵢ / Σfᵢ
  • Median = l + [(n/2 − cf)/f] × h
  • Mode = l + [(f₁ − f₀)/(2f₁ − f₀ − f₂)] × h
  • Empirical: 3 Median = Mode + 2 Mean

Mind map

  • Grouped data → class marks
  • Mean via 3 methods
  • Median: locate class via cumulative frequency
  • Mode: locate modal class
  • Empirical relation among mean, median, mode

Tricks & shortcuts

  • Step-deviation method is fastest when class widths are equal.
  • Locate median class by scanning cumulative frequency for n/2.
  • If distribution is nearly symmetric, mean ≈ median ≈ mode.

Common mistakes to avoid

  • Wrong cf class for median (must be BEFORE median class).
  • Confusing f₀, f₁, f₂ in mode formula.
  • Using inclusive classes without adjusting to continuous form.

Competency-based questions & answers

  1. Q. Modal class of frequencies 10, 15, 25, 20, 18 is which?
    A. Highest frequency = 25 ⇒ 3rd class is the modal class.
  2. Q. For a symmetric distribution, mean = 30. What is the median?
    A. Approximately 30 (since 3M = Mode + 2·Mean and mode ≈ mean ⇒ M ≈ mean).