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Schooling · Class 11 · Mathematics · Statistics · Chapter on a page

CBSE Class 11 · Mathematics · NCERT Mathematics

Statistics: chapter on a page

A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).

About the notes: Written by Shishya's AI on 27 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.

Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain.

Mind map

Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.

Mind map of Statistics: Key ideas, Formulas and facts, Worked examples, Watch out forKey ideasCentral tendency alone is not enoughRangeMean deviation about the meanVarianceStandard deviationFor grouped data with frequencies…Coefficient of variation (CV)Formulas and factsRangeMean deviation about meanVarianceStandard deviationShort-cut formula for variance…For grouped data with frequencies fᵢ…Alternate formWorked examplesUngrouped dataGrouped dataComparing two data sets using CVWatch out forUsing deviations without absolute…Forgetting to multiply by frequency…Confusing variance and standard…Comparing standard deviations of…Calculating range but reporting it…Statistics
The mind map as a list
  • Key ideas
    • Central tendency alone is not enough
    • Range
    • Mean deviation about the mean
    • Variance
    • Standard deviation
    • For grouped data with frequencies…
    • Coefficient of variation (CV)
  • Formulas and facts
    • Range
    • Mean deviation about mean
    • Variance
    • Standard deviation
    • Short-cut formula for variance…
    • For grouped data with frequencies fᵢ…
    • Alternate form
  • Worked examples
    • Ungrouped data
    • Grouped data
    • Comparing two data sets using CV
  • Watch out for
    • Using deviations without absolute…
    • Forgetting to multiply by frequency…
    • Confusing variance and standard…
    • Comparing standard deviations of…
    • Calculating range but reporting it…

What this chapter is about

In earlier classes you learnt to organise data, draw bar graphs and calculate basic measures like mean, median and mode. Class 11 Statistics moves a step further: it asks how spread out or scattered the data points are around the central value. Two batches of students may have the same average marks, yet one batch may have marks tightly clustered near that average while the other has marks ranging widely. Measures of dispersion capture this difference.

Key ideas

  • Central tendency alone is not enough. Two distributions can share the same mean yet differ greatly in how the individual values scatter around it.
  • Range is the simplest dispersion measure: Range = Maximum value − Minimum value. It uses only two data points and ignores the rest.
  • Mean deviation about the mean averages the absolute differences of each observation from the mean. Mean deviation about the median is similarly defined and is smallest when taken about the median.
  • Variance averages the squared deviations from the mean; squaring removes negative signs and gives extra weight to outliers.
  • Standard deviation is the positive square root of variance; it has the same unit as the data, making it easier to interpret.
  • For grouped data with frequencies f₁, f₂, …, fₙ and class marks x₁, x₂, …, xₙ, the formulas involve weighted sums using these frequencies.
  • Coefficient of variation (CV) = (Standard deviation / Mean) × 100. It is a pure number (no unit) and lets you compare spread across data sets with different scales.
  • A smaller standard deviation or CV indicates more consistent or homogeneous data.

Formulas and facts to remember

  • Measure: Range · Ungrouped data (n observations x₁, x₂, …, xₙ with mean x̄): Maximum − Minimum
  • Measure: Mean deviation about mean · Ungrouped data (n observations x₁, x₂, …, xₙ with mean x̄): M.D.(x̄) = (1/n) × Σ |xᵢ − x̄|
  • Measure: Variance · Ungrouped data (n observations x₁, x₂, …, xₙ with mean x̄): σ² = (1/n) × Σ (xᵢ − x̄)²
  • Measure: Standard deviation · Ungrouped data (n observations x₁, x₂, …, xₙ with mean x̄): σ = √[(1/n) × Σ (xᵢ − x̄)²]
  • Short-cut formula for variance (ungrouped):σ² = (1/n) × Σ xᵢ² − (x̄)²
  • For grouped data with frequencies fᵢ (total N = Σ fᵢ):
  • Variance: σ² = (1/N) × Σ fᵢ (xᵢ − x̄)²
  • Alternate form: σ² = (1/N) × Σ fᵢ xᵢ² − (x̄)²
  • Coefficient of variation:CV = (σ / x̄) × 100
  • Meaning: CV expresses standard deviation as a percentage of the mean, allowing comparison when means or units differ.

Common mistakes

  • Using deviations without absolute value or squaring when finding mean deviation or variance → always use |xᵢ − x̄| for mean deviation and (xᵢ − x̄)² for variance.
  • Forgetting to multiply by frequency in grouped-data formulas → each term must carry its weight fᵢ.
  • Confusing variance and standard deviation units; variance has squared units → take the square root to get σ in the original unit.
  • Comparing standard deviations of data sets with very different means → use coefficient of variation instead.
  • Calculating range but reporting it as a measure of average → range measures spread, not central tendency.

Quick revision

  • Range = Max − Min; quick but ignores most data.
  • Mean deviation uses absolute differences: M.D. = (1/n) Σ |xᵢ − x̄|.
  • Variance uses squared differences: σ² = (1/n) Σ (xᵢ − x̄)².
  • Standard deviation σ = √(variance); same unit as data.
  • CV = (σ / x̄) × 100 removes units; use it to compare variability across different scales.
  • Lower dispersion → more consistent data.