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NCERT Mathematics Class 11 - Chapter 13: Statistics - Notes

CBSEClass 11Mathematicsसांख्यिकी

Learning Objectives

  • Understand measures of dispersion
  • Learn to calculate range, mean deviation, variance, and standard deviation
  • Study coefficient of variation for comparing data sets

Key Concepts

Measures of Dispersion

Dispersion measures the scatter or spread of data. Range: Maximum value - Minimum value (simplest but most affected by outliers).

Mean Deviation

Mean deviation about mean: MD(x̄) = Σ|xi - x̄| / n (ungrouped) or Σfi|xi - x̄| / N (grouped). Mean deviation about median: MD(M) = Σ|xi - M| / n. Mean deviation about median is minimum among all deviations about any point.

Variance and Standard Deviation

Variance (σ2): Average of squared deviations from mean. σ2 = Σ(xi - x̄)2 / n = (Σxi2/n) - (x̄)2.

Standard deviation (σ): σ = √(Variance). For grouped data: σ2 = Σfi(xi - x̄)2 / N.

Shortcut formula: σ2 = (Σfixi2/N) - (Σfixi/N)2. This avoids computing deviations.

Step deviation method: For grouped data with equal class width h: Let di = (xi - A)/h. Then σ = h × √[(Σfidi2/N) - (Σfidi/N)2].

Coefficient of Variation

CV = (σ/x̄) × 100. Used to compare variability of two data sets with different units or different means. Lower CV = more consistent data. Higher CV = more variable data.

Effect of Change of Origin and Scale

If yi = axi + b: Variance of y = a2 × Variance of x. SD of y = |a| × SD of x. Variance and SD are independent of change of origin (adding a constant) but dependent on change of scale (multiplying by a constant).

Summary

Dispersion measures how spread out data values are. Range is the simplest measure. Mean deviation uses absolute deviations. Variance and standard deviation use squared deviations and are the most widely used. Coefficient of variation enables comparison between different data sets.

Important Terms

  • Variance: Average of squared deviations from mean
  • Standard deviation: Square root of variance; same units as data
  • Coefficient of Variation: CV = (σ/x̄) × 100; for comparing variability
  • Mean deviation: Average of absolute deviations from central value
  • Range: Difference between maximum and minimum values

Quick Revision

  • σ2 = Σ(xi - x̄)2/n = (Σxi2/n) - x̄2
  • SD = √Variance
  • CV = (σ/x̄) × 100; lower CV = more consistent
  • Variance is independent of change of origin
  • If y = ax + b: Var(y) = a2Var(x); SD(y) = |a|SD(x)
  • Mean deviation about median is minimum
  • Step deviation method simplifies grouped data calculations
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NCERT Mathematics Class 11 - Chapter 13: Statistics - Notes | EduMunch