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Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 2 Chapter 3: Skewness

Short notes, key concepts, quick revision and practice MCQs with answers

Skewness describes the lack of symmetry in a distribution. The chapter compares the positions of mean, median and mode in symmetric and skewed distributions and measures skewness with Karl Pearson's and Bowley's coefficients.

Key concepts: Skewness

Skewness
Lack of symmetry in a frequency distribution. Two distributions may have the same mean and S.D. but different shapes.
Symmetric distribution
Observations equidistant from the centre have equal frequencies. Mean = Median = Mode, and Q₃ − Q₂ = Q₂ − Q₁.
Positive skewness
The curve has a longer tail to the right. Mean > Median > Mode.
Negative skewness
The curve has a longer tail to the left. Mean < Median < Mode.
Karl Pearson's coefficient
Skp = (Mean − Mode)/S.D. When the mode is ill-defined, use Mean − Mode = 3(Mean − Median), so Skp = 3(Mean − Median)/S.D. Most values lie between −1 and 1; all lie between −3 and 3.
Bowley's coefficient
Skb = (Q₃ + Q₁ − 2Q₂)/(Q₃ − Q₁). It is based on quartiles and always lies between −1 and 1. Skb > 0: positive, Skb < 0: negative, Skb = 0: symmetric.
Empirical relation
For a moderately skewed distribution, Mode = 3 Median − 2 Mean.

Quick revision: important points

Practice MCQs with answers

  1. In a symmetric distribution:

    1. Mean > Median > Mode
    2. Mean = Median = Mode
    3. Mean < Median < Mode
    4. Mean = Mode ≠ Median
    Show answer

    Answer: B. Mean = Median = Mode
    All three averages coincide.

  2. In a positively skewed distribution:

    1. Mean > Median > Mode
    2. Mean < Median < Mode
    3. Mean = Median = Mode
    4. Median > Mean > Mode
    Show answer

    Answer: A. Mean > Median > Mode
    The long right tail pulls the mean up.

  3. In a negatively skewed distribution:

    1. Mean > Median > Mode
    2. Mean < Median < Mode
    3. Mean = Mode
    4. Mode < Mean
    Show answer

    Answer: B. Mean < Median < Mode
    The long left tail pulls the mean down.

  4. If mean = 50, mode = 45 and S.D. = 10, Pearson's coefficient of skewness is:

    1. 0.5
    2. −0.5
    3. 5
    4. 0.05
    Show answer

    Answer: A. 0.5
    (50 − 45)/10 = 0.5.

  5. If mean = 40, median = 42 and S.D. = 12, Pearson's coefficient of skewness is:

    1. 0.5
    2. −0.5
    3. −0.17
    4. 0.17
    Show answer

    Answer: B. −0.5
    3(40 − 42)/12 = −6/12 = −0.5.

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