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Mathematics and Statistics · Class 11 Commerce

Ch 12Skewness — Class 11 Mathematics and Statistics, concept-first.

Earlier chapters of the Maharashtra State Board Std XI Mathematics and Statistics (Commerce) course measured where a set of data centres (mean, median, mode) and how widely it spreads (range, quartile deviation, standard deviation).

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Key concepts

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Karl Pearson's Coefficient of Skewness

Karl Pearson's coefficient of skewness is Sk = (Mean minus Mode) divided by the Standard Deviation, used whenever the distribution has a clear, well-defined mode.

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Chapter contents

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Meaning of Skewness

Earlier chapters of the Maharashtra State Board Std XI Mathematics and Statistics (Commerce) course measured where a set of data centres (mean, median, mode) and how widely it spreads (range, quartile…

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Types of Skewness

Skewness is classified by the side on which the longer tail lies and, equivalently, by the relative positions of the mean, median and mode.

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Karl Pearson's Coefficient of Skewness

Knowing the direction of skewness from the order of the averages is not enough; a business analyst needs a single number that gives its degree as well, and that can be compared across distributions.

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Bowley's (Quartile) Coefficient of Skewness

Karl Pearson's coefficient needs the standard deviation and the mean, both of which are badly disturbed by a single extreme value and cannot even be computed for an open-ended distribution (one whose…

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Interpreting and Comparing Skewness

Both coefficients are relative measures — pure numbers — and are read the same way regardless of the data's units.

Exercises

+Show 7 questions7 questions
  1. Q9The mean of a distribution is $32$, its mode is $35$ and its standard deviation is $6$. Find Karl Pearson's coefficient of skewness and stat…Free
  2. Q10For a distribution the mean is $29$, the median is $30$ and the standard deviation is $5$. The mode is ill-defined. Find Karl Pearson's coef…Free
  3. Q11For a distribution $Q_1 = 20$, $Q_2 = 26$ and $Q_3 = 36$. Find Bowley's coefficient of skewness.Free
  4. Q12Karl Pearson's coefficient of skewness of a distribution is $0.4$, its mean is $48$ and its standard deviation is $10$. Find the mode.Preview
  5. Q13In a positively skewed distribution, which of the following relationships between the averages is correct? (a) Mean = Median = Mode (b) Mean…Preview
  6. Q14For a distribution the median $Q_2 = 40$, the third quartile $Q_3 = 50$ and Bowley's coefficient of skewness is $-0.2$. Find the first quart…Preview
  7. Q15State what is meant by skewness of a frequency distribution, and distinguish between a symmetric distribution and a positively skewed one in…Preview

More questions

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  1. Example 1For a distribution the mean is $45$, the mode is $39$ and the standard deviation is $6$. Compute Karl Pearson's coefficient of skewness and…Free
  2. Example 2The mean of a distribution is $50$, its median is $48$ and its standard deviation is $10$. The mode cannot be determined reliably. Find Karl…Free
  3. Example 3For a distribution $Q_1 = 24$, $Q_2 = 30$ and $Q_3 = 40$. Compute Bowley's coefficient of skewness and comment on the type of skewness.Free
  4. Example 4Karl Pearson's coefficient of skewness of a distribution is $0.5$, its mode is $20$ and its standard deviation is $8$. Find the mean.Preview
  5. Example 5For a distribution the mean is $40$, the median is $42$ and Karl Pearson's coefficient of skewness is $-0.3$. Find the standard deviation an…Preview
  6. Example 6For a distribution $Q_1 = 15$, the median $Q_2 = 20$ and Bowley's coefficient of skewness is $0.2$. Find the third quartile $Q_3$.Preview
  7. Example 7Compute Karl Pearson's coefficient of skewness for the following distribution and comment on its shape. | $x$ | 5 | 10 | 15 | 20 | 25 | |---…Preview
  8. Example 8For a certain distribution the mean, median and mode are all equal to $60$. What is its coefficient of skewness, and what does this tell you…Preview