Q.Distinguish between Karl Pearson's coefficient of skewness and Bowley's coefficient of skewness. Under what circumstances would you prefer Bowley's measure?

Karl Pearson's coefficient uses the mean, mode (or median) and standard deviation, while Bowley's coefficient uses only the three quartiles — so Bowley's is preferred whenever the mean or standard deviation cannot be reliably computed.

Basis of computation. Karl Pearson's coefficient, $Sk_P=\dfrac{\bar{x}-Z}{\sigma}$, is built from the arithmetic mean, the mode (or median) and the standard deviation — every single observation in the dataset enters the calculation, because both the mean and the standard deviation use every value. Bowley's coefficient, $Sk_B=\dfrac{Q_3+Q_1-2M_d}{Q_3-Q_1}$, is built entirely from the three quartiles — it depends only on the middle 50% of the ranked data and completely ignores how far the extreme 25% on either side actually stretch.

Sensitivity to extreme values. Because the mean and standard deviation are both pulled by every value, however extreme, Karl Pearson's coefficient is sensitive to outliers — a single very large or very small value can shift it noticeably. Bowley's coefficient, depending only on quartile positions, is far less affected by a handful of extreme values.

Open-end classes. A distribution with an open-end class (for example, 'above 60' with no upper limit) makes it impossible to compute a mid-value for that class, and therefore impossible to compute the mean or standard deviation at all — so Karl Pearson's coefficient cannot be calculated. Quartiles, however, can usually still be located even with an open-end class (as long as the quartile itself does not fall inside the open class), so Bowley's coefficient remains computable.

Range. Karl Pearson's coefficient can theoretically range from $-3$ to $+3$ (practically almost always between $-1$ and $+1$); Bowley's coefficient always lies strictly between $-1$ and $+1$.

When to prefer Bowley's measure:

1. When the distribution has an open-end class.
2. When a few extreme outliers would distort the mean/SD and hence Karl Pearson's coefficient.
3. When only a quick, approximate measure of skewness is needed, since Bowley's needs only three quartile values and no standard-deviation computation.

When to prefer Karl Pearson's measure: when the full dataset (not just the middle half) should influence the result, and the mean, mode and SD are already available or easy to compute — it does not discard the extreme quarter of observations on either side the way Bowley's coefficient does.

A common error is stating that Bowley's coefficient is simply 'less accurate' — it is not less accurate, it is measuring skewness using different information (quartiles rather than mean/mode/SD), which makes it more robust in some situations (outliers, open-end classes) and less complete in others (it discards the extreme 25% on each side).