Q.Explain positive and negative skewness with reference to the relative positions of mean, median and mode. What is the empirical relationship among these three averages, and when is it used?

Positive skewness means a longer right-hand tail with the bulk of data at lower values, while negative skewness means a longer left-hand tail with the bulk of data at higher values.

Positively skewed distribution. The right tail (higher-value end) of the frequency curve is longer and thinner than the left; most observations cluster at the lower end, and a relatively small number of unusually large values stretch the tail to the right. Because the mean is dragged toward these large values more than the median, and the median more than the mode:

Negatively skewed distribution. The left tail (lower-value end) is longer and thinner; most observations cluster at the higher end, and a small number of unusually small values stretch the tail to the left. The inequality reverses:

Why the mode stays put. The mode simply marks where observations are most concentrated — it is completely indifferent to how far a handful of extreme values stretch a tail. The mean, by contrast, incorporates every observation, however extreme, into its calculation, so it moves the most toward a long tail. The median depends only on the position (rank) of the middle observation, so it moves partway between the two.

The empirical relationship. For a moderately skewed, unimodal distribution, Karl Pearson observed an approximate relationship among the three averages:

This is used whenever the mode cannot be located directly from the data — for example, when the distribution has an open-end class, is bimodal or multimodal, or the modal class is not clearly the single highest-frequency class. Substituting this estimated mode into Karl Pearson's coefficient formula is exactly what gives the alternative median-based version of that formula, $Sk_P = \dfrac{3(\text{Mean}-\text{Median})}{\sigma}$.

Students often try to memorise the inequality direction instead of reasoning it out from 'the mean moves furthest toward the long tail' — this makes it easy to reverse the order under exam pressure. Also common: misapplying the empirical relationship as Mode = 3 Mean minus 2 Median (the coefficients are the wrong way round) — the median's coefficient is always 3 and the mean's is always 2.