Q.For a distribution, Mean = Median = Mode = 50. What can you conclude about the skewness of this distribution, without doing any further calculation?

When all three averages are exactly equal, both Karl Pearson's and Bowley's numerators reduce to zero, so no calculation is even needed to conclude the distribution is symmetric. …

A symmetric distribution is defined precisely by the condition Mean = Median = Mode (Section 1). Since all three are given as equal to 50 here, this condition is satisfied exactly, so the distribution must be symmetric — there is no need to compute the standard deviation or quartiles to know that Karl Pearson's coefficient will be zero:

$Sk_P=\dfrac{\text{Mean}-\text{Mode}}{\sigma}=\dfrac{50-50}{\sigma}=0$ …

Do not assume that Mean = Median = Mode alone guarantees the entire frequency curve is a perfect mirror image (a real, oddly-shaped distribution could coincidentally have equal averages while still being non-symmetric in its tails) — for the purposes of this chapter and Karl Pearson' …