Class 11 Statistics Notes · GSEB

Absolute and Relative Measures

Absolute and Relative Measures of skewness — turn the unit-carrying absolute S_k into the unit-free coefficient j for cross-population comparison. GSEB Class 11 Statistics notes with both method conversions.

Last updated: 5 Oct 2026

Notes

Direction Is Not Degree

“Measures of skewness indicates the tendency of the group to depart from symmetry.” — Connor

What the curve cannot tell you

A frequency curve shows whether the skew is positive or negative — but not how large it is. Two curves can both lean right while one barely tilts and the other drags a kilometre-long tail. Pinning a number on that lean needs two kinds of measures: an absolute one and a relative one.

Absolute measure of skewness (Sₖ)

Expressed in the same units of the population variable… denoted by Sₖ. Obtained as a difference of averages (Karl Pearson) or a difference of quartiles (Bowley).

Coefficient of skewness (j)

The relative measure of skewness is called coefficient of skewness. It is denoted by j

j = Sₖ ÷ an appropriate dispersion measure — the division is what strips the unit off. Values cluster near 0 when the curve is close to symmetry, and grow as the tail stretches.

Why Units Block Comparison

Try ranking an income curve against a height curve using their absolute measures — then switch to the coefficients and watch the comparison unlock.

Absolute · income

Sₖ = ₹2.37 lakh

carries the data’s own unit — rupees

Absolute · heights

Sₖ = 4.2 inches

a completely different unit

Disabled — ₹2.37 lakh vs 4.2 inches is not a ranking, it is a category error.

Relative · income

j = 0.11

no units — the dispersion cancelled them out

Relative · heights

j = 0.36

also unit-free — and directly comparable

Compared ✓

0.36 > 0.11 — the heights curve leans harder from symmetry than the income curve, even though the two datasets share nothing except being numbers.

The same trick powers the coefficient of variation from the dispersion chapter — divide, and the unit disappears.

Where this matters

Only relative measures can compare two or more groups with different units. A Zerodha analyst comparing trading-volume skew across the NSE (₹ crore) and a crypto exchange (USDT) must go through j — the raw Sₖ values would be measuring rupees against dollars, not shape.

The comparison that looked rigorous — and was not

A consultancy once ranked “skewness” of ₹ revenue data (Sₖ = ₹2.37 lakh) against staff attendance data (Sₖ = 4.2 days) and declared revenue “much more skewed” because 2.37 lakh beats 4.2. It does not — the numbers were never commensurable. Divide each by its own dispersion first, and only then compare: 0.11 vs 0.36 tells a real story, the raw pair never could.

From Absolute to Relative

One conversion, two methods — each absolute measure becomes its coefficient by dividing by the matching dispersion measure:

Karl Pearson route: (x̄ − M₀) ÷ s → j (difference of averages ÷ standard deviation)

Bowley route: (Q₃ + Q₁ − 2M) ÷ (Q₃ − Q₁) → j (difference of quartiles ÷ quartile range)

Read the sign, size the tail

j > 0 → positive skew · j < 0 → negative skew · j ≈ 0 → close to symmetry. The sign gives direction, the size gives degree — exactly the two things the curve alone could not.

Quick Reference

Sₖ = absolute skewness — carries the data’s own unit

⭐ j = coefficient of skewness (unit-free) = Sₖ ÷ dispersion

The curve shows the sign of skewness but not its size — that is why numerical measures exist

Karl Pearson conversion: ÷ s · Bowley conversion: ÷ (Q₃ − Q₁)

j ≈ 0 → the distribution is close to symmetric

Words marked with a dotted underline carry the question they usually appear in — tap them to reveal it.

Key Takeaways

A curve tells you the direction of skew but never the degree — two right-leaning curves can differ wildly in how hard they lean, which is exactly why Sₖ and j exist.
Sₖ is absolute: mean − mode or Q₃ + Q₁ − 2M, measured in ₹, marks or inches — as incomparable across datasets as comparing a ₹2.37 lakh salary to a 4.2-inch rainfall anomaly.
j = Sₖ ÷ dispersion is the unit-free coefficient — divide, and rupees, inches and days all speak the same language, so a Zerodha volume curve can be ranked against anything.
Karl Pearson divides (x̄ − M₀) by s; Bowley divides (Q₃ + Q₁ − 2M) by (Q₃ − Q₁) — two routes, one destination: a number near 0 means near-symmetric.
Before trusting any "more skewed than" claim, check the units on both sides — if they differ, the comparison is meaningless until both sides become coefficients.