Class 11 Statistics Notes · GSEB

Comparison of Methods

Comparison of Methods — know when Karl Pearson beats Bowley, when Bowley is the only option, and why their values must never be cross-compared. GSEB Class 11 Statistics notes with the sign-clash case.

Last updated: 5 Oct 2026

Notes

Why the Two Coefficients Are Not Equal

Run Karl Pearson and Bowley on the same data and you will almost never get the same number — the two rest on different averages. Karl Pearson leans on the mean, median/mode and the standard deviation, which are built from every observation; Bowley leans on the quartiles and the median, which are built from positions. Different raw material, different answer.

Karl Pearson reads

x̄, M or M₀ and s — arithmetic averages, moved by every single value including the extremes.

Bowley reads

Q₁, M and Q₃ — positional averages, blind to everything outside the middle 50%.

Both can be right — they answer slightly different questions

A class's phone-bill data can show +0.5 by one method and +0.1 by the other. The first describes where the whole distribution leans; the second describes how the middle half is lopsided. The disagreement is information, not an error — as long as you never rank the two numbers against each other.

Karl Pearson vs Bowley

The two methods, attribute by attribute
Aspect	Karl Pearson	Bowley
Averages used	x̄, M, M₀ + s	Q₁, M, Q₃
Data used	All observations	Central 50% only
Absolute formula	x̄ − M₀, or 3(x̄ − M)	(Q₃ − M) − (M − Q₁) = Q₃ + Q₁ − 2M
Coefficient formula	(x̄ − M₀)/s, or 3(x̄ − M)/s	(Q₃ + Q₁ − 2M)/(Q₃ − Q₁)
Range	−1..1 in practice (−3..3 special)	−1..1 always
Reliability	⭐ More reliable	Less reliable — middle 50% only
Open-ended classes	Impossible (no mean/SD)	⭐ Only option
Calculation effort	Higher — A, d, fd, fd², then √	⭐ Easier — positions and one subtraction

The Sign Clash — Illustration 18

Illustration 18 — monthly incomes of two cities, both methods run

City A

x̄ = 300 · M = 284 · s = 60 · Q₁ = 124 · Q₃ = 390

Karl Pearson

+0.80

3(x̄ − M)/s

Bowley

−0.20

(Q₃+Q₁−2M)/(Q₃−Q₁)

Opposite signs on one dataset — KP says positive, Bowley says negative.

City B

x̄ = 280 · M = 310 · s = 110 · Q₁ = 160 · Q₃ = 520

Karl Pearson

−0.82

3(x̄ − M)/s

Bowley

+0.17

(Q₃+Q₁−2M)/(Q₃−Q₁)

Opposite signs on one dataset — KP says negative, Bowley says positive.

+0.8 vs −0.2 on the SAME data — never cross-compare methods.

The book's verdict, verbatim: “it is not appropriate to compare the coefficients obtained by two methods. It is advisable to compare using only one method.”

The comparison that IS allowed

Never compare KP's j with Bowley's j — compare within one method.✓ Within Karl Pearson:City A +0.80 vs City B −0.82 → |0.80| < |0.82| → City B is more skewed. Same ruler, same method — that answer is safe.

✗ Wrong — one report, two rulers

“Revenue skew is +0.80 by Karl Pearson but user-spend skew is −0.20 by Bowley, so revenue is far more skewed.” Two different windows on two different questions — the comparison says nothing.

✓ Right — one ruler per report

Pick Karl Pearson for everything in that report, then rank by |j|: City A |0.80| vs City B |0.82| → City B wins. Or pick Bowley everywhere and rank |0.20| vs |0.17| → City A wins. Either is valid — mixing is not.

Who Sees More of the Data?

Reliability Gauge — Karl Pearson reads everything, Bowley reads the middle

Same distribution, two windows

One extreme wage can move x̄ and s; it cannot move a quartile at all. That is the reliability gap — and also why the two coefficients disagree.

Open-ended classes — who survives?

Less than 4	4 – 7	7 – 10	10 – 13	13 – 16	16 – 20	20 and above
26	119	198	86	39	20	12

Karl Pearson — locked

No mid-value for the open classes → x̄ and s cannot exist.

Bowley — unlocked, the only option

The cf column is complete → Q₁, M, Q₃ all computable (j = +0.10).

All four coefficients on one −1 … +1 line

Karl Pearson BowleyCity A: +0.80 vs −0.20 · City B: −0.82 vs +0.17 — opposite sides, same cities.

Rank cities within one colour only: rose vs rose, sky vs sky. Comparing a rose pin with a sky pin is the cross-method mistake.

Easier vs More Reliable ⭐

⭐ Bowley — easier to calculate

No assumed mean, no d, no fd², no square root. Locate three positions in the cf column, interpolate, subtract. Ill 15 even hands you the sum and difference directly.

⭐ Karl Pearson — more reliable

Mean and SD digest every observation, so a single outlier cannot hide from the coefficient. Bowley never sees beyond Q₁ and Q₃.

The reliability question

Karl Pearson is more reliable — all observations versus the central 50%. When a question asks for reliability, name Karl Pearson; when it asks for ease or for open-ended data, name Bowley.

j = 0 Means Symmetric — Mostly

j = 0 ⇒ symmetric (absence of skewness)

For Bowley: Q₃ + Q₁ = 2M. For Karl Pearson: x̄ = M = M₀. Either way the seesaw balances.

The caveat the book adds

A zero coefficient says the measuresbalance — yet the frequency curve can still lean. The book puts it plainly: “it is possible in some cases that the frequency curve of the distribution has positive or negative skewness even if j = 0.” The coefficient summarises averages; the eye sees the tails.

Worked Example Gallery

Both methods, one dataset

1 problems

Full question

The measures from the data about the monthly income of residents of two cities are obtained as follows. Find coefficient of skewness using Karl Pearson's and Bowley's method from these data.

Details	City A	City B
Mean x̄	300	280
Median M	284	310
Standard deviation s	60	110
First quartile Q₁	124	160
Third quartile Q₃	390	520

Karl Pearson — j = 3(x̄ − M)/s

City A: 3(300 − 284)/60 = 48/60 = +0.80 · City B: 3(280 − 310)/110 = −90/110 = −0.82

Bowley — j = (Q₃ + Q₁ − 2M)/(Q₃ − Q₁)

City A: (390 + 124 − 568)/(390 − 124) = −54/266 = −0.20

City B: (520 + 160 − 620)/(520 − 160) = 60/360 = +0.17

Conclusion

Opposite signs inside each city → the two coefficients must never be compared with each other. Within one method: KP gives |0.80| < |0.82| → City B more skewed; Bowley gives |0.20| > |0.17| → City A more skewed. Each statement is valid for its own method only.

Different windows on the same data — the disagreement is the lesson, not an error.

Quick Reference

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

Aspect	Karl Pearson	Bowley
Averages used	x̄, M, M₀ + s	Q₁, M, Q₃
Data used	All observations	Central 50% only
Easier to compute	No	⭐ Yes
More reliable	⭐ Yes	No
Open-ended classes	Impossible	⭐ Only option
Range	−1..1 (±3 special)	−1..1 always

⭐ Never compare KP's j with Bowley's j — compare within one method.
The two methods rest on different averages: KP on mean/median/mode + s, Bowley on quartiles + median.
Bowley is easier to calculate; Karl Pearson is more reliable (all observations vs central 50%).
When comparing degrees of skewness, compare |j| — ignore the sign
j = 0 ⇒ symmetric (absence of skewness) — with the curve caveat above.

Key Takeaways

Karl Pearson and Bowley generally return different numbers because they average different things: all observations vs the central 50%.
Bowley is quicker to compute; Karl Pearson is the more reliable of the two — and it is the answer to "which method is more reliable?"
Open-ended classes eliminate Karl Pearson entirely — Bowley becomes the only option, not merely the easier one.
The two coefficients can carry opposite signs on one dataset (Ill 18: +0.8 vs −0.2), which is exactly why cross-method comparison is meaningless.
Compare within one method, rank by |j|, and treat j = 0 as symmetric while remembering the curve can still lean.
So what? A dashboard that mixes both coefficients — say a startup KPI card showing KP for revenue skew and Bowley for user-spend skew — is comparing apples to oranges; pick one ruler per report.