ISS 2016 Statistics Paper-1 Solution: Question 49 (Regression Coefficients and Correlation)

This post continues our ongoing question-by-question walkthrough of the ISS Statistics Paper-1 previous year papers, where every problem is worked out from first principles rather than just stating the final option. Today we're on the 2016 paper, and the question tests a classical identity relating the correlation coefficient to the two regression coefficients — one that rewards careful algebra rather than memorised shortcuts.

Quick Summary

• Topic: Relationship between the correlation coefficient and the two regression coefficients (bYX and bXY)

• Question Reference: ISS 2016, Statistics Paper-1, Question 49

• Official Key Answer: Option (b)

• Our independent derivation: Option (c) — with the discrepancy explained in full below

The Question

Consider the following statements:

Statement I:The correlation coefficient between two variables X and Y is the geometric mean of the two regression coefficients βYX and βXY.

Statement II:The arithmetic mean of the two regression coefficients βYX and βXY is greater than or equal to the correlation coefficient between the variables X and Y.

Which one of the following is correct in respect of the above two statements?

• (a) Both Statement I and Statement II are true and Statement II is the correct explanation for Statement I.

• (b) Both Statement I and Statement II are true but Statement II is not the correct explanation for Statement I.

• (c) Statement I is true but Statement II is false.

• (d) Statement I is false but Statement II is true.

Step 1: Write Down What bYX and bXY Actually Are

For two variables X and Y with standard deviations σX and σY and correlation coefficient r, the two regression coefficients are defined as:

βYX = r · (σY / σX)   and   βXY = r · (σX / σY)

Notice both coefficients are r multiplied by a strictly positive quantity (a ratio of two standard deviations). That means βYX and βXY always carry the same sign as r — both positive when r is positive, both negative when r is negative, and both zero when r is zero. This single observation is the key to the whole question, so keep it in mind as we go.

Step 2: Check Statement I — Is r the Geometric Mean of the Two Coefficients?

Multiply the two regression coefficients together:

βYX · βXY = [r · (σY/σX)] · [r · (σX/σY)] = r2 · (σY/σX) · (σX/σY) = r2

The σX and σY terms cancel completely, leaving a clean result: the product of the two regression coefficients is always exactly r2, regardless of what σX and σY happen to be. Taking the square root gives √(βYX · βXY) = |r|.

Since we already established that βYX and βXY always share the same sign as r, the geometric mean is conventionally taken with that sign attached (this is the standard convention used in every statistics textbook that states this result) — so the signed geometric mean equals r itself, not just |r|.Statement I is true.

Step 3: Check Statement II — Is the Arithmetic Mean Always ≥ r?

This is where the question gets interesting, and where it's worth slowing down rather than pattern-matching to a half-remembered rule. We want to know whether:

(βYX + βXY) / 2 ≥ r   always holds.

Let's compute the difference directly instead of guessing:

βYX + βXY − 2r = r(σY/σX) + r(σX/σY) − 2r = r [ (σY2 + σX2) / (σXσY) − 2 ]

The bracketed term simplifies using a perfect square:

(σY2 + σX2 − 2σXσY) / (σXσY) = (σX − σY)2 / (σXσY)

This fraction is a square divided by a positive number, so it is always ≥ 0. That gives us the full, honest result:

βYX + βXY − 2r = r · [ (σX − σY)2 / (σXσY) ]

The right-hand side is r multiplied by something non-negative.That means the sign of (AM − r) matches the sign of r itself — not a fixed direction. When r is positive, AM − r ≥ 0, so the arithmetic mean is indeed at least r. But when r is negative, r times a non-negative number is non-positive, so AM − r ≤ 0, meaning the arithmetic mean is at most r, not at least r.

Step 4: A Concrete Counterexample for Negative r

Let's test this with actual numbers rather than leaving it abstract. Take r = −0.5, σX = 1, σY = 4 (a perfectly valid, achievable correlation and pair of standard deviations):

βYX = (−0.5)(4/1) = −2    βXY = (−0.5)(1/4) = −0.125

Arithmetic mean = (−2 + (−0.125)) / 2 = −1.0625

Compare this to r = −0.5. Is −1.0625 ≥ −0.5? No — it is considerably smaller. So in this case the arithmetic mean of the regression coefficients is well below the correlation coefficient, directly contradicting Statement II as written. Since a single valid counterexample is enough to disprove a "for all" claim,Statement II is false in general — it only holds when r ≥ 0.

Pro Tip: Whenever a statement-based MCQ gives you a "universal" inequality involving the correlation coefficient, don't just verify it for a positive-correlation example and move on — correlation can be negative, and many textbook inequalities quietly assume r ≥ 0 without saying so. The habit worth drilling until it's automatic is to re-derive the expression symbolically (as we did with βYX + βXY − 2r above) and check what happens to the sign when r flips negative, rather than trusting a memorised one-line "fact." This is exactly the kind of step-by-step checking that's emphasised when these problems are taught properly, rather than just handing out the final formula.

Step 5: Matching Against the Official Answer Key

The official ISS 2016 Statistics Paper-1 answer key (Series A) lists the answer to Question 49 as option (b) — both statements true, with Statement II not explaining Statement I. Our derivation agrees completely with Statement I being true, and agrees with the "AM ≥ r" relationship whenever r is positive. But we found a clean, verifiable counterexample showing Statement II fails for negative r, which makes the mathematically rigorous answer (c): Statement I true, Statement II false, since Statement II as worded contains no restriction to r ≥ 0.

This is a genuine case worth flagging rather than quietly overriding: the "arithmetic mean of regression coefficients ≥ r" line appears in many statistics textbooks stated as an unconditional theorem, and it's likely that's the version the paper-setters had in mind when the key was prepared — most worked examples used to illustrate it happen to use positive correlation. For exam-scoring purposes the official key (b) is what's binding, but the algebra above shows precisely where that general statement breaks down, which is worth understanding regardless of which option you'd bubble in.

Final Answer

Statement I is true (r is the signed geometric mean of βYX and βXY, always). Statement II is true only for r ≥ 0 and false for negative r, as shown by direct derivation and a numeric counterexample. Going purely by the rigorous mathematics, the correct option is (c). The official key marks this question as (b)— we've shown exactly why and where that answer's underlying assumption (implicitly restricting to non-negative correlation) comes apart.

Why This Question Matters

This question is a great stress-test of whether you actually understand the algebra behind the bYX, bXY, and r relationship, or whether you've only memorised the standard "AM ≥ GM" soundbite that gets repeated in most coaching material. The product relationship βYX·βXY = r2 is extremely useful and shows up repeatedly in correlation-regression questions across ISS, IIT JAM, and GATE Statistics papers, so it's worth being able to derive it on the spot rather than recalling it as a fact. The inequality half of the question is a good reminder that in statistics, "always true" claims deserve a quick sign-check before you trust them in an exam setting.

Frequently Asked Questions

Why do βYX and βXY always have the same sign as the correlation coefficient r?

Because each regression coefficient equals r multiplied by a ratio of two standard deviations, and standard deviations are always positive. Multiplying r by a positive number never changes its sign, so both βYX and βXY inherit the sign of r exactly.

Is it correct to say r equals the geometric mean of the two regression coefficients even when r is negative?

Yes, provided you use the signed geometric mean rather than the unsigned (positive-root) definition. The unsigned square root of βYX·βXY always gives |r|, but since βYX and βXY share r's sign, the conventionally "signed" geometric mean recovers r itself, including its sign.

Where exactly does the "arithmetic mean ≥ correlation coefficient" claim break down?

It breaks down whenever r is negative and the two standard deviations σX and σY are unequal. The derivation shows βYX + βXY − 2r equals r times a non-negative quantity, so for negative r that difference becomes negative, meaning the arithmetic mean actually falls below r instead of staying above it.

Does this inequality issue mean the AM-GM inequality itself is wrong here?

No, the AM-GM inequality is perfectly valid — it's just being applied to the magnitudes |βYX| and |βXY|, which does give AM(|βYX|, |βXY|) ≥ GM(|βYX|, |βXY|) = |r| correctly. The subtlety is that converting that magnitude inequality back into a statement about the signed arithmetic mean versus signed r requires extra care once r is negative.

How should I treat this kind of discrepancy between my own working and the official answer key during exam preparation?

Always re-derive the result symbolically rather than trusting a one-line memorised claim, exactly as done above, and if your derivation and the key still disagree after a careful recheck, note the discrepancy and move on — for scoring purposes the official key is what counts, but understanding exactly where and why a "standard" claim fails is far more valuable for later, harder questions than simply memorising the answer letter.

Will similar "Statement I / Statement II" correlation-regression questions appear again in future ISS papers?

Very likely — the βYX·βXY = r2 identity and related inequalities are recurring favourites in ISS, IIT JAM, and GATE Statistics papers because they test algebraic fluency rather than rote recall, so it's worth being comfortable deriving both the product and sum relationships from scratch.

If anything in this derivation feels off or you'd solve it differently, drop a comment below — working through the disagreement is often where the real learning happens. And if you know a fellow ISS aspirant who's grinding through the same papers, feel free to share this one with them.