ISS 2016 Statistics Paper-1 Solution: Question 30 (Regression Lines and Their Properties)
Continuing our question-by-question walk through the ISS 2016 Statistics Paper-1, this post tackles Question 30 — a problem that looks intimidating because it hides three separate claims inside one question, but which becomes very manageable once you remember one simple fact about regression lines. As always, every algebraic step is shown in full so you can follow along even if regression is still a new topic for you.
Quick Summary
Topic:Regression lines of Y on X and X on Y — slopes, correlation coefficient, ratio of variances, and the mean point
Question Reference:ISS 2016, Statistics Paper-1, Question 30
Correct Answer:(d) 1, 2 and 3
The Question As It Appeared In The Paper
Let the regression lines of Y on X and X on Y be Y = aX + b and X = cY + d respectively. Consider the following statements:
The ratio of the variances of X and Y is c/a.
The correlation coefficient between X and Y is √(ac).
The values of X̄ and Ȳ are (cb + d)/(1 − ac) and (ad + b)/(1 − ac) respectively.
Which of the above are correct?
(a) 1 and 2 only
(b) 2 and 3 only
(c) 1 and 3 only
(d) 1, 2 and 3
Step 1: Write Down The Standard Form Of Both Regression Lines
Before touching this question, it helps to recall the textbook form of the two regression lines for a bivariate data set (X, Y), where X̄ and Ȳ are the means, σx and σy are the standard deviations, and r is the correlation coefficient between X and Y.
Regression line of Y on X:
Y − Ȳ = bYX (X − X̄), where the slope bYX = r · (σy/σx)
Regression line of X on Y:
X − X̄ = bXY (Y − Ȳ), where the slope bXY = r · (σx/σy)
Now compare these to the two lines given in the question, Y = aX + b and X = cY + d. Matching the coefficient of X in the first line and the coefficient of Y in the second line, we get:
a = bYX = r · (σy/σx) and c = bXY = r · (σx/σy)
These two relationships are the key to checking statements 1 and 2, so keep them handy.
Step 2: Check Statement 2 — Is r = √(ac)?
Multiply the two slopes a and c together:
a × c = [r · (σy/σx)] × [r · (σx/σy)]
Notice that σy/σx and σx/σy are reciprocals of each other, so they cancel out completely, leaving:
ac = r²
Taking the square root of both sides gives r = √(ac), where the sign of r is understood to match the (common) sign of a and c, since both slopes always carry the same sign as the correlation coefficient. So statement 2 is correct.
Step 3: Check Statement 1 — Is The Ratio Of Variances Equal To c/a?
We already have:
a = r · (σy/σx) and c = r · (σx/σy)
Now simply divide c by a:
c/a = [r · (σx/σy)] / [r · (σy/σx)]
The r in the numerator and denominator cancels, and dividing (σx/σy) by (σy/σx) is the same as multiplying by its reciprocal:
c/a = (σx/σy) × (σx/σy) = σx²/σy²
Since σx² is the variance of X and σy² is the variance of Y, this shows c/a = Var(X)/Var(Y), which is exactly the ratio of variances of X and Y. So statement 1 is also correct.
Step 4: Check Statement 3 — Solving For X̄ And Ȳ
This is the step where most students lose marks, not because the algebra is hard, but because it's easy to forget the one fact that unlocks it: every regression line, by construction, passes through the mean point (X̄, Ȳ). That means we can simply substitute X = X̄ and Y = Ȳ into both given equations and solve them as a pair of simultaneous equations.
Substituting into Y = aX + b:
Ȳ = aX̄ + b ...(i)
Substituting into X = cY + d:
X̄ = cȲ + d ...(ii)
Now substitute equation (ii) into equation (i) to eliminate X̄:
Ȳ = a(cȲ + d) + b = acȲ + ad + b
Bring the acȲ term to the left side:
Ȳ − acȲ = ad + b
Ȳ(1 − ac) = ad + b
Dividing both sides by (1 − ac):
Ȳ = (ad + b)/(1 − ac)
Now substitute this value of Ȳ back into equation (ii) to get X̄:
X̄ = c · [(ad + b)/(1 − ac)] + d
Bring both terms to a common denominator of (1 − ac):
X̄ = [c(ad + b) + d(1 − ac)] / (1 − ac)
Expand the numerator:
c(ad + b) + d(1 − ac) = acd + bc + d − acd
The acd term and the −acd term cancel each other out, leaving:
Numerator = bc + d
So:
X̄ = (bc + d)/(1 − ac) = (cb + d)/(1 − ac)
These are exactly the expressions given in statement 3, so statement 3 is correct as well.
Pro Tip: Whenever a question gives you two regression lines in "mixed up" form like Y = aX + b and X = cY + d, don't try to memorise a ready-made formula for X̄ and Ȳ. Just remember that both lines pass through (X̄, Ȳ), write that substitution down first, and solve the two resulting linear equations by ordinary elimination. This is the way it's usually taught to beginners — get the "lines pass through the mean point" habit fixed in your head, and every variant of this question (no matter how the letters are swapped) becomes a two-line simultaneous-equation problem instead of something to memorise.
Final Answer
Since statement 1, statement 2, and statement 3 are all individually correct, the right choice is (d) 1, 2 and 3. This matches the official ISS 2016 answer key, which also lists the answer to Question 30 as (d).
Why This Question Matters
Regression lines are one of the most heavily tested topics across ISS Statistics Paper-1 year after year, and this particular question is a great example of why. It doesn't just test whether you know the formula bYX = rσy/σx by rote — it tests whether you can combine that formula with two other ideas (the product of the two regression coefficients equalling r², and both lines passing through the mean point) inside a single problem. Questions like this reward students who understand where the standard regression formulas come from rather than those who have only memorised the end results, and building that habit of re-deriving results from first principles is exactly what pays off when the letters in a question get rearranged, as they did here with a, b, c, d instead of the usual bYX and bXY.
Frequently Asked Questions
Why do both regression lines always pass through the point (X̄, Ȳ)?
Both regression lines are derived by minimising the sum of squared errors, and one of the standard results of that minimisation (from the normal equations) is that the fitted line must pass through the point of means. This is true for the regression of Y on X and for the regression of X on Y, which is exactly why we could substitute X̄ and Ȳ into both given equations in Step 4.
Why is r = √(ac) and not simply r = ac?
The product of the two regression coefficients, a and c, works out to r², not r itself, because each coefficient individually carries one factor of r along with a ratio of standard deviations that cancels only when you multiply the two coefficients together. Taking the square root of r² is what recovers r, with the sign taken to match the common sign of a and c.
Can the correlation coefficient be negative even though we write r = √(ac)?
Yes. The symbol √(ac) here denotes the magnitude of r; the actual sign of r is always the same as the sign of both regression coefficients a and c (they can never have opposite signs, since both are proportional to r). If a and c are both negative, r is negative, and you would write r = −√(ac).
What is the difference between bYX and bXY in the regression equations?
bYX is the slope of the regression line of Y on X, used to predict Y from a given X, and equals rσy/σx. bXY is the slope of the regression line of X on Y, used to predict X from a given Y, and equals rσx/σy. They are generally different numbers (unless σx = σy), and in this question they correspond to a and c respectively.
Is it safe to assume a multi-statement question like this always has "all correct" or "all wrong" as the answer?
No, that would be a risky shortcut. Options like "1 and 2 only" or "2 and 3 only" are deliberately included to catch students who verify only one or two of the statements and guess the rest. Each statement needs to be derived independently, exactly as done in Steps 2 to 4 above, before you pick an option.
How often does this kind of regression-line question appear in ISS Statistics Paper-1?
Very often. Correlation and regression together form one of the highest-weightage topics in the objective paper across almost every year of the ISS exam, and questions frequently test the relationships between bYX, bXY, r, variances, and the mean point in different combinations, exactly as this one does.
If any step above felt unclear, or if you solved this differently, drop a comment below — and if you know a fellow ISS aspirant who's revising correlation and regression right now, consider sharing this post with them so they can work through it too.
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