ISS 2016 Statistics Paper-1 Solution: Question 52 (Third-Order Divided Difference)

Continuing our question-by-question walkthrough of the ISS Statistics Paper-1 archive, this post picks up right where we left off and tackles the next item on the list: a numerical methods question on divided differences. If you've been following this series from Question 1, you'll notice we're now deep into the numerical analysis portion of the 2016 paper.

Quick Summary

• Topic:Numerical Analysis — Divided Differences

• Question Reference:ISS 2016, Statistics Paper-1, Question 52

• Correct Answer:Option (a) — 1

The Question

What is the third order divided difference with arguments 2, 4, 9 and 10 of the function f(x) = x3 − 2x ?

• (a) 1

• (b) 2

• (c) 3

• (d) 4

Step 1: Set Up the Function Values

Divided differences are built from the actual values of the function at the given arguments (points), so the first thing to do — always — is compute f(x) at every one of the four given points: x0 = 2, x1 = 4, x2 = 9, x3 = 10.

We're given f(x) = x3 − 2x. Substitute each argument one at a time:

• f(2) = 23 − 2(2) = 8 − 4 = 4

• f(4) = 43 − 2(4) = 64 − 8 = 56

• f(9) = 93 − 2(9) = 729 − 18 = 711

• f(10) = 103 − 2(10) = 1000 − 20 = 980

So we now have the table of (x, f(x)) pairs: (2, 4), (4, 56), (9, 711), (10, 980).

Step 2: Recall the Divided Difference Formula

The first order divided difference between two points xi and xj is defined as:

f[xi, xj] = (f(xj) − f(xi)) / (xj − xi)

Once you have first order differences, you build second order divided differences out of them the same way — by dividing the difference of two first order differences by the spread between the outer arguments. The same pattern repeats to get third order differences from second order ones. This "difference of differences, divided by the spread of x-values" idea is the entire engine behind the method — every higher order is built the same way, one level at a time, so there's no shortcut to memorise, just the pattern to repeat carefully.

Step 3: Compute the First Order Divided Differences

With four points x0, x1, x2, x3 we get three first order divided differences, one for each consecutive pair:

f[x0, x1] = (f(x1) − f(x0)) / (x1 − x0) = (56 − 4) / (4 − 2) = 52 / 2 = 26

f[x1, x2] = (f(x2) − f(x1)) / (x2 − x1) = (711 − 56) / (9 − 4) = 655 / 5 = 131

f[x2, x3] = (f(x3) − f(x2)) / (x3 − x2) = (980 − 711) / (10 − 9) = 269 / 1 = 269

Step 4: Compute the Second Order Divided Differences

Now combine consecutive first order differences, dividing by the gap between the two outermost x-values involved:

f[x0, x1, x2] = (f[x1, x2] − f[x0, x1]) / (x2 − x0) = (131 − 26) / (9 − 2) = 105 / 7 = 15

f[x1, x2, x3] = (f[x2, x3] − f[x1, x2]) / (x3 − x1) = (269 − 131) / (10 − 4) = 138 / 6 = 23

Step 5: Compute the Third Order Divided Difference

Finally, combine the two second order differences the same way, dividing by the spread of the full set of four arguments:

f[x0, x1, x2, x3] = (f[x1, x2, x3] − f[x0, x1, x2]) / (x3 − x0) = (23 − 15) / (10 − 2) = 8 / 8 = 1

So the third order divided difference comes out to 1, which is option (a).

Pro Tip: Whenever a question gives you a polynomial and asks for its divided difference of the

Step 6: Confirm Against the Official Answer Key

Our independently derived value, computed purely from the definition of divided differences step by step, is 1. Checking the official ISS 2016 Statistics Paper-1 answer key (Series A) for Question 52 confirms the marked correct option is also (a) 1 — so our derivation matches the official key exactly.

Why This Question Matters

Divided differences aren't just an abstract numerical-methods curiosity — they are the foundation of Newton's divided difference interpolation formula, which lets you construct a polynomial through unequally spaced data points. Questions like this one test two things simultaneously: whether you can mechanically apply the recursive definition without making an arithmetic slip across three levels of calculation, and whether you understand the deeper structural fact that ties divided differences to the degree of the underlying polynomial. Both skills come up repeatedly across the numerical analysis section of the ISS syllabus, from interpolation to numerical differentiation and integration, so a solid grip on this one definition pays off well beyond this single question.

Frequently Asked Questions

What exactly is a divided difference in numerical analysis?

A divided difference is a recursively defined quantity built from function values at a set of (possibly unequally spaced) points. The first order divided difference is simply the slope between two points, and each higher order is formed by taking the difference of two divided differences one order lower and dividing by the spread of the outermost arguments involved.

Why do divided differences work even when the x-values are not equally spaced?

Unlike simple forward or backward differences, which assume a constant step size h, divided differences explicitly divide by the actual gap between arguments at every stage. This built-in normalization is exactly what allows them to handle irregularly spaced data, which is why they underlie Newton's divided difference interpolation formula rather than the equal-spacing-only Newton forward/backward formulas.

Is there a faster way to solve this type of question in the exam?

Yes — if the order of the divided difference asked for matches the degree of the given polynomial, the answer is simply the leading coefficient of that polynomial, regardless of which points are used. For f(x) = x3 − 2x and a third order divided difference, the answer is the coefficient of x3, which is 1. This shortcut only applies when the order equals the polynomial's degree, so it's worth double-checking that condition before relying on it.

What happens if the order of the divided difference is higher than the degree of the polynomial?

Any divided difference of order strictly greater than the degree of a polynomial is always zero. For example, a fourth order divided difference of our cubic f(x) = x3 − 2x, computed with any five points, would come out to exactly 0.

Does the order in which the points are listed affect the final answer?

No. A key property of divided differences is that they are symmetric functions of their arguments — permuting the order of x0, x1, x2, x3 and redoing the calculation will always give the same final third order divided difference, even though the intermediate first and second order values along the way will look different.

How many marks is a question like this typically worth, and is it negatively marked?

Like every objective question in ISS Paper-1, this item carries equal marks out of the paper's total of 200 across 80 questions, with a one-third penalty deducted for a wrong attempt and no penalty for leaving it unattempted. Given how mechanical and verifiable this calculation is, it is exactly the kind of question worth attempting confidently once the method is clear.

If anything in this derivation feels unclear, or you'd like to see the leading-coefficient shortcut proved more formally, drop a comment below — and if this helped, do share it with a fellow ISS aspirant working through the same paper.