ISS 2016 Statistics Paper-1 Solution: Question 51 (True Statements on Divided Differences)
Continuing our question-by-question walk through the ISS Statistics Paper-1 archive, we now arrive at Question 51 of the 2016 paper. This series exists so that no question gets skimmed over — every statement is checked from first principles rather than taken on faith, which is exactly the habit this particular question rewards.
Quick Summary
Topic:Numerical Analysis — properties of divided differences
Question Reference:ISS 2016, Statistics Paper-1, Question 51 (Series A)
Correct Answer:Option (a) — Statements 2, 3 and 4 only
The Question as Asked
Consider the following statements:
Divided difference of kth order involves k arguments.
Divided difference is a symmetric function of its argument values.
Divided difference can be expressed as the ratio of two determinants of (n + 1) order each.
Divided difference is applicable even when arguments are not equispaced.
Which of the above statements are correct?
(a) 2, 3 and 4 only
(b) 1, 3 and 4 only
(c) 1, 2 and 4 only
(d) 1, 2 and 3 only
Step 1: Recall What a Divided Difference Actually Is
Before judging any statement, we need the exact definition in front of us, because this whole question is really a test of definitions.
For a function f(x) known at distinct points x0, x1, x2, ..., the divided differences are built up step by step:
Zeroth order:f[x0] = f(x0)
First order:f[x0, x1] = (f[x1] − f[x0]) / (x1 − x0)
Second order:f[x0, x1, x2] = (f[x1, x2] − f[x0, x1]) / (x2 − x0)
and in general, the kth order divided difference is written f[x0, x1, ..., xk]. Count the arguments inside the bracket: there are k + 1 of them (x0 through xk), not k. This single counting detail is the key to the whole question.
Step 2: Check Statement 1
Statement 1 claims: "Divided difference of kth order involves k arguments."
From the definitions above, a first order divided difference (k = 1) already needs 2 points, x0 and x1. A second order divided difference (k = 2) needs 3 points. In general, the kth order divided difference needs k + 1 points, not k. So Statement 1 is false — it is off by exactly one argument.
Pro Tip: This "off by one" trap is one of the most common ways ISS and similar exams test whether you truly understand divided differences versus having just memorized the name. The way this distinction is usually drilled into students is by writing out f[x₀], then f[x₀,x₁], then f[x₀,x₁,x₂] side by side and literally counting the subscripts against the order — do that once by hand and you will never fall for this trap again.
Step 3: Check Statement 2 — The Symmetry Property
Statement 2 claims divided differences are a symmetric function of their arguments, meaning the value of f[x0, x1, ..., xk] does not change if we shuffle the order of x0, x1, ..., xk among themselves.
This follows from the explicit closed-form expression for the kth order divided difference:
f[x0, x1, ..., xk] = Σi=0k [ f(xi) / Πj≠i (xi − xj) ]
Look carefully at the right-hand side: it is just a sum over all the points, and swapping the labels of any two points xp and xq only reshuffles which term in the sum corresponds to which point — the total sum itself is unchanged. That is precisely the definition of a symmetric function. So Statement 2 is true.
Step 4: Check Statement 3 — The Determinant Ratio Form
Statement 3 claims the divided difference can be written as a ratio of two determinants, each of order (n + 1).
To see where this comes from, recall that fitting a degree-n polynomial through (n + 1) points x0, x1, ..., xn by solving for its coefficients directly (the "method of undetermined coefficients") leads to a system of (n + 1) linear equations in (n + 1) unknowns. By Cramer's Rule, the leading coefficient of that polynomial — which is exactly equal to the nth order divided difference f[x0, x1, ..., xn] — can be written as:
f[x0, ..., xn] = (determinant of the matrix built from function values and the Vandermonde-type powers) / (determinant of the plain Vandermonde matrix in x0, ..., xn)
Both the numerator and denominator here are determinants of an (n + 1) × (n + 1) matrix, since there are (n + 1) data points. So the statement's claim of "(n + 1) order each" matches this construction exactly. Statement 3 is true.
Step 5: Check Statement 4 — No Equal Spacing Required
Statement 4 claims divided differences work even when the x-values are not equally spaced.
Look back at the very first-order formula: f[x0, x1] = (f(x1) − f(x0)) / (x1 − x0). Nowhere does this formula require x1 − x0 to equal some fixed spacing h. Contrast this with forward differences (Δf(x) = f(x+h) − f(x)) or backward differences, both of which are built around a constant step size h and break down the moment the data points are unevenly spaced.This is actually the entire reason divided differences exist as a separate tool — they generalize finite differences to irregularly spaced data, which is common in real survey and experimental datasets. So Statement 4 is true.
Step 6: Combine and Confirm
Putting it together: Statement 1 is false, while Statements 2, 3 and 4 are all true. That matches option (a) 2, 3 and 4 only, which is also exactly what the official ISS 2016 answer key (Series A) lists for Question 51. Our independent derivation and the official key agree.
Why This Question Matters
Questions like this one are less about computation and more about precision of definitions — a skill that pays off across the entire numerical analysis portion of the Statistics paper, from Newton's divided difference interpolation formula to error bounds in Lagrange interpolation. Students who drill through each property statement-by-statement, rather than relying on a vague "I remember this is true" feeling, consistently score better on these conceptual MCQs because the exam is deliberately designed to insert one small, plausible-sounding error (like "k arguments" instead of "k + 1 arguments") into an otherwise correct list of facts.
Frequently Asked Questions
What is the difference between divided differences and ordinary finite differences?
Ordinary finite differences (forward, backward, central) assume the data points are equally spaced by a fixed step h, and the formulas are written directly in terms of h. Divided differences drop this requirement entirely and instead divide by the actual gap between the chosen x-values, which is why they work for any set of distinct points, spaced however they like.
Why is the symmetry property of divided differences useful?
Symmetry means you get the same divided difference no matter which order you list your data points in. Practically, this lets you build a divided difference table using the points in whatever order they were collected, without worrying about re-sorting them first, and it also simplifies several theoretical proofs in interpolation theory.
How many points are needed for a 3rd order divided difference?
A kth order divided difference always needs k + 1 points, so a 3rd order divided difference f[x₀, x₁, x₂, x₃] needs exactly 4 points. This is the same counting rule that makes Statement 1 in this question false.
Is the determinant-ratio representation of divided differences commonly asked in exams?
It appears less frequently than the recursive definition, but ISS and similar statistics exams do test it as a "second representation" to check whether a candidate understands divided differences conceptually rather than just being able to build a table mechanically. It is worth remembering as a cross-check method.
Where are divided differences used after this exam, in practice?
They form the backbone of Newton's divided difference interpolation formula, which is the standard way to interpolate a value from unevenly spaced data — something that comes up constantly in actual survey statistics, where observations are rarely recorded at perfectly equal intervals.
What is the fastest way to double-check a "which statements are true" question like this one?
Write out the exact formal definition for the concept being tested before reading the options, then test each statement against that definition one at a time, marking it true or false in isolation. Resist the temptation to judge options by "does this sound right" — that is exactly the instinct this kind of question is designed to catch out.
If any step above felt unclear, drop a comment below and we'll work through it together — and if you know a fellow ISS aspirant who's revising numerical analysis right now, sharing this with them costs nothing and might save them from the exact trap in Statement 1.
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