ISS 2016 Statistics Paper-2 Solution: Question 61 (Fisher's Ideal Index Number)

Continuing our question-by-question walk through the ISS Statistics Paper-2 archive, today we land on a question from the Index Numbers section of the 2016 paper. This topic is a perennial favourite with examiners because it mixes a definition-based "statement" question with a subtle conceptual trap, so it rewards careful reading far more than memorisation.

Quick Summary

• Topic: Index Numbers — Fisher's Ideal Index Number

• Question Reference: ISS 2016, Statistics Paper-2, Question 61

• Correct Answer: Option (c) — Statement-I is true but Statement-II is false

The Question

Consider the following statements:

Statement-I: Fisher's Index number is known as ideal Index number.

Statement-II: Fisher's Index number is the arithmetic mean of Laspeyre's Index number and Paasche's Index number which show upward bias and downward bias respectively.

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 Laspeyres and Paasche Price Indices

Before we can judge Statement-II, we need the exact formulas for the two building-block indices. Let p 0,q 0 denote price and quantity in the base year, and p 1,q 1 denote price and quantity in the current year.

Laspeyres Price Index (uses base-year quantities as weights):

P01L = ( Σ p1q0 / Σ p0q0 ) × 100

Paasche Price Index (uses current-year quantities as weights):

P01P = ( Σ p1q1 / Σ p0q1 ) × 100

Both numbers try to measure the same thing — how much prices have moved between two periods — but because they use different quantity weights, they almost never come out exactly equal.

Step 2: Recall the Exact Definition of Fisher's Index

Fisher's Index number is explicitly constructed by combining the above two indices. The formula is:

P01F = √( P01L × P01P )

In words, you multiply the Laspeyres index and the Paasche index together, and then take the square root of the product. This is, by definition, the geometric mean of the two numbers — not the arithmetic mean. If it were the arithmetic mean, the formula would instead read P01F = (P01L + P01P) / 2, which is a completely different expression.

This single word — "arithmetic" versus "geometric" — is exactly where Statement-II goes wrong.

Step 3: Check the Bias Direction Claim in Statement-II

To be fair to the statement, let's verify the other half of its claim, about bias direction. In most real markets, price and quantity move in opposite directions (when a good gets costlier, people buy less of it — the substitution effect). Under this typical negative price-quantity relationship:

• Laspeyres index tends to overstate the true price rise, because it keeps using the old (higher) base-period quantities as weights even after consumers have shifted away from the now-expensive goods. This is the "upward bias."

• Paasche index tends to understate the true price rise, because it weights by current-period quantities, which already reflect the shift away from expensive goods. This is the "downward bias."

So the bias-direction part of Statement-II (upward bias for Laspeyres, downward bias for Paasche) is actually a correct and commonly cited property. The statement as a whole, however, is still false, because it misdescribes the averaging method as "arithmetic mean" when it is in fact the geometric mean.

Pro Tip: Many students skim statements like this and only check the "big idea" — here, that Fisher's index somehow combines Laspeyres and Paasche — without checking the precise type of average used. In ISS and similar exams, this is exactly where examiners plant the trap. The habit worth drilling until it's automatic is to isolate every mathematical keyword in a statement (mean, median, upward, downward, necessary, sufficient) and verify each one separately against the formula, rather than judging the statement as a whole on vibes. This is the kind of line-by-line verification habit that's typically drilled hard in a structured Index Numbers class, such as the way it's taught at Sunrise Classes, precisely because this trap repeats across years.

Step 4: Why Fisher Called It the "Ideal" Index Number

Now let's verify Statement-I. Irving Fisher proposed his index and called it "ideal" because, unlike Laspeyres or Paasche individually, it satisfies two important consistency tests simultaneously:

• Time Reversal Test:P01 × P10 = 1. If you compute the index from year 0 to year 1, and then from year 1 back to year 0, multiplying the two results should bring you back to 1 (i.e., no net change). Fisher's index satisfies this exactly, because it is built as a geometric mean, and geometric means have this reciprocal property.

• Factor Reversal Test:P01 × Q01 = Σp1q1 / Σp0q0. This says that if you multiply the price index by the corresponding quantity index (constructed the same way), you should get the ratio of current to base total value. Fisher's index satisfies this test too.

Since Fisher's Index number is the only commonly used index that satisfies both the Time Reversal Test and the Factor Reversal Test, it earned the name "ideal" index number in index-number theory. Statement-I is therefore correct.

Step 5: Combine the Two Findings and Match the Option

Putting it together:

• Statement-I ("Fisher's Index number is known as ideal Index number") —True, because of the dual-test property shown above.

• Statement-II ("Fisher's Index number is the arithmetic mean of Laspeyres and Paasche...") —False, because Fisher's index is actually the geometric mean, not the arithmetic mean, even though the bias-direction part of the sentence is correct on its own.

Since Statement-I is true and Statement-II is false, the correct option is (c). This matches the official ISS 2016 Paper-2 answer key, which lists option (c) as the correct response for Question 61.

Why This Question Matters

Index number theory shows up repeatedly across ISS Paper-2, both as direct numerical questions (compute Laspeyres, Paasche, Fisher for given data) and as these conceptual "statement" questions. Getting comfortable with the exact formulas — and training yourself to notice when a statement subtly swaps "geometric" for "arithmetic," or reverses a bias direction — pays off across multiple questions in every year's paper, not just this one.

Frequently Asked Questions

Is Fisher's Index number always strictly between Laspeyres and Paasche?

Yes. Since the geometric mean of two positive numbers always lies between them (and equals the arithmetic mean only when the two numbers are equal), Fisher's index always falls between the Laspeyres and Paasche values for the same data, touching one of them only in the special case where Laspeyres equals Paasche.

Why is the geometric mean preferred over the arithmetic mean here?

Because Laspeyres tends to overstate price change and Paasche tends to understate it, averaging them tends to cancel out much of the bias. The geometric mean is the natural choice for combining ratios and index numbers (which are themselves ratios), and it is also what makes the Time Reversal Test hold exactly — something the arithmetic mean cannot guarantee.

What other properties make an index number "ideal" in general?

Besides the Time Reversal and Factor Reversal Tests, index number theory also discusses the Circular Test. Interestingly, Fisher's index does not satisfy the Circular Test, which is sometimes asked as a follow-up trap in other questions — so "ideal" refers specifically to the two tests it does satisfy, not to every consistency property an index could have.

Does the direction of bias (upward for Laspeyres, downward for Paasche) always hold?

No, it holds under the typical assumption that price and quantity are negatively related (the usual substitution effect in consumer behaviour). If prices and quantities happen to move together instead, the bias directions can reverse. Exam statements usually assume the typical case unless stated otherwise.

How should I remember the Fisher formula quickly in an exam?

Think of it as "square root of the product," the same structure as the geometric mean of any two numbers: P01F = √(P01L × P01P). If you can recall that Fisher's index is a geometric mean rather than an arithmetic mean, you will catch almost every variant of this trap question.

Is this topic important for other exams like GATE(ST) or IIT JAM(MS) too?

Index numbers are more of an applied/official-statistics topic, so they appear more prominently in ISS Paper-2 than in GATE(ST) or IIT JAM(MS), but the underlying averaging concepts (arithmetic vs. geometric mean, weighted averages) are tested across all these exams in various forms.

If anything in this derivation felt unclear, drop a comment below and we'll walk through it further — and if you found this useful, consider sharing it with a fellow ISS aspirant who's also working through the Paper-2 archive.