ISS 2016 Statistics Paper-2 Solution: Question 47 (Standard Error of Sample Mean)

Continuing our question-by-question walk through the ISS Statistics Paper-2 archive, we're now at Question 47 of the 2016 paper. This one looks tiny on the page, but it hides a classic sampling-theory idea that trips up a lot of first-time test takers — so we'll slow right down and build it from first principles rather than just quoting a formula.

Quick Summary

• Topic:Standard error of the sample mean, infinite population

• Question Reference:ISS 2016, Statistics Paper-2, Question 47

• Correct Answer:Option (b) — σ⁄5

The Question, As Asked

If population variance of an infinite population is σ² and a sample of 25 items is selected from this population, then what is the standard error of sample mean?

• (a) σ²/25

• (b) σ/5

• (c) σ/25

• (d) σ

Step 1: Set Up What's Given

We're told two things, and both matter:

• The population is infinite (or at least large enough to be treated that way).

• A random sample of size n = 25 is drawn from it, and the population variance is σ².

The quantity we want is the standard error of the sample mean, usually written SE(X̅). Standard error is simply the standard deviation of the sampling distribution of a statistic — here, the statistic is the sample mean X̅.

Step 2: Find the Variance of the Sample Mean

Let X1, X2, ..., X25 be the 25 observations drawn independently from the population, each with variance σ². The sample mean is

X̅ = (X1 + X2 + ... + X25) / 25

Because the population is infinite, each draw is independent of the others (there's no "using up" of the population as you sample, unlike sampling without replacement from a small finite population). For independent random variables, the variance of a sum is the sum of the variances. So:

Var(X1 + X2 + ... + X25) = Var(X1) + Var(X2) + ... + Var(X25) = 25σ²

Now use the rule Var(aY) = a²Var(Y) with a = 1/25:

Var(X̅) = Var[(1/25)(X1 + ... + X25)] = (1/25)² × 25σ² = (1/625) × 25σ² = σ²/25

So in general, for a sample of size n from an infinite population:

Var(X̅) = σ² / n

Step 3: Convert Variance to Standard Error

Standard error is the square root of this variance (just like standard deviation is the square root of variance for any random variable):

SE(X̅) = √(σ²/n) = σ/√n

Substituting n = 25:

SE(X̅) = σ/√25 = σ/5

Pro Tip: The single most common slip on this type of question is dividing σ² by n and stopping there — which hands you option (a), a trap built directly into the choices. Standard error is a standard

Step 4: Why "Infinite Population" Matters Here

If the population were finite of size N, and we sampled without replacement, the formula would pick up a finite population correction (FPC)factor:

Var(X̅) = (σ²/n) × [(N − n)/(N − 1)]

As N becomes very large (effectively infinite) relative to n, the correction factor (N − n)/(N − 1) approaches 1, and we're left with the simple formula σ²/n used above. The question explicitly says "infinite population" precisely so you know to skip the FPC term — it's a deliberate signal, not a throwaway phrase.

Final Answer

SE(X̅) = σ/√25 =σ/5, which is option (b).

This matches the official ISS 2016 Statistics Paper-2 answer key (Series A), which also lists (b) as the correct response for Question 47.

Why This Question Matters

The standard error of the mean is one of the most frequently reused building blocks in the entire ISS statistics syllabus — it shows up again inside confidence intervals, hypothesis tests for the mean, sample-size determination problems, and even in questions about efficiency comparisons between estimators. Getting comfortable deriving σ/√n from scratch, rather than just recalling it, means you won't freeze when a question dresses the same idea up differently (for instance, asking for the sample size needed to achieve a target standard error, which is this exact formula rearranged).

Frequently Asked Questions

What's the difference between standard deviation and standard error?

Standard deviation (σ) measures how spread out individual observations are around the population mean. Standard error measures how spread out a statistic, like the sample mean, is across repeated samples. Standard error is always smaller than σ for n > 1, because averaging reduces variability.

Why does increasing the sample size reduce the standard error?

Because SE = σ/√n, and √n grows as n grows, so dividing σ by a larger number gives a smaller result. Intuitively, averaging more observations smooths out random fluctuations, so the sample mean lands closer to the true population mean on average.

Do we ever need the finite population correction on the ISS exam?

Yes — whenever a question explicitly gives you a finite population size N and says sampling is done without replacement, you must include the factor (N − n)/(N − 1). If the population is called "infinite," or n is small compared to N, or sampling is with replacement, the plain σ²/n formula applies.

Is this formula only valid for a Normal population?

No. Var(X̅) = σ²/n holds for any population distribution with finite variance σ², as long as the observations are independent. What does depend on normality (or large-sample approximations via the Central Limit Theorem) is the shape of the sampling distribution of X̅, not the value of its standard error.

How is this related to confidence intervals?

A confidence interval for the population mean is typically built as X̅ ± (critical value) × SE(X̅). So the exact quantity computed in this question, σ/√n, is literally the building block multiplied by a z or t value to get the margin of error in those problems.

What if the population variance σ² is unknown?

In practice σ² is rarely known, so it's estimated using the sample variance s², and the standard error is estimated as s/√n. This "estimated standard error" is what leads into t-distribution-based inference when the sample size is small.

Have a doubt about any step in this derivation, or want to see how this idea extends to the finite population case? Drop a comment below — and if you found this walkthrough useful, do share it with a fellow ISS aspirant who's grinding through the same paper.