ISS 2016 Statistics Paper-1 Solution: Question 26 (Nonparametric Tests for Skewed Populations)
Continuing our question-by-question walkthrough of the ISS Statistics Paper-1 previous year papers, today we pick up right where the last post left off. This time the topic shifts away from probability and into a corner of the syllabus that trips up a lot of first-time aspirants: choosing the right hypothesis test when the textbook assumptions don't quite hold.
Quick Summary
Topic: Nonparametric (distribution-free) tests — Run test and Sign test
Question Reference: ISS 2016, Statistics Paper-1, Question 26
Correct Answer: (b) 3 and 4 only
The Question
Which of the following tests should be employed for deciding whether or not two samples come from the same skewed parent population?
Paired t-test
Unpaired t-test
Run test
Sign test
Which of the above statements are correct?
(a) 1 and 3 only
(b) 3 and 4 only
(c) 2, 3 and 4 only
(d) 1, 3 and 4 only
Step 1: Read the Question Slowly — the Word "Skewed" Is the Whole Question
Before touching any test, notice exactly what is being asked. We are not just testing whether two samples come from the same population in general — we are told upfront that the parent population is skewed, i.e. not symmetric, not bell-shaped, not normal. That single adjective is doing all the work in this question. It tells us we must reject any test that secretly assumes normality, no matter how familiar or comfortable that test feels.
So our job has two parts: (1) list every test that is valid regardless of the shape of the population, and (2) eliminate every test that silently leans on normality.
Step 2: Why the Paired t-test Is Ruled Out
The paired t-test is used when you have two measurements on the same subject (say, before-and-after readings) and you want to test whether the mean difference is zero. The test statistic is built as
t = (d̄ − 0) / (sd / √n)
where d̄ is the mean of the paired differences and sd is their standard deviation. This statistic follows a t-distribution with (n − 1) degrees of freedom only when the differences themselves are normally distributed. That is the entire derivation — Student built the t-distribution starting from the assumption that the underlying observations are drawn from a normal population. If the population is skewed, the differences won't be normal either (except by some coincidence), so the t-distribution no longer describes the sampling behaviour of our statistic correctly, especially in the small-sample ISS-exam sense where we can't lean on the Central Limit Theorem to bail us out. So Statement 1 is eliminated.
Step 3: Why the Unpaired (Independent Samples) t-test Is Ruled Out Too
The unpaired t-test compares the means of two independent samples using
t = (X̄1 − X̄2) / (sp √(1/n1 + 1/n2))
Exactly the same problem applies here: this statistic is derived under the assumption that both populations are normal (and usually that they share a common variance, for the pooled estimate sp to make sense). A skewed parent population violates the starting assumption directly, so the exact t-distribution result no longer holds. Statement 2 is eliminated as well.
Pro Tip: Whenever an MCQ explicitly tells you the population is "skewed," "non-normal," or gives you a distribution like exponential or a heavily asymmetric histogram, treat that as a giant red flag against every test with "t", "F", or "Z" in its name unless the sample size is described as large. The way this distinction is usually drilled into students is by pairing every parametric test with the one assumption it quietly depends on — do that pairing exercise for t-test, F-test and Z-test until it's automatic, and questions like this one stop being tricky and start being free marks.
Step 4: Why the Run Test Works
The Run test (formally, the Wald–Wolfowitz runs test) is built for exactly this situation: you pool the two samples together, arrange all the observations in increasing order, and label each value by which original sample it came from — something like AABAABBBA. A "run" is a maximal sequence of the same label in a row. If the two samples really come from the same population, the labels should be well mixed, giving neither too few runs (heavy clustering) nor too many runs (an unnatural, overly-alternating pattern). The test statistic is simply the number of runs, and its sampling distribution is worked out purely from combinatorics — counting how many ways the labels could arrange themselves — with no appeal to normality, skewness, or any specific shape at all. That is precisely what makes it "distribution-free": it works whether the parent population is normal, skewed, bimodal, or anything else. Statement 3 is correct.
Step 5: Why the Sign Test Works
The Sign test looks at paired observations and asks only one question of each pair: is the difference positive or negative? It throws away the actual magnitude of the difference and keeps only its sign. Under the null hypothesis that the two samples come from the same population, a positive and a negative sign should each be equally likely, so the number of positive signs simply follows a Binomial(n, 0.5) distribution — a result that depends only on counting plus and minus signs, never on the shape of the underlying population. Because it never uses the actual values, skewness in the parent population is completely irrelevant to its validity. Statement 4 is correct.
Step 6: Putting It Together
Only the Run test (3) and the Sign test (4) remain valid once we insist on a skewed, non-normal parent population — both are nonparametric procedures whose derivations never assume a particular distributional shape. The Paired t-test (1) and Unpaired t-test (2) both silently assume normality and must be eliminated.
That gives us option (b) 3 and 4 only, which also matches the official answer key exactly, so there is no discrepancy to flag here.
Why This Question Matters
ISS Paper-1 loves to test whether a candidate actually understands why a formula works, not just the formula itself. A huge number of marks across the paper hinge on correctly matching a testing situation to its assumptions — normal vs. skewed, paired vs. independent, large-sample vs. small-sample. Getting comfortable with the assumption checklist behind t-tests, and knowing your standard nonparametric toolkit (Run test, Sign test, Median test, Mann-Whitney U test, Wilcoxon signed-rank test) as a reflex rather than something you have to derive under exam pressure, pays off across dozens of similar questions in every year's paper.
Frequently Asked Questions
Why can't we just use a t-test if the sample size is large enough?
With a sufficiently large sample, the Central Limit Theorem does let the sample mean approach normality even from a skewed population, which makes the t-test approximately valid. However, ISS MCQs of this type are testing the exact, small-sample assumption behind the t-test, and the question gives no indication of a large sample, so we treat the t-test as invalid here.
What exactly is a "run" in the Run test?
A run is an unbroken sequence of identical labels once you merge and order both samples together. For example, in the sequence AAABBAAB there are 4 runs: AAA, BB, AA, B. Too few runs suggests the two samples are clustered apart (different populations); too many suggests unnatural over-alternation.
Is the Sign test the same as the Wilcoxon signed-rank test?
No. Both are nonparametric tests for paired data, but the Sign test only uses the direction (sign) of each difference, while the Wilcoxon signed-rank test also uses the relative magnitude of the differences by ranking them. The signed-rank test is more powerful when the underlying distribution is roughly symmetric, but the plain Sign test makes no such assumption at all.
Does "skewed" always mean we must reject t-tests in every ISS question?
Not automatically — always check whether the question also mentions a large sample size, in which case approximate normality via the Central Limit Theorem may still justify a t-test or Z-test. When no sample size is given, or the sample is explicitly small, treat skewness as disqualifying for parametric tests based on the normal or t-distribution.
Why does the Sign test only look at signs and not actual values?
That is exactly what makes it distribution-free. By discarding magnitude and keeping only whether a difference is positive or negative, the test statistic becomes a simple count that follows a Binomial(n, 0.5) distribution under the null hypothesis, regardless of what the parent population actually looks like.
Are there other nonparametric alternatives to the unpaired t-test worth knowing for ISS?
Yes — the Mann-Whitney U test (also called the Wilcoxon rank-sum test) is the most common nonparametric substitute for the unpaired/independent-samples t-test, and it appears frequently across ISS papers alongside the Run test and Sign test, so it is worth adding to the same mental checklist.
If any step above felt shaky, drop a comment with exactly where you got stuck — working through the doubt out loud is often the fastest way to make it stick. And if you know a fellow ISS aspirant who's grinding through the same previous-year papers, sharing this one with them costs you nothing and might save them the confusion this question causes most first-time solvers.
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