ISS 2016 Statistics Paper-2 Solution: Question 4 (F-Ratio in One-Way ANOVA)
Continuing our question-by-question walk through the ISS Statistics Paper-2 previous year papers, today we pick up right where the last post left off and tackle Question 4 from the 2016 paper. This one sits comfortably in one of the most exam-friendly corners of the syllabus — Analysis of Variance — and once you see the pattern, you'll be able to solve any similar question in under a minute.
Quick Summary
Topic: One-Way Analysis of Variance (ANOVA) — computing the F-ratio
Question Reference: ISS 2016, Statistics Paper-2, Question 4
Correct Answer: Option (b) 1.48
The Question As It Appeared In The Paper
The following data pertain to a test involving analysis of variance:
Estimate of population variance based on variance among the sample means = 18.50 Estimate of the population variance based on the variance within the samples = 12.50
The F-ratio for the sample is equal to:
(a) 0.68
(b) 1.48
(c) 6.00
(d) 31.00
Step 1: Understand What The Two Given Numbers Actually Mean
Before touching any formula, it helps to slow down and translate the words into symbols, because that's really where this question is testing you.
In one-way ANOVA, we split every observation's total variation into two separate sources:
Variation among (or "between") the sample means — this measures how spread out the different group averages are from each other. If the groups genuinely differ, this number tends to be large.
Variation within the samples — this measures the natural, everyday scatter of individual observations around their own group's mean. This is basically random noise or error, and it exists no matter what.
Statisticians call these two quantities Mean Square Between (MSB, sometimes written MST for "treatment") and Mean Square Within (MSW, sometimes written MSE for "error"). Both of them are, individually, unbiased estimates of the same underlying population variance σ² — but only when the null hypothesis (that all the group means are truly equal) is actually true.
So from the question:
MSB (variance among the sample means) = 18.50
MSW (variance within the samples) = 12.50
Step 2: Recall The Definition Of The F-Ratio
The F-statistic in ANOVA is defined simply as:
F = MSB / MSW = (Mean Square Between groups) / (Mean Square Within groups)
The logic behind this ratio is worth internalising, not just memorising. If the group means really are all equal, MSB and MSW are both just independently estimating the same σ², so their ratio should hover close to 1. But if the group means are genuinely different, the between-groups variation gets inflated by that real difference, while the within-groups variation stays as pure error — so the ratio climbs above 1. The bigger F gets, the stronger the evidence against "all means are equal."
Notice something important here: the question gives us the two variance estimates directly, so we don't even need the raw data, the sums of squares, or the degrees of freedom to answer it. We only need those extra pieces if we were asked to test significance against a critical F-value from tables — here we're only asked for the computed ratio itself.
Step 3: Substitute The Values
F = 18.50 / 12.50
Let's do this division by hand, the long way, so there's no ambiguity:
First, clear the decimals by multiplying both numerator and denominator by 100: 18.50 × 100 = 1850, and 12.50 × 100 = 1250. So F = 1850 / 1250.
Both numbers are divisible by 10: 1850 / 10 = 185, and 1250 / 10 = 125. So F = 185 / 125.
Both are divisible by 5: 185 / 5 = 37, and 125 / 5 = 25. So F = 37 / 25.
Now divide 37 by 25 directly: 25 goes into 37 once, leaving a remainder of 12. Bring down a decimal zero: 120 divided by 25 is 4, with remainder 20 (since 25 × 4 = 100). Bring down another zero: 200 divided by 25 is exactly 8.
Putting the digits together: 37 / 25 = 1.48.
So F = 1.48, exactly matching option (b) 1.48.
Pro Tip: the single easiest mistake on a question like this is flipping the ratio — putting the within-groups variance on top and the between-groups variance on the bottom. If you did that here, you'd get 12.50 / 18.50 = 0.6757, which rounds to roughly 0.68 — and notice that's sitting right there as option (a), placed deliberately to catch exactly that slip. The way this is usually taught in a good ANOVA class is to fix the rule as a small phrase you say to yourself every time: "between on top, within on the bottom, because we're testing whether the groups differ." Say it, don't just remember the letters MSB and MSW — under exam pressure, letters get swapped far more easily than a sentence does.
Step 4: Cross-Check Against The Official Answer Key
The official ISS 2016 Statistics Paper-2 answer key (Series A) lists the answer to Question 4 as option (b), which is 1.48. This matches our independently derived value exactly, so there's no discrepancy to flag here — our step-by-step computation and the official key are in full agreement.
Why This Question Matters
This question looks tiny, but it's really testing whether you understand the architecture of ANOVA rather than just a formula. Once you're comfortable identifying which number is "between" and which is "within," you can handle every variant ISS throws at you — one-way ANOVA, two-way ANOVA, and even the F-test used to compare two independent sample variances, all of which lean on this same numerator-over-denominator logic. Students who drill this kind of simple substitution until it becomes automatic tend to move faster and make fewer careless errors on the harder, multi-step ANOVA table questions later in the paper, where you have to compute MSB and MSW yourself from raw sums of squares before even getting to this final division.
Frequently Asked Questions
What exactly is the F-ratio used for in ANOVA?
The F-ratio is a test statistic used to check whether the means of several groups are all equal or whether at least one group differs. It compares the variation explained by differences between group means against the variation that's just natural scatter within each group. A large F-ratio suggests the groups are not all the same.
Why is "variance between samples" always placed in the numerator?
Because that's the quantity we're actually testing — we want to know if it's unusually large compared to the baseline noise level given by the within-groups variance. Putting it in the numerator means the F-ratio grows as evidence against equal means grows, which is the direction we care about.
Do we always need the degrees of freedom to compute the F-ratio?
Not to compute the ratio itself if the two mean squares (variance estimates) are already given to you, as in this question. Degrees of freedom become essential only when you want to compare your computed F-value against a critical value from the F-distribution table to decide whether to reject the null hypothesis at a given significance level.
Can the F-ratio ever be less than 1?
Yes, absolutely — it happens whenever the within-groups variance estimate turns out larger than the between-groups one, which is common when the sample group means are genuinely close together. A value below 1 gives you no reason to reject the assumption that all population means are equal.
Is this F-ratio the same F-statistic used in regression analysis?
It's the same underlying F-distribution and the same core idea of comparing two independent variance estimates, but in regression the numerator is usually the "explained" (regression) mean square and the denominator is the residual mean square. The ANOVA table format is nearly identical in both cases — that overlap is exactly why ISS Statistics Paper-2 tests both angles.
What's the most common silly mistake students make on questions like this?
Besides inverting the ratio (covered in the Pro Tip above), the other frequent slip is forgetting to double-check whether the question is asking for the raw sums of squares or the already-divided mean squares (variance estimates). Here, both quantities given were already variance estimates, so no further division by degrees of freedom was needed before taking the ratio.
That wraps up Question 4 of the ISS 2016 Statistics Paper-2 series. If any step felt unclear or you spotted something worth discussing, drop a comment below — and if this helped, do share it with a fellow ISS aspirant who's grinding through the same papers.
When F statistic is less than 1, is it always because the sample means are genuinely close together? Can't it be because the individual observations within a group is highly spread out increasing the variation in the denominator?