Assistant Provident Fund Commissioner (APFC) 2016 recruitment test, advertisement 170. Transcribed from the Commission's own booklet (series A, B-CRHE-O-PAPA); answer as per the official UPSC final answer key. This question is in Part B. Answer last reviewed 24 August 2026.

The mean and standard deviation of a set of 16 non-zero positive numbers in an observation are 26 and 3·5 respectively. The mean and standard deviation of another set of 24 non-zero positive numbers without changing the circumstances of both sets of observations, are 29 and 3, respectively. The mean and standard deviation of their combined set of observations will respectively be

(a)27·8 and 3·21
(b)26·2 and 3·32
(c)27·8 and 3·32
(d)26·2 and 3·21

Answer

Why

Correct — A, (a) 27·8 and 3·21.

DO THE MEAN FIRST — IT SETTLES HALF THE ITEM WITHOUT ANY WORK ON THE SPREAD. The combined mean of two groups is not the average of the two means; it is the average WEIGHTED BY HOW MANY OBSERVATIONS EACH GROUP CONTRIBUTES.

combined mean = (n1 x m1 + n2 x m2) / (n1 + n2) = (16 x 26 + 24 x 29) / 40 = (416 + 696) / 40 = 1112 / 40 = 27·8

Two of the four choices print 26·2 as the mean, and both are gone at this line. Only 27·8 survives, so the item reduces to a choice between 3·21 and 3·32 for the spread.

A SANITY CHECK BEFORE ANY ARITHMETIC ON THE SPREAD. The two groups have standard deviations of 3·5 and 3, and the larger group — 24 of the 40 observations — is the one with the SMALLER spread. Whatever combination is used, the result must sit nearer 3 than 3·5, that is below the halfway mark of 3·25. Of the two figures offered, 3·21 does and 3·32 does not.

NOW THE SPREAD ITSELF. Standard deviations are never averaged directly; their SQUARES, the variances, are what combine. Weighting the two variances by their group sizes:

s^2 = (n1 x s1^2 + n2 x s2^2) / (n1 + n2) = (16 x 3·5^2 + 24 x 3^2) / 40 = (16 x 12·25 + 24 x 9) / 40 = (196 + 216) / 40 = 412 / 40 = 10·3

s = square root of 10·3 = 3·2094, which is 3·21 to two decimal places.

That is the POOLED STANDARD DEVIATION of the two sets — the single measure of spread WITHIN the groups, obtained by treating both sets on the same footing, which is what the stem's clause 'without changing the circumstances of both sets of observations' points at. It is the figure the Commission's key takes, and it is the only one of the two remaining numbers that answers to a defined calculation on the data given: 3·32 is not produced by any standard combination of 26, 29, 3·5 and 3.

ONE REFINEMENT WORTH CARRYING, BECAUSE EXAMINERS SOMETIMES WANT IT. When the two group means differ, the observations of the combined set are spread not only about their own group means but also about the common mean of 27·8, and a full combined-variance formula adds that second term, using d1 = 26 - 27·8 = -1·8 and d2 = 29 - 27·8 = +1·2. Learn both expressions and read the stem to see which is being asked for; on the printed choices here, 3·21 is the figure in play.

Why the others are wrong

(b)26·2 and 3·32 — Both halves fail, and the mean fails in a way worth remembering. A weighted average of 26 and 29 must lie BETWEEN 26 and 29, and it must lie nearer the mean of the bigger group. Here the bigger group is the 24-strong set with mean 29, so the answer must be above the simple midpoint of 27·5 — which 27·8 is. To land on 26·2 the 16-strong set would have to carry about ninety-three per cent of the weight, roughly 37 of the 40 observations, and it carries 16 of them, that is forty per cent. So 26·2 cannot come from these two group sizes under any weighting. The spread figure fails too: 3·32 sits above the halfway mark between 3 and 3·5, which would require the SMALLER group, the one with the wider spread, to dominate. Reversing the two group sizes in the variance formula gives about 3·31, and that is the nearest thing to a route to this number.
(c)27·8 and 3·32 — The mean is right, so this is the option a candidate reaches after doing the easy half correctly and then guessing at the spread. The figure 3·32 answers to no standard combination of the numbers given. Weighting the two VARIANCES by group size gives 10·3 and a standard deviation of 3·21; weighting the two STANDARD DEVIATIONS directly, which is not a correct procedure but is the commonest shortcut, gives (16 x 3·5 + 24 x 3) / 40 = 3·20, which still rounds nowhere near 3·32. The one mis-step that lands close is swapping the two group sizes — putting 24 with the 3·5 group and 16 with the 3 group — which returns about 3·31. Since the bigger group is the one with the narrower spread, the combined figure has to be pulled towards 3, and 3·32 is pulled the wrong way.
(d)26·2 and 3·21 — The spread figure is the one the key takes, and the mean is wrong — which makes this the most dangerous option on the page, because it rewards the harder half of the calculation and punishes the easier one. It is where a candidate lands who computes the pooled standard deviation carefully and then either averages the two means as (26 + 29) / 2, which gives 27·5 and not 26·2, or attaches the weights to the wrong groups. Neither route actually reaches 26·2: as with option (b), that value would need the 16-strong set to carry about ninety-three per cent of the weight. Do the mean first, precisely because it is quick and because it eliminates half the option set on its own; an item that offers each combination of a right and a wrong half is built to punish an answer assembled out of order.

Concept

COMBINING TWO SETS OF OBSERVATIONS. Two quantities are asked for, and they behave quite differently.

THE MEAN COMBINES LINEARLY, WEIGHTED BY COUNT. If the first set has n1 observations with mean m1 and the second has n2 with mean m2, the total of all the values is n1 x m1 + n2 x m2, and the combined mean is that total divided by n1 + n2. The mean of the means is correct only when the two groups are the same size. This is the same idea as an average of marks over unequal classes or an average price over unequal quantities: the count is the weight.

THE SPREAD DOES NOT COMBINE LINEARLY. Standard deviations cannot be averaged; the VARIANCE, which is the square of the standard deviation, is the additive quantity, because variance is a mean of squared deviations and sums of squares are what add. So every combination question is worked in variances and square-rooted only at the end.

POOLED VARIANCE, the spread within the groups: s^2 = (n1 x s1^2 + n2 x s2^2) / (n1 + n2)

COMBINED VARIANCE OF THE MERGED SET, which additionally accounts for the two group means sitting at different distances d1 and d2 from the common mean: s^2 = [ n1 x (s1^2 + d1^2) + n2 x (s2^2 + d2^2) ] / (n1 + n2)

The second expression reduces to the first when the two group means are equal, because then d1 and d2 are zero. Read the stem to see which is wanted, and be able to write both.

WHAT A STANDARD DEVIATION IS. Take each observation's deviation from the mean, square them so that positive and negative deviations do not cancel, average the squares — that is the variance — and take the square root so the answer is back in the original units. It is the most used measure of dispersion because it uses every observation and has properties that make it tractable in further work, unlike the range, which uses only two.

ABSOLUTE AND RELATIVE MEASURES. Range, quartile deviation, mean deviation and standard deviation are ABSOLUTE measures: they carry the units of the data, so a spread in rupees cannot be compared with a spread in kilograms. The COEFFICIENT OF VARIATION, which is 100 x standard deviation / mean, is a RELATIVE measure, a pure number, and it is what allows two series in different units or of very different size to be compared for consistency. The series with the smaller coefficient of variation is the more consistent.

Quantitative aptitude, data handling and statistics together form the largest strand on this APFC paper — roughly a third of Part B — and the descriptive-statistics corner of it recurs across EPFO papers because an Assistant Provident Fund Commissioner reads tabulated returns for a living.

The way to save time on an item like this one is to notice that it asks for TWO numbers and that the options pair them in every combination. That structure is an invitation to work in stages: compute the cheaper quantity, strike out every option that disagrees with it, and only then attack the expensive one. Here the mean takes one line and removes half the page.

The second habit is to know which quantities add and which do not. Counts add. Totals add. Sums of squares add. Means and standard deviations do NOT add, and averaging them directly is the single most common error in this topic. Anyone who remembers only 'work in variances, square-root at the end' will get most of these items out.

The third is the sanity check. Before computing, ask where the answer must lie. A weighted mean must fall between the two means and be pulled towards the bigger group; a combined measure of spread must fall between the two spreads and be pulled the same way. Two of the four options here can be rejected on that reasoning alone, without arithmetic, and on a paper with a large quantitative section that kind of elimination is worth more than speed at calculation.

One feature of the printing is worth naming so the numbers are not misread: throughout this booklet the decimal point is set as a RAISED MIDDLE DOT, so 3·5 means three point five.

Key facts

Combined mean of two groups = (n1 x m1 + n2 x m2) / (n1 + n2) — weighted by the number of observations, not the simple average of the two means.
Here: (16 x 26 + 24 x 29) / 40 = 1112 / 40 = 27·8.
A weighted mean always lies between the two group means and is pulled towards the group with more observations.
Standard deviations are never averaged. Variances — the squares — are what combine, and the square root is taken only at the end.
Pooled variance = (n1 x s1^2 + n2 x s2^2) / (n1 + n2); here (16 x 12·25 + 24 x 9) / 40 = 10·3, and its square root is 3·21.
The full combined variance of a merged set adds a term for the distance of each group mean from the common mean: [n1(s1^2 + d1^2) + n2(s2^2 + d2^2)] / (n1 + n2).
That extra term vanishes when the two group means are equal, which is when the two formulas agree.
Range, quartile deviation, mean deviation and standard deviation are ABSOLUTE measures of dispersion and carry the units of the data.
The coefficient of variation, 100 x standard deviation / mean, is a RELATIVE measure and is used to compare the consistency of series in different units.
Variance and standard deviation are unchanged if every observation is shifted by a constant, and are scaled by that constant if every observation is multiplied by one.

Study next

Weighted averages in every form they are asked — average marks over unequal classes, average speed over unequal distances, average price over unequal quantities.The four measures of dispersion — range, quartile deviation, mean deviation and standard deviation — and what each uses and ignores.Coefficient of variation, and why comparing consistency requires a relative rather than an absolute measure.Mean, median and mode: how each is computed and which is least disturbed by an extreme value.Effect on the mean and the standard deviation of adding a constant to every observation or multiplying every observation by one.Grouped-data calculations — class marks, cumulative frequency and the step-deviation method for a mean.Skewness and the empirical relation between mean, median and mode, which is where descriptive statistics usually ends on these papers.

Common traps

Averaging the two means when the groups are of different sizes. Only equal group sizes make that correct.
Averaging the two standard deviations. Variances combine; standard deviations do not.
Attaching the weights to the wrong groups, which here would move the mean from 27·8 to 27·2.
Forgetting to take the square root at the end, and reporting the variance as though it were the standard deviation.
Answering the half of the item you did first without checking the other half — the option set pairs a right mean with a wrong spread and the reverse.
Comparing the spreads of two series in different units using an absolute measure, where the coefficient of variation is what is needed.
Misreading the raised middle dot in this booklet's decimals as a multiplication sign or a full stop in a list.

Descriptive statistics on EPFO papers is asked either as a definition — which measure is absolute, which is relative, which is unaffected by extreme values — or as a short two-group computation of exactly this shape. The computational version almost always asks for a mean, or a mean and a spread together, and the option set pairs the correct value of one with the incorrect value of the other, so a candidate who does half the work carefully still has a one-in-two chance of losing the item. The reliable method is fixed: identify what is being combined, do the mean by weighting on counts, eliminate, then work the spread in variances and square-root at the end. Rounding is usually to two decimal places, and the choices are close enough that the arithmetic has to be carried through rather than estimated.

Related PYQs

EPFO_APFC_2016_Q61

There are 20 girls and 30 boys in a class, and their respective average marks are found to be 55 and 58. The average marks of the entire class are

(a) 56·5
(b) 56·6
(c) 56·7
(d) 56·8

Answer(d) 56·8

Twenty girls and thirty boys with different average marks, asking for the class average — the weighted-mean half of this item asked on its own, and the same trap of averaging the two means.

EPFO_APFC_2016_Q107

Which of the following is not an absolute measure of dispersion ?

(a) Range
(b) Mean Deviation
(c) Quartile Deviation
(d) Coefficient of Variation

Answer(d) Coefficient of Variation

Which measure of dispersion is not an absolute one — the definitional companion to this computation, turning on the difference between an absolute measure and the coefficient of variation.

EPFO_APFC_2023_Q37

P, Q, R, S and T are five friends. The mean of weights of P, Q, R and S is 50 kg, whereas the mean of weights of Q, R, S and T is also 50 kg. Which of the following statements is/are correct? 1. The weight of T is 50 kg. 2. The weights of P and T are equal. Select the correct answer using the code given below.

(a) 1 only
(b) 2 only
(c) Both 1 and 2
(d) Neither 1 nor 2

Answer(b) 2 only

Two overlapping groups of friends with equal means, asking what follows about the members — combining group means asked as reasoning rather than arithmetic.

Practice

practice — not a real PYQ
The mean of 30 observations is 20 and the mean of another 20 observations is 25. What is the mean of all 50 observations taken together ?
- (a)21
- (b)22
- (c)22·5
- (d)23
Answer(b) 22 — the combined mean is (30 x 20 + 20 x 25) / 50 = (600 + 500) / 50 = 22. The simple average of 20 and 25 is 22·5, which is offered as option (c) and is what a candidate gets by ignoring the unequal group sizes.
practice — not a real PYQ
A series has a mean of 50 and a standard deviation of 5. Its coefficient of variation is
- (a)5%
- (b)10%
- (c)20%
- (d)25%
Answer(b) 10% — the coefficient of variation is 100 x standard deviation / mean = 100 x 5 / 50 = 10 per cent. It is a pure number, which is exactly why it can compare the consistency of two series measured in different units.

Why