Out of a total of n observations, the mean (average) of m observations is n and the mean of the remaining (n − m) observations is m. What is the mean of all n observations in terms of m and n?

Introduction / Context:
This statistics question tests understanding of how means (averages) combine when we have subgroups within a data set. We are given the mean of two disjoint groups of observations and asked to find the overall mean of all observations. Instead of plugging in concrete numbers, we work algebraically with n and m. This type of question is common in aptitude exams and helps build intuition about weighted averages.

Given Data / Assumptions:

• There are n total observations.
• m of these observations have mean n.
• The remaining n − m observations have mean m.
• All observations are real numbers, and n and m are positive integers with n ≥ m.
• We need to find the overall mean of all n observations.

Concept / Approach:
The overall mean of all observations is found by dividing the total sum of all observations by the total number of observations. When we know the means of subgroups, we can reconstruct the sums for each subgroup by multiplying each mean by the number of observations in that group. Then we add these sums to get the total sum, and finally divide by n to get the combined mean. This is a direct application of the definition of mean and the idea of weighted averages.

Step-by-Step Solution:
Step 1: For the first group of m observations with mean n, the total sum of these observations is S1 = m × n.Step 2: For the second group, there are n − m observations with mean m, so the total sum is S2 = (n − m) × m.Step 3: Compute S2 explicitly: S2 = mn − m^2.Step 4: The total sum of all n observations is S = S1 + S2 = mn + (mn − m^2) = 2mn − m^2.Step 5: The overall mean of all n observations is S divided by the total number of observations n.Step 6: Therefore overall mean = (2mn − m^2)/n.Step 7: Simplify the expression by dividing numerator and denominator: overall mean = 2m − m^2/n.

Verification / Alternative check:
Take a simple numeric example to verify the formula. Let n = 10 and m = 4, so there are 10 observations in total. The first 4 observations have mean 10, and the remaining 6 have mean 4. Then S1 = 4 × 10 = 40 and S2 = 6 × 4 = 24, giving total sum 64. The overall mean is 64/10 = 6.4. Now apply the derived expression: 2m − m^2/n = 2 × 4 − 16/10 = 8 − 1.6 = 6.4, which matches the direct computation.

Why Other Options Are Wrong:
The expression 2m − m/n ignores the squared term and does not correctly represent the weighting of the second group, so it gives wrong values in general. The option 2m is only correct in special contrived cases and fails for most choices of n and m. The option 2m − m simplifies to m, which clearly cannot represent the combined mean in general. The option m^2/n is far too small and does not involve the combined effect of both subgroup means. Only 2m − m^2/n matches the correct derivation from the definition of mean.

Common Pitfalls:
Many learners mistakenly average the two given means n and m directly, for example by computing (n + m)/2, which is incorrect because the groups may have different sizes. Others forget to multiply each mean by the corresponding number of observations, leading to an incorrect total sum. Remember that means from subgroups must be weighted by their group sizes when combining them.

Final Answer:
The mean of all n observations is 2m − m^2/n.