Q.The marks scored by 5 first-year Commerce students in an Economics unit test (out of 100) are: 45, 50, 55, 60, 70. Calculate the arithmetic mean by the direct method, and verify your answer using the short-cut (assumed mean) method.

The arithmetic mean of an individual series is simply the sum of all values divided by how many there are.

Step 1 — Direct method.

Step 2 — Verify by the short-cut (assumed mean) method. Take $A=55$ (one of the given values, for convenience) and find the deviation $d=X-A$ of every item:

Both methods agree exactly.

Independent cross-check using a different assumed mean. Taking $A=50$ instead: deviations are $-5,0,5,10,20$, so $\Sigma d = 30$ and $\bar X = 50 + 30/5 = 50+6=56$ — the same answer regardless of which value is chosen as the assumed mean, confirming the arithmetic.

Forgetting to divide $\Sigma d$ by $N$ before adding it back to $A$ (i.e., writing $\bar X = A + \Sigma d$ instead of $A + \Sigma d/N$) is the single most common slip in the short-cut method; also, choosing an assumed mean far from the data makes the deviations larger and more error-prone to add, even though the final answer is unaffected by which value is chosen.