What we see: a yes or no Data Sufficiency question about spread. We do not need the exact standard deviation; we only need to know for sure whether it is more than 2.
Step 1. Remember what standard deviation measures.
What we see: \(S\) is the standard deviation of 5 items.
What it means: it measures how far the items sit from their average. Items close together give a small \(S\); items spread out give a big \(S\); 5 equal items give \(S = 0\).
What we do and why: for a yes or no question, a statement is enough when every set it allows gives the same answer. So we test the tightest set and the widest set each statement allows.
Step 2. Statement (1): five different even numbers between 1 and 19.
What we see: the items come from 2, 4, 6, ..., 18, and no two are the same.
What it means: the numbers cannot sit closer than 2 apart. The tightest possible set is five even numbers in a row, such as 2, 4, 6, 8, 10.
What we do and why: find \(S\) for that tightest set. The average is 6. The distances from 6 are 4, 2, 0, 2, 4. Square them: 16, 4, 0, 4, 16, which add to 40. Divide by 5: 8. So \(S = \sqrt{8} \approx 2.83\), which is more than 2. Every other allowed set is more spread out, so its \(S\) is even bigger. The answer is always yes. Sufficient.
Step 3. Statement (2): the smallest number is 10.
What we see: only the smallest item is fixed.
What it means: the set could be 10, 10, 10, 10, 10, with \(S = 0\) (no). It could be 10, 20, 30, 40, 50, with a large \(S\) (yes).
What we do and why: two different answers, so it is not sufficient.
Step 4. Decide.
Statement (1) alone gives a definite yes. Statement (2) alone does not.
The trap: E or C, from thinking you need the actual numbers. A yes or no question only needs a sure answer, and the tightest set under statement (1) already clears 2.
Faster way: five even numbers in a row are just 1, 2, 3, 4, 5 doubled. The standard deviation of 1 to 5 is \(\sqrt{2} \approx 1.41\), and doubling every number doubles it, to about 2.83.
Takeaway: when a statement limits how close the numbers can be, test the tightest set it allows. If even that set passes, every set passes.
Answer: A
