✅ Correct Answer: σ = √([1/12] (n² – 1))

📝 Note: We can derive the answer but we do not have that much time in the exam. We will use a shortcut method here. We will try to find out the standard deviation for the first 5 numbers and check which formula from the option fits the answer.

📘 Explanation:

Step 1: Find the Average

To start, we need to find the “average” of the numbers. For the first n numbers, we can find the average by adding up all the numbers and dividing by how many numbers there are.

For example, for the first 5 numbers: 1, 2, 3, 4, and 5:
Average = [1+2+3+4+5] / 5 = 15 / 5 = 3

So, the average is 3.

Step 2: Find How Far Each Number Is from the Average

Next, we look at how far each number is from the average. This is called the “difference.”

For example, for the first 5 numbers:

• How far is 1 from 3? It’s 2 steps away.
• How far is 2 from 3? It’s 1 step away.
• How far is 3 from 3? It’s 0 steps away.
• How far is 4 from 3? It’s 1 step away.
• How far is 5 from 3? It’s 2 steps away.

Step 3: Find the “Squared” Differences

Now, to make sure we don’t get negative numbers when we look at differences, we “square” each difference. Squaring means multiplying the number by itself.

For example:

• The difference for 1 was 2, so 2×2=4.
• The difference for 2 was 1, so 1×1=1.
• The difference for 3 was 0, so 0×0=0.
• The difference for 4 was 1, so 1×1=1.
• The difference for 5 was 2, so 2×2=4.

Step 4: Find the Average of the Squared Differences

Now we find the average of all these squared differences. This helps us see, on average, how far the numbers are from the average.

For example, the squared differences are 4, 1, 0, 1, and 4. We add them up:
4 + 1 + 0 + 1 + 4 = 10

Then, we divide by how many numbers there are (5 numbers):
Average of squared differences = 10 / 5 ​= 2

Step 5: Take the Square Root

Finally, to get the standard deviation, we take the square root of this average. The square root is just the opposite of squaring a number. So, for our example:

2^0.5 ​≈1.41

This is the standard deviation for the first 5 numbers.

Put n = 5 in all the alternatives and you can verify that only the right answer gives the value of sqrt(2).

✅ Final Answer: σ = √([1/12] (n² – 1))