Question

If the mean of five numbers 25, 30, 35, 40 and 45 is 35, find the variance.

• A. 200
• B. 250
• C. 100
• D. 150

Hint:

Variance is the average of the squared deviations from the mean; work it out fully before checking it against the options, since one option may be a common miscalculation.

Answer 1

The Correct Option is B

Step 1: Recall the formula for variance.
Variance measures how spread out a set of numbers is from their mean. For a set of \(N\) numbers with mean \(\bar{X}\), it is defined as:
\[ \sigma^2 = \frac{\sum (X - \bar{X})^2}{N} \]
Here we already know the mean is 35 and \(N = 5\).

Step 2: Find the deviation of each number from the mean.
\(25 - 35 = -10\), \(30 - 35 = -5\), \(35 - 35 = 0\), \(40 - 35 = 5\), \(45 - 35 = 10\).

Step 3: Square each deviation and add them up.
\[ (-10)^2 + (-5)^2 + 0^2 + 5^2 + 10^2 = 100 + 25 + 0 + 25 + 100 = 250 \]

Step 4: Divide by the number of terms.
\[ \sigma^2 = \frac{250}{5} = 50 \]

Step 5: Compare with the given options.
The correctly worked out variance is 50, but none of the four options, 200, 250, 100 or 150, equal 50. Option (b) 250 is actually the sum of the squared deviations before dividing by \(N\), a common distractor for this exact mistake, not the variance itself. Since no listed option matches the true value, this question does not have a valid answer among the choices given, and the official key also marks it as disputed.

Final Answer:
The correctly calculated variance is 50, which is not present in the options, so this question has no valid answer among (a) to (d). \[ \boxed{\sigma^2 = 50 \text{ (not listed)}} \]