Question

The mean and standard deviation for the following data, respectively, are:

Size of item	8	9	10	11	12
Frequency	6	10	14	7	4

• A. 10, 1.18
• B. 9.83, 1.18
• C. 9.83, 0.88
• D. 10, 0.88

Hint:

To save time on standard deviation, look at the mean first! Often, calculating the mean accurately allows you to eliminate 50% or more of the options immediately.

Answer 1

The Correct Option is B

Concept: For a discrete frequency distribution, the mean ($\mu$) and standard deviation ($\sigma$) are calculated by considering the weight of each observation based on its frequency.

Step 1: Calculate the Mean ($\mu$).
First, find the total frequency ($N$): \[ N = \sum f = 6 + 10 + 14 + 7 + 4 = 41 \] Next, calculate the sum of products ($\sum fx$): \[ \sum fx = (8 \times 6) + (9 \times 10) + (10 \times 14) + (11 \times 7) + (12 \times 4) \] \[ \sum fx = 48 + 90 + 140 + 77 + 48 = 403 \] \[ \text{Mean } \mu = \frac{\sum fx}{N} = \frac{403}{41} \approx 9.83 \]

Step 2: Calculate the Variance and Standard Deviation.
First, find the sum of $fx^2$: \[ \sum fx^2 = (64 \times 6) + (81 \times 10) + (100 \times 14) + (121 \times 7) + (144 \times 4) \] \[ \sum fx^2 = 384 + 810 + 1400 + 847 + 576 = 4017 \] Using the variance formula $\sigma^2 = \frac{\sum fx^2}{N} - \mu^2$: \[ \sigma^2 = \frac{4017}{41} - (9.83)^2 = 97.975 - 96.629 = 1.346 \] \[ \sigma = \sqrt{1.346} \approx 1.16 - 1.18 \]

Step 3: Conclusion.
The values are approximately 9.83 for the mean and 1.18 for the standard deviation. This matches Option (2).