Question

If both mean and variance of 50 observations $x_1, x_2, \ldots, x_{50}$ are equal to 16 and 256 respectively, then mean of $(x_1-5)^2, (x_2-5)^2, \ldots, (x_{50}-5)^2$ is

• A. 357
• B. 367
• C. 377
• D. 387

Hint:

Remember the formulas for mean ($\bar{x} = \frac{\sum x_i}{n}$) and variance ($\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$). When calculating the mean of a transformed series, expand the terms algebraically and use the sums of the original series.

Answer 1

The Correct Option is C

Step 1: Use the given mean to find the sum of observations. Given that the mean of 50 observations $x_1, x_2, \ldots, x_{50}$ is 16. The formula for the mean is $\bar{x} = \frac{\sum x_i}{n}$. Here, $\bar{x} = 16$ and $n = 50$. \[ 16 = \frac{\sum_{i=1}^{50} x_i}{50} \] Multiply both sides by 50: \[ \sum_{i=1}^{50} x_i = 16 \times 50 = 800 \]
Step 2: Use the given variance to find the sum of squares of observations. Given that the variance of 50 observations is 256. The formula for variance is $\sigma^2 = \frac{\sum x_i^2}{n} - (\bar{x})^2$. Here, $\sigma^2 = 256$, $n = 50$, and $\bar{x} = 16$. \[ 256 = \frac{\sum_{i=1}^{50} x_i^2}{50} - (16)^2 \] \[ 256 = \frac{\sum_{i=1}^{50} x_i^2}{50} - 256 \] Add 256 to both sides: \[ 256 + 256 = \frac{\sum_{i=1}^{50} x_i^2}{50} \] \[ 512 = \frac{\sum_{i=1}^{50} x_i^2}{50} \] Multiply both sides by 50: \[ \sum_{i=1}^{50} x_i^2 = 512 \times 50 = 25600 \]
Step 3: Calculate the mean of the new set of observations $(x_i - 5)^2$. The new observations are $(x_1-5)^2, (x_2-5)^2, \ldots, (x_{50}-5)^2$. The mean of these new observations is $\frac{\sum_{i=1}^{50} (x_i - 5)^2}{n}$. Expand the term $(x_i - 5)^2$: $(x_i - 5)^2 = x_i^2 - 10x_i + 25$. So, the sum of the new observations is: \[ \sum_{i=1}^{50} (x_i - 5)^2 = \sum_{i=1}^{50} (x_i^2 - 10x_i + 25) \] Using the linearity of summation: \[ = \sum_{i=1}^{50} x_i^2 - 10 \sum_{i=1}^{50} x_i + \sum_{i=1}^{50} 25 \] \[ = \sum_{i=1}^{50} x_i^2 - 10 \sum_{i=1}^{50} x_i + 50 \times 25 \]
Step 4: Substitute the values found in Step 1 and Step 2. We have $\sum x_i = 800$ and $\sum x_i^2 = 25600$. \[ \sum_{i=1}^{50} (x_i - 5)^2 = 25600 - 10(800) + 1250 \] \[ = 25600 - 8000 + 1250 \] \[ = 17600 + 1250 \] \[ = 18850 \]
Step 5: Compute the mean of the new observations. The mean of the new observations is: \[ \text{New Mean} = \frac{18850}{50} \] \[ = 377 \]
Step 6: State the final answer. The mean of $(x_1-5)^2, (x_2-5)^2, \ldots, (x_{50}-5)^2$ is 377.