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If both the mean and the standard deviation of 50 observations

$x_1,x_2,x_3,.................,x_{50}$ are equal to 16 , then the mean

of $(x_1-4)^{2},(x_2-4)^{2},............,(x_{50}-4)^{2}$ is :

Option 1)
400

Option 2)
380

Option 3)
525

Option 4)
480

Answers (1)

V Vakul

50 observations

$x_1,x_2,x_3,.................,x_{50}$

$mean,\bar{x}=\frac{\sum x_i}{50}=16$ .....................(1)

$variance,\sigma ^{2}=\frac{\sum x_i^{2}}{50}-(\bar{x})^{2}=16^{2}$

$=>\frac{\sum x_i^{2}}{50}=16^{2}+(\bar{x})^{2}=16^{2}+(16)^{2}=512$ ......................(2)

So, mean value of $(x_1-4)^{2},(x_2-4)^{2},.........................,(x_{50}-4)^{2}$

$\\=> \sum\frac{(x_i-4)^{2}}{50}=\frac{\sum x_i^2-8\sum x_i+16\times 50}{50}\\\frac{\sum x_i^2}{50}-\frac{8\sum x_i}{50}+16\\512-8\times16+16\\=400$

Option 1)

400

Option 2)

380

Option 3)

525

Option 4)

480

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