Population Standard Deviation Matching Explained

This question involves matching items from List - I with the correct values from List - II. List - I provides the population mean ($\mu$) and the mean of the squares ($\frac{1}{N}\sum{x_i^2}$), while List - II provides possible values for the population standard deviation ($\sigma$). We need to calculate $\sigma$ for each case in List - I and find its corresponding match in List - II.

Formula for Population Standard Deviation

The core concept here is understanding how to calculate the population standard deviation ($\sigma$) when given the population mean ($\mu$) and the sum of the squares of the observations divided by the number of observations ($\frac{1}{N}\sum{x_i^2}$).

The formula for population variance ($\sigma^2$) is:

$$ \sigma^2 = \left( \frac{\sum{x_i^2}}{N} \right) - \mu^2 $$

From this, the population standard deviation ($\sigma$) is the square root of the variance:

$$ \sigma = \sqrt{\sigma^2} = \sqrt{\left( \frac{\sum{x_i^2}}{N} \right) - \mu^2} $$

Step-by-Step Calculation and Matching

Let's calculate the population standard deviation ($\sigma$) for each pair of values in List - I:

Calculation for Item A:

Given values:

• Population Mean ($\mu$) = 9
• Mean of Squares ($\frac{1}{N}\sum{x_i^2}$) = 117

Calculate the population variance ($\sigma^2$):

$$ \sigma^2 = 117 - (9)^2 = 117 - 81 = 36 $$

Calculate the population standard deviation ($\sigma$):

$$ \sigma = \sqrt{36} = 6 $$

The calculated value of $\sigma$ is 6. This matches value III in List - II.

Calculation for Item B:

Given values:

• Population Mean ($\mu$) = 10
• Mean of Squares ($\frac{1}{N}\sum{x_i^2}$) = 200

Calculate the population variance ($\sigma^2$):

$$ \sigma^2 = 200 - (10)^2 = 200 - 100 = 100 $$

Calculate the population standard deviation ($\sigma$):

$$ \sigma = \sqrt{100} = 10 $$

The calculated value of $\sigma$ is 10. This matches value I in List - II.

Calculation for Item C:

Given values:

• Population Mean ($\mu$) = 11
• Mean of Squares ($\frac{1}{N}\sum{x_i^2}$) = 202

Calculate the population variance ($\sigma^2$):

$$ \sigma^2 = 202 - (11)^2 = 202 - 121 = 81 $$

Calculate the population standard deviation ($\sigma$):

$$ \sigma = \sqrt{81} = 9 $$

The calculated value of $\sigma$ is 9. This matches value IV in List - II.

Calculation for Item D:

Given values:

• Population Mean ($\mu$) = 12
• Mean of Squares ($\frac{1}{N}\sum{x_i^2}$) = 160

Calculate the population variance ($\sigma^2$):

$$ \sigma^2 = 160 - (12)^2 = 160 - 144 = 16 $$

Calculate the population standard deviation ($\sigma$):

$$ \sigma = \sqrt{16} = 4 $$

The calculated value of $\sigma$ is 4. This matches value II in List - II.

Summary of Matches

After performing the calculations, we can establish the correct matches between List - I and List - II:

Thus, the correct combination is A-III, B-I, C-IV, D-II.