Population Standard Deviation Calculation

To find the standard deviation for each population, we use the formula for population standard deviation ($\sigma$):

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

Calculating Standard Deviation for Population A

Given:

• $n = 10$
• $\sum x_i = 450$
• $\sum x_i^2 = 24250$

First, calculate the mean ($\mu_A$):

$\mu_A = \frac{450}{10} = 45$

Next, calculate the variance ($\sigma_A^2$):

$\sigma_A^2 = \frac{24250}{10} - (45)^2 = 2425 - 2025 = 400$

Finally, calculate the standard deviation ($\sigma_A$):

$\sigma_A = \sqrt{400} = 20$

Calculating Standard Deviation for Population B

Given:

• $n = 8$
• $\sum x_i = 104$
• $\sum x_i^2 = 1424$

Mean ($\mu_B$):

$\mu_B = \frac{104}{8} = 13$

Variance ($\sigma_B^2$):

$\sigma_B^2 = \frac{1424}{8} - (13)^2 = 178 - 169 = 9$

Standard Deviation ($\sigma_B$):

$\sigma_B = \sqrt{9} = 3$

Calculating Standard Deviation for Population C

Given:

• $n = 5$
• $\sum x_i = 120$
• $\sum x_i^2 = 3600$

Mean ($\mu_C$):

$\mu_C = \frac{120}{5} = 24$

Variance ($\sigma_C^2$):

$\sigma_C^2 = \frac{3600}{5} - (24)^2 = 720 - 576 = 144$

Standard Deviation ($\sigma_C$):

$\sigma_C = \sqrt{144} = 12$

Calculating Standard Deviation for Population D

Given:

• $n = 7$
• $\sum x_i = 28$
• $\sum x_i^2 = 140$

Mean ($\mu_D$):

$\mu_D = \frac{28}{7} = 4$

Variance ($\sigma_D^2$):

$\sigma_D^2 = \frac{140}{7} - (4)^2 = 20 - 16 = 4$

Standard Deviation ($\sigma_D$):

$\sigma_D = \sqrt{4} = 2$

Ascending Order of Standard Deviations

We have calculated the standard deviations for each population:

• Population A: $\sigma_A = 20$
• Population B: $\sigma_B = 3$
• Population C: $\sigma_C = 12$
• Population D: $\sigma_D = 2$

Arranging these values in ascending order:

$2, 3, 12, 20$

This corresponds to the populations in the order: D, B, C, A.