Determine which formula should be used to calculate the standard deviation for a neighborhood.

A. [tex] s^2 = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n-1} [/tex]

B. [tex] s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n-1}} [/tex]

C. [tex] \sigma^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_n - \mu)^2}{n} [/tex]

D. [tex] \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_n - \mu)^2}{n}} [/tex]



Answer :

To determine which formula represents standard deviation, we need to analyze each given option:

1. Formula 1:
[tex]\[ s^2 = \frac{\left(x_1 - \bar{x}\right)^2 + \left(x_2 - \bar{x}\right)^2 + \ldots + \left(x_n - \bar{x}\right)^2}{n - 1} \][/tex]
This formula calculates the sample variance. Sample variance is a measure of the dispersion of the sample data points around the mean of the sample.

2. Formula 2:
[tex]\[ s = \sqrt{\frac{\left(x_1 - \bar{x}\right)^2 + \left(x_2 - \bar{x}\right)^2 + \ldots + \left(x_n - \bar{x}\right)^2}{n - 1}} \][/tex]
This formula calculates the sample standard deviation. Sample standard deviation is the square root of the sample variance, providing a measure of the dispersion of the sample data points around the mean in the same units as the data.

3. Formula 3:
[tex]\[ \sigma^2 = \frac{\left(x_1 - \mu\right)^2 + \left(x_2 - \mu\right)^2 + \ldots + \left(x_n - \mu\right)^2}{n} \][/tex]
This formula calculates the population variance. Population variance measures the dispersion of the population data points around the population mean.

4. Formula 4:
[tex]\[ \sigma = \sqrt{\frac{\left(x_1 - \mu\right)^2 + \left(x_2 - \mu\right)^2 + \ldots + \left(x_n - \mu\right)^2}{n}} \][/tex]
This formula calculates the population standard deviation. Population standard deviation is the square root of the population variance, providing a measure of the dispersion of the population data points around the mean in the same units as the data.

Based on the analysis, the formulas for standard deviation, whether for a sample or population, are:

- Sample standard deviation:
[tex]\[ s = \sqrt{\frac{\left(x_1 - \bar{x}\right)^2 + \left(x_2 - \bar{x}\right)^2 + \ldots + \left(x_n - \bar{x}\right)^2}{n - 1}} \][/tex]

- Population standard deviation:
[tex]\[ \sigma = \sqrt{\frac{\left(x_1 - \mu\right)^2 + \left(x_2 - \mu\right)^2 + \ldots + \left(x_n - \mu\right)^2}{n}} \][/tex]