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]
