To solve this question, we need to identify which formula correctly represents the standard deviation of a sample data set.
1. The formula for the sample standard deviation is:
[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]
2. The second formula given:
[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]
represents the variance of a population and not the sample standard deviation.
3. The third formula:
[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]
is the formula for the population standard deviation.
4. The fourth formula:
[tex]\[ s = \frac{\left(x_1 - \bar{x}\right)^2 + \left(x_2 - \bar{x}\right)^2 + \ldots + \left(x_n - \bar{x}\right)^2}{n-1} \][/tex]
is incorrect because it lacks the square root which is necessary for calculating the standard deviation (it resembles the sample variance formula).
Therefore, the formula used to calculate the standard deviation of sample data is:
[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]
