To determine which formula is used to calculate the standard deviation of sample data, we should carefully examine the given options and understand the correct formula for standard deviation in the context of a sample.
Standard deviation measures the amount of variation or dispersion of a set of values. For sample data, it is denoted as [tex]\( s \)[/tex] and calculated using the formula:
[tex]\[ s = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n-1}} \][/tex]
Here:
- [tex]\( x_i \)[/tex] represents each value in the sample.
- [tex]\( \bar{x} \)[/tex] is the sample mean.
- [tex]\( n \)[/tex] is the number of observations in the sample.
Now, let's inspect each provided option:
1. [tex]\( 5 - \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \cdots + (x_n - \bar{x})^2}{n-1}} \)[/tex]
- This option is incorrect because it contains an additional term "5 -" which is not part of the standard deviation formula.
2. [tex]\( \sigma^2=\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \cdots + (x_N - \mu)^2}{N} \)[/tex]
- This option represents the variance for an entire population (not for a sample), as indicated by using [tex]\(\sigma^2\)[/tex] and [tex]\(\mu\)[/tex].
3. [tex]\( \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}} \)[/tex]
- This option represents the standard deviation for an entire population, using [tex]\(\sigma\)[/tex] and [tex]\(\mu\)[/tex] which refer to population parameters.
4. [tex]\( s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \cdots + (x_n - \bar{x})^2}{n-1}} \)[/tex]
- This option corresponds exactly to the standard deviation formula for a sample, using the sample mean [tex]\(\bar{x}\)[/tex] and sample size [tex]\(n\)[/tex].
Based on the above analysis, the correct formula used to calculate the standard deviation of sample data is:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \cdots + (x_n - \bar{x})^2}{n-1}} \][/tex]
Thus, the answer is the third option:
[tex]\[ \boxed{4} \][/tex]
