To determine which formula is used to calculate the standard deviation of sample data, let's analyze each given option:
1. [tex]\( s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \)[/tex]
2. [tex]\( a^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \)[/tex]
3. [tex]\( \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}} \)[/tex]
4. [tex]\( s = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \)[/tex]
### Definitions & Symbols:
- [tex]\( s \)[/tex] is the standard deviation of a sample.
- [tex]\( \sigma \)[/tex] is the standard deviation of a population.
- [tex]\( \bar{x} \)[/tex] is the sample mean.
- [tex]\( \mu \)[/tex] is the population mean.
- [tex]\( n \)[/tex] is the sample size.
- [tex]\( N \)[/tex] is the population size.
### Explanation of Each Option:
1. Option 1:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \][/tex]
This is the formula for the standard deviation of a sample. The divisor is [tex]\( n - 1 \)[/tex] which is known as Bessel's correction and it is used to correct the bias in the estimation of the population variance and standard deviation when sample data is used.
2. Option 2:
[tex]\[ a^2 = \frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N} \][/tex]
This formula computes the population variance ([tex]\( \sigma^2 \)[/tex]), not the standard deviation, and uses the population mean ([tex]\( \mu \)[/tex]) and population size ([tex]\( N \)[/tex]).
3. Option 3:
[tex]\[ \sigma = \sqrt{\frac{(x_1 - \mu)^2 + (x_2 - \mu)^2 + \ldots + (x_N - \mu)^2}{N}} \][/tex]
This is the formula for the population standard deviation. It uses the population mean ([tex]\( \mu \)[/tex]) and population size ([tex]\( N \)[/tex]).
4. Option 4:
[tex]\[ s = \frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1} \][/tex]
This is not a standard deviation formula, but rather an expression for the sample variance without taking the square root.
### Conclusion:
The correct formula to calculate the standard deviation of sample data is:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \][/tex]
Therefore, the correct option is:
[tex]\[ \boxed{1} \][/tex]
