To determine which formula is used to calculate the standard deviation of sample data, we first need to understand the standard deviation itself. Standard deviation is a measure of the amount of variation or dispersion in a set of values. Specifically, for sample data, the formula should account for the fact that the data is just a sample of the entire population by dividing by [tex]\( n-1 \)[/tex], which is known as Bessel's correction.
Let's review the provided formulas:
1. [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. [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]
3. [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]
4. [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]
To determine which formula is correct for calculating the standard deviation of sample data, we observe the following details:
- Formula 1 calculates the standard deviation [tex]\( s \)[/tex] of sample data using the sample mean [tex]\( \bar{x} \)[/tex] and corrects by dividing by [tex]\( n-1 \)[/tex], then taking the square root. This is the correct method for standard deviation of a sample.
- Formula 2 calculates the variance [tex]\( \sigma^2 \)[/tex] of population data using the population mean [tex]\( \mu \)[/tex] and divides by [tex]\( N \)[/tex], the number of values in the entire population. This is for population variance, not for sample data.
- Formula 3 calculates the standard deviation [tex]\( \sigma \)[/tex] of population data using [tex]\( \mu \)[/tex] and divides by [tex]\( N \)[/tex]. This is for the entire population standard deviation, not for a sample.
- Formula 4 describes the sum of squared deviations divided by [tex]\( n-1 \)[/tex], but does not take the square root, hence it is the formula for the sample variance, not the standard deviation.
The correct formula for calculating 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]
Thus, the correct answer is:
[tex]\[ \boxed{1} \][/tex]
