To determine which formula accurately calculates the standard deviation of sample data, let's examine each formula in detail:
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]
- This formula calculates the sample standard deviation.
- [tex]\( x_1, x_2, \ldots, x_n \)[/tex] are the data points in the sample.
- [tex]\( \bar{x} \)[/tex] is the sample mean.
- [tex]\( n \)[/tex] is the sample size.
- The denominator [tex]\( n-1 \)[/tex] indicates that this is a sample standard deviation, incorporating Bessel's correction to provide an unbiased estimate.
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]
- This formula calculates the population variance.
- [tex]\( x_1, x_2, \ldots, x_N \)[/tex] are the data points in the population.
- [tex]\( \mu \)[/tex] is the population mean.
- [tex]\( N \)[/tex] is the population size.
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]
- This formula calculates the population standard deviation.
- Similar to the previous formula, but it includes the square root to provide the standard deviation rather than the variance.
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]
- This is incorrect because it lacks the square root, making it actually calculate the sample variance rather than the sample standard deviation.
After examining these formulas, the correct one 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]
This formula accounts for the sample mean and uses [tex]\( n-1 \)[/tex] in the denominator to adjust for the sample size, making it suitable for computing the sample standard deviation.
Thus, the correct formula to use for calculating the sample standard deviation is the first one. The index of the correct formula is:
[tex]\[ 1 \][/tex]
