To calculate the standard deviation of sample data, the correct formula is given by:
[tex]\[ s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}} \][/tex]
Here's the step-by-step explanation:
1. Mean Calculation:
- First, calculate the sample mean ([tex]\(\bar{x}\)[/tex]), which is the average of all sample data points.
- [tex]\(\bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}\)[/tex]
2. Deviation from the Mean:
- Compute the deviation of each data point [tex]\(x_i\)[/tex] from the sample mean [tex]\(\bar{x}\)[/tex].
- [tex]\(x_i - \bar{x}\)[/tex] for all [tex]\(i\)[/tex].
3. Square of Deviations:
- Square each of these deviations.
- [tex]\((x_i - \bar{x})^2\)[/tex] for all [tex]\(i\)[/tex].
4. Sum of Squared Deviations:
- Sum up all the squared deviations.
- [tex]\(\sum_{i=1}^{n} (x_i - \bar{x})^2 = (x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2\)[/tex]
5. Divide by [tex]\(n-1\)[/tex]:
- Divide the summed squared deviations by [tex]\(n - 1\)[/tex], where [tex]\(n\)[/tex] is the number of data points. This division by [tex]\(n - 1\)[/tex] instead of [tex]\(n\)[/tex] is known as Bessel's correction, which corrects the bias in the estimation of the population variance and standard deviation from a sample.
- [tex]\(\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}\)[/tex]
6. Square Root:
- Finally, take the square root of the result to obtain the standard deviation.
- [tex]\(s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}}\)[/tex]
Given the presented options, the correct formula for 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]
Thus, the correct choice is the first one in the given options.
