To calculate the standard deviation of a sample data set, we use the following formula:
[tex]\[
s = \sqrt{\frac{(x_1 - \bar{x})^2 + (x_2 - \bar{x})^2 + \ldots + (x_n - \bar{x})^2}{n - 1}}
\][/tex]
Let's break down the formula step-by-step:
1. Mean of Sample Data ([tex]\(\bar{x}\)[/tex]):
- Calculate the mean (average) of the sample data. This is done by summing up all the sample values and then dividing by the number of samples ([tex]\(n\)[/tex]).
[tex]\[
\bar{x} = \frac{x_1 + x_2 + \ldots + x_n}{n}
\][/tex]
2. Deviations from Mean:
- Find the deviation of each sample value from the mean. This is done by subtracting the mean from each sample value.
[tex]\[
x_i - \bar{x} \qquad \text{for } i \in 1 \text{ to } n
\][/tex]
3. Square the Deviations:
- Square each of these deviations to get rid of negative values and amplify larger differences.
[tex]\[
(x_i - \bar{x})^2 \qquad \text{for } i \in 1 \text{ to } n
\][/tex]
4. Sum of Squared Deviations:
- Sum all these 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 Degrees of Freedom:
- Instead of dividing by [tex]\(n\)[/tex], we divide by [tex]\(n - 1\)[/tex] to account for the estimation of the sample mean (this adjustment is known as Bessel's correction).
[tex]\[
\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}
\][/tex]
6. Square Root:
- Finally, take the square root of the result to bring the squared units back to the original units.
[tex]\[
s = \sqrt{\frac{\sum_{i=1}^n (x_i - \bar{x})^2}{n - 1}}
\][/tex]
Thus, the formula for calculating 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]
