A standard deviation example would typically have a set of observations x_1, x_2, x_3, …… x_n. The number of observations being n. In general it can be written like this: x_i, where i = 1, 2, 3, … n. To calculate the standard deviation of such a problem the following steps are followed:
Step 1: List all the observations in the first column of a table. The title of column would be x_i and the observations would be x_1, x_2, x_3, …… x_n.
Step 2: Find the mean of the observations. If we denote the mean by x ¯, then the formula for calculating the sample mean would be: x ¯ = (x_1+ x_2 + x_3+ ……+ x_n)/n = [∑x_i]/n
Step 3: We now come to the second column. In this column we find the deviation of each of the observation from the mean. For that we calculate the value of x_i - x ¯ for each of the observations.
Note that in this step it is possible that we get negative values. We shall take care of the negatives in the next step.
Step 4: Now we construct the third column. In this column we square the results of column (3) for each of the observations. That means we find the value of (x_i - x ¯)^2 for each of the n observations. Note here that since we squared the difference, the negatives are got rid of so we need not worry about the negative numbers canceling out the positive numbers and affecting our end result.
Step 5: Now we find the standard deviation for each of the observation. For that we compute the following value: (x_i - x ¯)^2 / n. That would be placed in our fourth column.
Step 6: Adding standard deviations for each of the observations we get the value of the expression : ∑((x_i - x ¯)^2 / n). That will give us our sample standard deviation formula. The symbol for standard deviation is the Greek small alphabet ‘s’ pronounced as ‘sigma’. Therefore we say that:
s = sqrt[∑((x_i - x ¯)^2 / n)] or s = sqrt [∑(x_i - x ¯)^2/n]
