Which of the following is not a step used in calculating the standard deviation $\sigma$ of a data set?

• A. Find the mean $\overline{x}$ of the data set
• B. Take the square root $\sqrt{}$ of the average of the squared differences $\frac{{x-\overline{x}}^2}{n}$
• C. Multiply all data values together $\prod x$
• D. Subtract the mean $\overline{x}$ from each data value $x$ and square the result ${x-\overline{x}}^2$

Answer: (C)

Verified step by step guidance

1. Understand that calculating the standard deviation involves measuring how spread out the data values are around the mean.
2. Step 1: Find the mean (average) of the data set by summing all data values and dividing by the number of values, using the formula \(\text{mean} = \frac{\sum x_i}{n}\).
3. Step 2: Subtract the mean from each data value to find the deviation of each value from the mean, i.e., calculate \(x_i - \text{mean}\) for each data point.
4. Step 3: Square each of these deviations to ensure all values are positive and to emphasize larger deviations, resulting in \((x_i - \text{mean})^2\).
5. Step 4: Find the average of these squared deviations (this is called the variance), by summing them and dividing by \(n\) (for population) or \(n-1\) (for sample), then take the square root of this average to get the standard deviation, expressed as \(\sqrt{\frac{\sum (x_i - \text{mean})^2}{n}}\) or \(\sqrt{\frac{\sum (x_i - \text{mean})^2}{n-1}}\).