Multiple Choice
A sample of scores has a mean of and a sample variance of . What is the sample standard deviation?
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3. Describing Data Numerically
Standard Deviation
Multiple Choice
Which of the following is not a step used in calculating the standard deviation of a data set?
A
Find the mean of the data set
B
Take the square root of the average of the squared differences
C
Multiply all data values together
D
Subtract the mean from each data value and square the result
Verified step by step guidance1
Understand that calculating the standard deviation involves measuring how spread out the data values are around the mean.
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 \(\text{mean} = \frac{\sum x_i}{n}\).
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.
Step 3: Square each of these deviations to ensure all values are positive and to emphasize larger deviations, resulting in \((x_i - \text{mean})^2\).
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 \(\sqrt{\frac{\sum (x_i - \text{mean})^2}{n-1}}\).
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