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In statistics, what does the standard deviation of a data set measure?
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3. Describing Data Numerically
Standard Deviation
Multiple Choice
Which data set is the most spread out from its mean (i.e., has the largest standard deviation)?
A
D: 10, 10, 10, 10, 10
B
C: 9, 10, 10, 10, 11
C
B: 0, 5, 10, 15, 20
D
A: 8, 9, 10, 11, 12
검증된 단계별 안내1
Understand that the standard deviation measures how spread out the numbers in a data set are from the mean. A larger standard deviation means the data points are more spread out.
Calculate the mean (average) of each data set using the formula: \(\text{mean} = \frac{\sum x_i}{n}\), where \(x_i\) are the data points and \(n\) is the number of points.
For each data set, find the squared differences from the mean: calculate \((x_i - \text{mean})^2\) for each data point.
Compute the variance by averaging these squared differences: \(\text{variance} = \frac{\sum (x_i - \text{mean})^2}{n}\) (for population variance) or \(\frac{\sum (x_i - \text{mean})^2}{n-1}\) (for sample variance).
Take the square root of the variance to get the standard deviation: \(\text{standard deviation} = \sqrt{\text{variance}}\). Compare the standard deviations of all data sets to determine which one is the most spread out.
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