In computing skewness, which of the following is true?
I. It uses the cubed deviation from the mean.
II. It raises the deviation from the mean to the third power.
III. The formula preserves the direction of the deviation.
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A. B. C. D.C
I and II are true and equivalent. III is true because the cube of a negative number is still a negative number: -2 * -2 * -2 = -8.
The correct answer is A. I and III.
Skewness is a measure of the asymmetry of a probability distribution. It helps to understand the shape of the distribution and whether it is skewed to the left or right. Skewness can be calculated using different formulas, but in this question, we are asked to identify the true statements about computing skewness.
I. It uses the cubed deviation from the mean. This statement is true. Skewness is calculated by taking the average of the cubed deviations from the mean. The formula for skewness involves subtracting the mean from each data point, cubing the result, and then taking the average of those cubed deviations. By cubing the deviations, the formula emphasizes the impact of extreme values on the skewness measure.
II. It raises the deviation from the mean to the third power. This statement is not true. Skewness does not raise the deviation from the mean to the third power. It raises the individual deviations from the mean to the third power and then takes the average of those cubed deviations. The purpose of cubing the deviations is to eliminate any negative signs and emphasize the magnitude of the deviations rather than the direction.
III. The formula preserves the direction of the deviation. This statement is true. Skewness considers the direction of the deviation from the mean. If the majority of the data points are concentrated to the left of the mean, the skewness will be negative, indicating a left-skewed distribution. Conversely, if the majority of the data points are concentrated to the right of the mean, the skewness will be positive, indicating a right-skewed distribution. Thus, the formula preserves the direction of the deviation.
To summarize, the correct statements about computing skewness are that it uses the cubed deviation from the mean (statement I) and that the formula preserves the direction of the deviation (statement III). Therefore, the correct answer is A. I and III.