Reading 7: Statistical Concepts and Market Returns-LOS j 习题 2011, interpret, describe, relative, meaning

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Session 2: Quantitative Methods: Basic Concepts
Reading 7: Statistical Concepts and Market Returns

LOS j: Define and interpret skewness, explain the meaning of a positively or negatively skewed return distribution, and describe the relative locations of the mean, median, and mode for a nonsymmetrical distribution.

 

 

A distribution with a mode of 10 and a range of 2 to 25 would most likely be:

A) normally distributed.

B) negatively skewed.

C) positively skewed.

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The distance to the left from the mode to the beginning of the range is 8. The distance to the right from the mode to the end of the range is 15. Therefore, the distribution is skewed to the right, which means that it is positively skewed.

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Which of the following statements about statistical concepts is least accurate?

A) The coefficient of variation is useful when comparing dispersion of data measured in different units or having large differences in their means.

B) For a normal distribution, only 95% of the observations lie within ±3 standard deviations from the mean.

C) For any distribution, based on Chebyshev’s Inequality, 75% of the observations lie within ±2 standard deviations from the mean.

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For a normal distribution, 95% of the observations lie within ±2 standard deviations of the mean while 99% of the observations lie within plus or minus three standard deviations of the mean. Both remaining statements are true. Note that 75% of observations for any distribution lie within ±2 standard deviations of the mean using Chebyshev’s inequality.

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Which of the following statements regarding skewness is least accurate?

A) A distribution that is not symmetrical has skew not equal to zero.

B) In a skewed distribution, 95% of all values will lie within plus or minus two standard deviations of the mean.

C) A positively skewed distribution is characterized by many small losses and a few extreme gains.

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For a normal distribution, the mean will be equal to its median and 95% of all observations will fall within plus or minus two standard deviations of the mean. For a skewed distribution, because it is not symmetrical, this may not be the case. Chebyshev’s inequality tells us that at least 75% of observations will lie within plus or minus two standard deviations from the mean.

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If a distribution is skewed:

A) it will be more or less peaked reflecting a greater or lesser concentration of returns around the mean.

B) each side of a return distribution is the mirror image of the other.

C) the magnitude of positive deviations from the mean is different from the magnitude of negative deviations from the mean.

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Skewness is caused by the magnitude of positive deviations from the mean being either larger or smaller than the magnitude of negative deviations from the mean. Each side of a skewed distribution is not a mirror image of the other. Peakedness of a distribution is measured by kurtosis.

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Twenty Level I CFA candidates in a study group took a practice exam and want to determine the distribution of their scores. When they grade their exams they discover that one of them skipped an ethics question and subsequently filled in the rest of his answers in the wrong places, leaving him with a much lower score than the rest of the group. If they include this candidate’s score, their distribution will most likely:

A) have a mode that is less than its median.

B) be positively skewed.

C) have a mean that is less than its median.

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With the low outlier included, the distribution will be negatively skewed. For a negatively skewed distribution, the mean is less than the median, which is less than the mode.

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If a distribution is positively skewed, then generally you could say:

A) mean < median < mode.

B) mean > median > mode.

C) mean > median < mode.

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When a distribution is positively skewed the right side tail is longer than normal due to outliers. The mean will exceed the median, and the median will generally exceed the mode because large outliers falling to the far right side of the distribution can dramatically influence the mean.

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In a positively skewed distribution, the:

A) median equals the mean.

B) mean is greater than the median.

C) mean is less than the median.

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In a right-skewed distribution, there are large positive outliers. These outliers increase the mean of the distribution but have little effect on the median. Therefore, the mean is greater than the median.

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In a negatively skewed distribution, what is the order (from lowest value to highest) for the distribution’s mode, mean, and median values?

A) Median, mode, mean.

B) Mode, mean, median.

C) Mean, median, mode.

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In a negatively skewed distribution, the mean is less than the median, which is less than the mode.

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Consider the following graph of a distribution for the prices of various bottles of champagne.

Which of the following statements regarding the distribution is least accurate?

A) The distribution is negatively skewed.

B) The mean value will be less than the mode.

C) Point A represents the mode.

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The graph represents a negatively skewed distribution, and thus Point A represents the mean. By definition, mean < median < mode describes a negatively skewed distribution.

Both remaining statements are true. Chebyshev’s Inequality states that for any set of observations (normally distributed or skewed), the proportion of observations that lie within k standard deviations of the mean is at least 1 – 1 / k². Here, 1 – (1 / 1.3²) = 1 ? 0.59172 = 0.40828, or 40%.

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In a normal distribution, the:

A) mean is less than the mode.

B) median equals the mode.

C) mean is greater than the median.

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In a normal distribution, the mean, median, and mode are all equal.