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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Consider the following graph of a distribution for the prices for various bottles of California-produced wine.

Which of the following statements about this distribution is least accurate?

A) The graph could be of the sample $16, $12, $15, $12, $17, $30 (ignore graph scale).

B) The distribution is positively skewed.

C) Approximately 68% of observations fall within one standard deviation of the mean.

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This statement is true for the normal distribution. The above distribution is positively skewed. Note: for those tempted to use Chebyshev’s inequality to determine the percentage of observations falling within one standard deviation of the mean, the formula is valid only for k > 1.

The other statements are true. When we order the six prices from least to greatest: $12, $12, $15, $16, $17, $30, we observe that the mode (most frequently occurring price) is $12, the median (middle observation) is $15.50 [(15 + 16)/2], and the mean is $17 (sum of all prices divided by number in the sample). Time-Saving Note: Just by ordering the distribution, we can see that it is positively skewed (there are large, positive outliers). By definition, mode < median < mean describes a positively skewed distribution.

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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.

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A distribution with a mean that is less than its median most likely:

A) is negatively skewed.

B) is positively skewed.

C) has negative excess kurtosis.

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A distribution with a mean that is less than its median is a negatively skewed distribution. A negatively skewed distribution is characterized by many small gains and a few extreme losses. Note that kurtosis is a measure of the peakedness of a return distribution.

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If a distribution is positively skewed:

A) the mode is greater than the median.

B) the mean is greater than the median.

C) the mode is greater than the mean.

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For a positively skewed distribution, the mode is less than the median, which is less than the mean (the mean is greatest). Remember that investors are attracted to positive skewness because the mean return is greater than the median return.

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

A) Mean, median, mode.

B) Mode, mean, median.

C) Mode, median, mean.

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

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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.