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Comparing Rate of Returns: Bond vs. Stock Mutual Funds
Chapter 9, Problem 45AYU(choose chapter or problem)
The data set on the left represents the annual rate of return (in percent) of eight randomly sampled bond mutual funds, and the data set on the right represents the annual rate of return (in percent) of eight randomly sampled stock mutual funds.
(a) Determine the mean and standard deviation of each data set.
(b) Based only on the standard deviation, which data set has more spread?
(c) What proportion of the observations is within one standard deviation of the mean for each data set?
(d) The coefficient of variation, CV, is defined as the ratio of the standard deviation to the mean of a data set, so
\(\mathrm{CV}=\frac{\text { standard deviation }}{\text { mean }}\)
The coefficient of variation is unitless and allows for comparison in spread between two data sets by describing the amount of spread per unit mean. After all, larger numbers will likely have a larger standard deviation simply due to the size of the numbers. Compute the coefficient of variation for both data sets. Which data set do you believe has more “spread”?
(e) Let’s take this idea one step further. The following data represent the height of a random sample of 8 male college students. The data set on the left has their height measured in inches, and the data set on the right has their height measured in centimeters.
For each data set, determine the mean and the standard deviation. Would you say that the height of the males is more dispersed based on the standard deviation of height measured in centimeters? Why? Now, determine the coefficient of variation for each data set. What did you find?
Questions & Answers
QUESTION:
The data set on the left represents the annual rate of return (in percent) of eight randomly sampled bond mutual funds, and the data set on the right represents the annual rate of return (in percent) of eight randomly sampled stock mutual funds.
(a) Determine the mean and standard deviation of each data set.
(b) Based only on the standard deviation, which data set has more spread?
(c) What proportion of the observations is within one standard deviation of the mean for each data set?
(d) The coefficient of variation, CV, is defined as the ratio of the standard deviation to the mean of a data set, so
\(\mathrm{CV}=\frac{\text { standard deviation }}{\text { mean }}\)
The coefficient of variation is unitless and allows for comparison in spread between two data sets by describing the amount of spread per unit mean. After all, larger numbers will likely have a larger standard deviation simply due to the size of the numbers. Compute the coefficient of variation for both data sets. Which data set do you believe has more “spread”?
(e) Let’s take this idea one step further. The following data represent the height of a random sample of 8 male college students. The data set on the left has their height measured in inches, and the data set on the right has their height measured in centimeters.
For each data set, determine the mean and the standard deviation. Would you say that the height of the males is more dispersed based on the standard deviation of height measured in centimeters? Why? Now, determine the coefficient of variation for each data set. What did you find?
ANSWER:
Step 1 of 6
Given, the data set on the left represents the annual rate of return (in percent) of eight randomly sampled bond mutual funds, and the data set on the right represents the annual rate of return (in percent) of eight randomly sampled stock mutual funds.
\(\begin{array}{|l|l|l|l|l|l|} \hline \begin{array}{l} \text { Bond mutual } \\ \text { funds(x) } \end{array} & \begin{array}{l} \text { Stock } \\\text { mutual } \\ \text { funds(y) } \end{array} & \left(x_{i}-\bar{x}\right) & \left(x_{i}-\bar{x}\right)^{2} & \left(y_{i}-\bar{y}\right) & \left(y_{i}-\bar{y}\right)^{2} \\ \hline 2.0 & 8.4 & -0.375 & 0.140625 & 0.3875 & 0.150156 \\ \hline 3.2 & 7.4 & 0.825 & 0.680625 & -0.6125 & 0.375156 \\ \hline 1.6 & 9.1 & -0.775 & 0.600625 & 1.0875 & 1.182656 \\ \hline 1.9 & 7.2 & -0.475 & 0.225625 & -0.8125 & 0.660156 \\ \hline 2.4 & 6.9 & 0.025 & 0.000625 & -1.1125 & 1.237656 \\ \hline 2.7 & 8.1 & 0.325 & 0.105625 & 0.0875 & 0.007656 \\ \hline 1.8 & 7.6 & -0.575 & 0.330625 & -0.4125 & 0.170156 \\ \hline 3.4 & 9.4 & 1.025 & 1.050625 & 1.3875 & 1.925156 \\ \hline \text { Total = } 19 & \text { Total }=64.1 & & \sum_{i=1}^{n}\left(x_{i}-\bar{x}\right){ }^{2}=3.135 & & \sum_{i=1}^{n}\left(y_{i}-\bar{y}\right)^{2}=5.7087 \\ \hline \end{array} \)