Answer
The mean and the standard deviation of the dividend yield are 1.2 and 0.88.
The approximate mean and standard deviation are nearly equal to the actual mean and standard deviation.
The following steps are followed to obtain the mean and the standard deviation for the provided data.
Step 1: Obtain the midpoint of the class interval by adding the consecutive two lower class boundaries and divide the sum by 2.
Step 2: The formula used to obtain the mean is:
Work Step by Step
Therefore, the mean of the dividend yield is:
\[\begin{align}
& \bar{x}=\frac{\sum{{{x}_{i}}{{f}_{i}}}}{\sum{{{f}_{i}}}} \\
& =\frac{34}{28} \\
& =1.2
\end{align}\]
And the standard deviation is calculated as:
\[\begin{align}
& s=\sqrt{\frac{\sum{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}{{f}_{i}}}}{\sum{{{f}_{i}}}-1}} \\
& =\sqrt{\frac{21}{28-1}} \\
& =0.88
\end{align}\]
The actual mean is calculated as:
\[\begin{align}
& \bar{x}=\frac{\sum{{{x}_{i}}}}{n} \\
& =\frac{1.7+2.83+\cdots +0.41}{28} \\
& =1.13
\end{align}\]
The actual standard deviation is calculated as:
\[\begin{align}
& \sigma =\sqrt{\frac{1}{n}\sum{{{\left( {{x}_{i}}-\bar{x} \right)}^{2}}}} \\
& =\sqrt{\frac{1}{28}\left( {{\left( 1.7-1.13 \right)}^{2}}+{{\left( 2.83-1.13 \right)}^{2}}+\cdots +{{\left( 0.41-1.13 \right)}^{2}} \right)} \\
& =0.91
\end{align}\]
Therefore, the actual mean and standard deviation of the dividend yield is nearly equal to the approximate mean and standard deviation, respectively.
