Let's go through the steps to solve each part of the question.
### Part a: Finding the Mean of the Earnings Per Share
The mean of a set of numbers is calculated by summing all the numbers and then dividing by the number of numbers.
Given the earnings per share for the 12 companies are:
[tex]\[ 0.59, 0.62, 1.33, 7.35, 4.36, 3.86, 0.69, 7.83, 3.05, 3.51, 7.89, 8.37 \][/tex]
The mean (average) of these numbers is calculated as follows:
[tex]\[
\text{Mean} = \frac{0.59 + 0.62 + 1.33 + 7.35 + 4.36 + 3.86 + 0.69 + 7.83 + 3.05 + 3.51 + 7.89 + 8.37}{12}
\][/tex]
Summing the numbers:
[tex]\[
0.59 + 0.62 + 1.33 + 7.35 + 4.36 + 3.86 + 0.69 + 7.83 + 3.05 + 3.51 + 7.89 + 8.37 = 49.49
\][/tex]
Dividing by the number of companies (12):
[tex]\[
\text{Mean} = \frac{49.49}{12} \approx 4.12
\][/tex]
So, the mean of the earnings per share is:
[tex]\[ \boxed{4.12} \][/tex]
### Part b: Finding the Standard Deviation of the Earnings Per Share
Standard deviation measures the amount of variation or dispersion in a set of values. The formula for standard deviation is:
[tex]\[
\sigma = \sqrt{\frac{\sum (x_i - \mu)^2}{N}}
\][/tex]
where [tex]\( \sigma \)[/tex] is the standard deviation, [tex]\( x_i \)[/tex] represents each value in the dataset, [tex]\( \mu \)[/tex] is the mean of the values, and [tex]\( N \)[/tex] is the number of values.
We already calculated the mean [tex]\( \mu = 4.12 \)[/tex].
Now, we calculate the squared differences from the mean for each value, sum them, and then take the square root of the average of these squared differences.
[tex]\[
\sum (x_i - \mu)^2 = (0.59 - 4.12)^2 + (0.62 - 4.12)^2 + (1.33 - 4.12)^2 + \ldots + (8.37 - 4.12)^2
\][/tex]
Calculating each squared difference:
[tex]\[
(0.59 - 4.12)^2 \approx 12.51 \\
(0.62 - 4.12)^2 \approx 12.24 \\
(1.33 - 4.12)^2 \approx 7.76 \\
(7.35 - 4.12)^2 \approx 10.43 \\
(4.36 - 4.12)^2 \approx 0.06 \\
(3.86 - 4.12)^2 \approx 0.07 \\
(0.69 - 4.12)^2 \approx 11.76 \\
(7.83 - 4.12)^2 \approx 13.81 \\
(3.05 - 4.12)^2 \approx 1.14 \\
(3.51 - 4.12)^2 \approx 0.37 \\
(7.89 - 4.12)^2 \approx 14.2 \\
(8.37 - 4.12)^2 \approx 18.02
\][/tex]
Summing these squared differences:
[tex]\[
12.51 + 12.24 + 7.76 + 10.43 + 0.06 + 0.07 + 11.76 + 13.81 + 1.14 + 0.37 + 14.2 + 18.02 = 102.37
\][/tex]
Average of the squared differences:
[tex]\[
\frac{102.37}{12} \approx 8.53
\][/tex]
Finally, taking the square root to find the standard deviation:
[tex]\[
\sigma \approx \sqrt{8.53} \approx 2.92
\][/tex]
So, the standard deviation of the earnings per share is:
[tex]\[ \boxed{2.92} \][/tex]
