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Listed below are the top 10 annual salaries​ (in millions of​ dollars) of TV personalities. Find the​ range, variance, and standard deviation for the sample data.

{ 39, 37, 36, 30, 20, 18, 15, 13,12.7, 11.2 }

The aim of this question is to understand the fundamental statistical analysis of the given sample data covering key concepts of mean, variance, and standard deviation.

The mean of sample data is defined as the sum of all the data point values divided by a number of data points. Mathematically:

\[ \mu \ = \ \dfrac{ x_1 \ + \ x_2 \ + \ x_3 \ + \ … \ … \ … \ + x_n }{ n } \]

\[ \mu \ = \ \dfrac{ \sum_{ i = 1 }^{ n } \ x_i }{ n } \]

The variance ( $ \sigma^2 $ ) and standard deviation ( $ \sigma $ ) of sample data is defined mathematically as follows:

\[ \sigma^2 \ = \ \dfrac{ \sum_{ i = 1 }^{ n } \ \bigg ( x_i \ – \ \mu \bigg )^2 }{ n -1 } \]

\[ \sigma \ = \ \sqrt{ \dfrac{ \sum_{ i = 1 }^{ n } \ \bigg ( x_i \ – \ \mu \bigg )^2 }{ n – 1 } } \]

Expert Answer

From the definition of mean:

\[ \mu \ = \ \dfrac{ \text{ 39 + 37 + 36 + 30 + 20 + 18 + 15 + 13 + 12.7 + 11.2 } }{ 10 } \]

\[ \mu \ = \ \dfrac{ 231.9 }{ 10 } \]

\[ \mu \ = \ 23.19 \]

Now to find the variance, we first need to find the $ ( x_i – \mu )^2 $ term against each data point:

\[ \begin{array}{ | c | c | c |} \hline \\ x_i & x_i – \mu & ( x_i – \mu )^2 \\ \hline \\ 39 & 15.81 & 249.96 \\ 37 & 13.81 & 190.72 \\36 & 12.81 & 164.10 \\ 30 & 6.81 & 46.38 \\20 & -3.19 & 10.18 \\18 & -5.19 & 26.94 \\15 & -8.19 & 67.08 \\13 & -10.19 & 103.84 \\12.7 & -10.49 & 110.04 \\11.2 & -11.99 & 143.76 \\ \hline \end{array} \]

From above table:

\[ \sum_{ i = 1 }^{ n } \ \bigg ( x_i \ – \ \mu \bigg )^2 \ = \ 1112.97 \]

From the definition of variance:

\[ \sigma^2 \ = \ \dfrac{ \sum_{ i = 1 }^{ n } \ \bigg ( x_i \ – \ \mu \bigg )^2 }{ n -1 } \]

\[ \sigma^2 \ = \ \dfrac{ 1112.97 }{ 9 } \]

\[ \sigma^2 \ = \ 123.66 \]

From the definition of standard deviation:

\[ \sigma \ = \ \sqrt{ \sigma^2 } \]

\[ \sigma \ = \ \sqrt{ 123.66 } \]

\[ \sigma \ = \ 11.12\]

Numerical Results

\[ \mu \ = \ 23.19 \]

\[ \sigma^2 \ = \ 123.66 \]

\[ \sigma \ = \ 11.12\]

Example

Given the following data, find the mean of the sample.

{ 10, 15, 30, 50, 45, 33, 20, 19, 10, 11 }

From the definition of mean:

\[ \mu \ = \ \dfrac{ \text{ 10 + 15 + 30 + 50 + 45 + 33 + 20 + 19 + 10 + 11 } }{ 10 } \]

\[ \mu \ = \ \dfrac{ 24.3 }{ 10 } \]

\[ \mu \ = \ 2.43\]

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