Sure! Let's walk through calculating the mean, mean absolute deviation (MAD), standard deviation, and variance step-by-step using the given treadmill times: [tex]\( 30, 45, 20, 60, 30, 30, 15 \)[/tex].
### Step-by-Step Solution:
1. Calculating the Mean:
The mean (average) is calculated by summing all the numbers and then dividing by the count of numbers.
[tex]\[
\text{Mean} = \frac{30 + 45 + 20 + 60 + 30 + 30 + 15}{7} = \frac{230}{7} \approx 32.857
\][/tex]
2. Calculating the Mean Absolute Deviation (MAD):
a. First, find the absolute differences between each number and the mean:
[tex]\[
\begin{align*}
|30 - 32.857| &\approx 2.857 \\
|45 - 32.857| &\approx 12.143 \\
|20 - 32.857| &\approx 12.857 \\
|60 - 32.857| &\approx 27.143 \\
|30 - 32.857| &\approx 2.857 \\
|30 - 32.857| &\approx 2.857 \\
|15 - 32.857| &\approx 17.857 \\
\end{align*}
\][/tex]
b. Then, calculate the mean of these absolute differences:
[tex]\[
\text{MAD} = \frac{2.857 + 12.143 + 12.857 + 27.143 + 2.857 + 2.857 + 17.857}{7} \approx 11.224
\][/tex]
3. Calculating the Standard Deviation:
a. First, find the squared differences between each number and the mean:
[tex]\[
\begin{align*}
(30 - 32.857)^2 &\approx 8.163 \\
(45 - 32.857)^2 &\approx 147.020 \\
(20 - 32.857)^2 &\approx 167.449 \\
(60 - 32.857)^2 &\approx 736.734 \\
(30 - 32.857)^2 &\approx 8.163 \\
(30 - 32.857)^2 &\approx 8.163 \\
(15 - 32.857)^2 &\approx 323.734 \\
\end{align*}
\][/tex]
b. Calculate the mean of these squared differences:
[tex]\[
\text{Variance} = \frac{8.163 + 147.020 + 167.449 + 736.734 + 8.163 + 8.163 + 323.734}{7} \approx 198.980
\][/tex]
c. The standard deviation is the square root of the variance:
[tex]\[
\text{Standard Deviation} = \sqrt{198.980} \approx 14.106
\][/tex]
4. Summary:
[tex]\[
\begin{align*}
\text{Mean} &= 32.857 \\
\text{MAD} &= 11.224 \\
\sigma &= 14.106 \\
\sigma^2 &= 198.980
\end{align*}
\][/tex]
So, the detailed solution gives us:
- Mean: [tex]\( 32.857 \)[/tex]
- Mean Absolute Deviation (MAD): [tex]\( 11.224 \)[/tex]
- Standard Deviation: [tex]\( 14.106 \)[/tex]
- Variance: [tex]\( 198.980 \)[/tex]
