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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]
### 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]
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