To find the mean absolute deviation (MAD) of the given data, we can follow these steps:
1. Calculate the Mean:
First, we need to find the average (mean) of the sample masses.
Given samples: [tex]\(10, 12, 7, 8, 5, 6\)[/tex]
Mean ([tex]\(\bar{x}\)[/tex]) is calculated by summing all the sample values and dividing by the number of samples.
[tex]\[
\bar{x} = \frac{10 + 12 + 7 + 8 + 5 + 6}{6}
\][/tex]
[tex]\[
\bar{x} = \frac{48}{6} = 8
\][/tex]
2. Calculate the Absolute Deviations:
Next, we find the absolute deviation of each sample from the mean. The absolute deviation of a sample [tex]\(x_i\)[/tex] from the mean [tex]\(\bar{x}\)[/tex] is given by [tex]\(|x_i - \bar{x}|\)[/tex].
For each sample:
[tex]\[
|10 - 8| = 2
\][/tex]
[tex]\[
|12 - 8| = 4
\][/tex]
[tex]\[
|7 - 8| = 1
\][/tex]
[tex]\[
|8 - 8| = 0
\][/tex]
[tex]\[
|5 - 8| = 3
\][/tex]
[tex]\[
|6 - 8| = 2
\][/tex]
The absolute deviations from the mean are [tex]\(2, 4, 1, 0, 3, 2\)[/tex].
3. Calculate the Mean of Absolute Deviations:
Finally, we find the mean of these absolute deviations. This is the mean absolute deviation.
[tex]\[
\text{MAD} = \frac{2 + 4 + 1 + 0 + 3 + 2}{6}
\][/tex]
[tex]\[
\text{MAD} = \frac{12}{6} = 2
\][/tex]
So, the mean absolute deviation of the masses of the 6 samples is [tex]\(2\)[/tex] grams.
