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