To find the mean absolute deviation (MAD) of the dataset \({21, 25, 25, 26, 26, 27, 28, 30}\), we can follow these steps:
- Calculate the mean of the dataset:
\[ \text{Mean} = \frac{\text{Sum of all data points}}{\text{Number of data points}} \]
Calculating the sum:
\[ 21 + 25 + 25 + 26 + 26 + 27 + 28 + 30 = 208 \]
The number of data points is \(8\). Thus, the mean is:
\[ \text{Mean} = \frac{208}{8} = 26 \]
- Calculate the absolute deviations from the mean:
For each data point, subtract the mean and take the absolute value:
\[ |21 - 26| = | -5 | = 5 \] \[ |25 - 26| = | -1 | = 1 \] \[ |25 - 26| = | -1 | = 1 \] \[ |26 - 26| = | 0 | = 0 \] \[ |26 - 26| = | 0 | = 0 \] \[ |27 - 26| = | 1 | = 1 \] \[ |28 - 26| = | 2 | = 2 \] \[ |30 - 26| = | 4 | = 4 \]
- List the absolute deviations:
The absolute deviations are:
\[ {5, 1, 1, 0, 0, 1, 2, 4} \]
- Calculate the mean of the absolute deviations:
Now, calculate the sum of the absolute deviations:
\[ 5 + 1 + 1 + 0 + 0 + 1 + 2 + 4 = 14 \]
Now, divide this sum by the number of data points to find the mean absolute deviation:
\[ \text{MAD} = \frac{14}{8} = 1.75 \]
Thus, the mean absolute deviation of the dataset is:
\[ \boxed{1.75} \]