To calculate the mean absolute deviation (MAD) of Lucia's Spanish quiz grades, we will follow these steps:
- Find the mean (average) of the grades.
- Calculate the absolute deviations from the mean.
- Find the average of those absolute deviations.
Step 1: Find the mean
The grades are: 74, 81, 86, 92, 74, 85, 90, 89, 78, 80.
First, we will sum these grades:
\[ 74 + 81 + 86 + 92 + 74 + 85 + 90 + 89 + 78 + 80 = 919 \]
Now, divide by the number of grades (10):
\[ \text{Mean} = \frac{919}{10} = 91.9 \]
Step 2: Calculate the absolute deviations from the mean
Now, we need to find the absolute deviations of each grade from the mean (91.9):
| Grade | Deviation from Mean (|Grade - Mean|) | |-------|---------------------------------------| | 74 | |74 - 91.9| = 17.9 | | 81 | |81 - 91.9| = 10.9 | | 86 | |86 - 91.9| = 5.9 | | 92 | |92 - 91.9| = 0.1 | | 74 | |74 - 91.9| = 17.9 | | 85 | |85 - 91.9| = 6.9 | | 90 | |90 - 91.9| = 1.9 | | 89 | |89 - 91.9| = 2.9 | | 78 | |78 - 91.9| = 13.9 | | 80 | |80 - 91.9| = 11.9 |
Now, we can sum the absolute deviations:
\[ 17.9 + 10.9 + 5.9 + 0.1 + 17.9 + 6.9 + 1.9 + 2.9 + 13.9 + 11.9 = 89.2 \]
Step 3: Calculate the mean absolute deviation
Now, divide the total absolute deviations by the number of grades:
\[ \text{MAD} = \frac{89.2}{10} = 8.92 \]
So, the mean absolute deviation of Lucia's Spanish quiz grades is 8.92.