To calculate the Mean Absolute Deviation (MAD) and the Interquartile Range (IQR) for the given dataset, we'll follow these steps:
Dataset:
1, 4, 6, 7, 9, 10, 10, 11, 12, 12, 14, 15, 17, 18, 19
Step 1: Calculate the Mean
First, we need to calculate the mean of the dataset.
\[ \text{Mean} = \frac{\text{sum of all values}}{\text{number of values}} = \frac{1 + 4 + 6 + 7 + 9 + 10 + 10 + 11 + 12 + 12 + 14 + 15 + 17 + 18 + 19}{15} \]
Sum = 1 + 4 + 6 + 7 + 9 + 10 + 10 + 11 + 12 + 12 + 14 + 15 + 17 + 18 + 19 = 19 + 6 + 36 + 62 = 10 + 78 + 63 = 96
\[
Mean = \frac{96}{15} = 6.4
\]
Step 2: Calculate the Mean Absolute Deviation (MAD)
Now, we calculate the absolute deviations from the mean and then the average of those deviations.
Absolute deviations:
- |1 - 11| = 10
- |4 - 11| = 7
- |6 - 11| = 5
- |7 - 11| = 4
- |9 - 11| = 2
- |10 - 11| = 1
- |10 - 11| = 1
- |11 - 11| = 0
- |12 - 11| = 1
- |12 - 11| = 1
- |14 - 11| = 3
- |15 - 11| = 4
- |17 - 11| = 6
- |18 - 11| = 7
- |19 - 11| = 8
The MAD is calculated by finding the average of these absolute deviations.
Total Absolute Deviation = 10 + 7 + 5 + 4 + 2 + 1 + 1 + 0 + 1 + 1 + 3 + 4 + 6 + 7 + 8 = 60
MAD = Total Absolute Deviation / Number of Values = 60 / 15 = 4
Step 3: Calculate the IQR
To find the IQR, we need to find the 1st quartile (Q1) and the 3rd quartile (Q3).
- Q1 (25th percentile): The median of the first half (1, 4, 6, 7, 9, 10, 10) is 7.
- Q3 (75th percentile): The median of the second half (11, 12, 12, 14, 15, 17, 18, 19) is 15.
Now we can calculate the IQR: \[ \text{IQR} = Q3 - Q1 = 15 - 7 = 8 \]
Summary of Results:
- MAD: 4
- IQR: 8
Correct Answer:
MAD: 4; IQR: 8