What is the MAD and IQR for the given dataset?
1, 4, 6, 7, 9, 10, 10, 11, 12, 12, 14, 15, 17, 18, 19
To find the MAD (median absolute deviation):
1. Find the median of the dataset. In this case, the median is 11.
2. Find the absolute difference between each data point and the median. For example, the absolute difference between 4 and 11 is 7.
3. Find the median of those absolute differences. In this case, the median absolute deviation is 4.
To find the IQR (interquartile range):
1. Find the median of the lower half of the dataset. In this case, the lower half is 1, 4, 6, 7, 9, and 10, so the median is 6.5.
2. Find the median of the upper half of the dataset. In this case, the upper half is 12, 12, 14, 15, 17, 18, and 19, so the median is 15.
3. Subtract the lower median from the upper median. In this case, 15 - 6.5 = 8.5. This is the IQR.
To find the MAD (Mean Absolute Deviation) and IQR (Interquartile Range) for the given dataset, follow these steps:
Step 1: Arrange the dataset in ascending order:
1, 4, 6, 7, 9, 10, 10, 11, 12, 12, 14, 15, 17, 18, 19
Step 2: Find the median (middle value) of the dataset:
The median of the dataset is 11.
Step 3: Find the median of the lower half of the dataset (Q1):
The lower half of the dataset consists of the values: 1, 4, 6, 7, 9, 10.
The median of the lower half is 6.
Step 4: Find the median of the upper half of the dataset (Q3):
The upper half of the dataset consists of the values: 12, 12, 14, 15, 17, 18, 19.
The median of the upper half is 15.
Step 5: Calculate the IQR:
IQR = Q3 - Q1
IQR = 15 - 6
IQR = 9
Step 6: Calculate the absolute deviation for each value in the dataset:
|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
Step 7: Calculate the mean of the absolute deviations:
MAD = sum of absolute deviations / number of values
MAD = (10 + 7 + 5 + 4 + 2 + 1 + 1 + 0 + 1 + 1 + 3 + 4 + 6 + 7 + 8) / 15
MAD ≈ 4.93
Therefore, the MAD for the given dataset is approximately 4.93 and the IQR is 9.