Here are the blood pressure readings for Dr. Jackson’s patients yesterday. The systolic blood pressure numbers (top numbers) are given. Which measure of variability—MAD or IQR—best represents the shape of the distribution? Enter 1 for MAD or 2 for IQR.
70, 79, 82, 90, 96, 110, 120, 134, 139, 142, 148, 150, 152, 155, 161
2 (IQR)
To determine whether the mean absolute deviation (MAD) or the interquartile range (IQR) best represents the shape of the distribution, we need to calculate both measures.
First, let's find the MAD:
1. Find the mean of the systolic blood pressure readings:
(70 + 79 + 82 + 90 + 96 + 110 + 120 + 134 + 139 + 142 + 148 + 150 + 152 + 155 + 161) / 15 = 121.8
2. Calculate the difference between each individual reading and the mean:
|70 - 121.8| = 51.8
|79 - 121.8| = 42.8
|82 - 121.8| = 39.8
|90 - 121.8| = 31.8
|96 - 121.8| = 25.8
|110 - 121.8| = 11.8
|120 - 121.8| = 1.8
|134 - 121.8| = 12.2
|139 - 121.8| = 17.2
|142 - 121.8| = 20.2
|148 - 121.8| = 26.2
|150 - 121.8| = 28.2
|152 - 121.8| = 30.2
|155 - 121.8| = 33.2
|161 - 121.8| = 39.2
3. Find the average of these absolute differences:
(51.8 + 42.8 + 39.8 + 31.8 + 25.8 + 11.8 + 1.8 + 12.2 + 17.2 + 20.2 + 26.2 + 28.2 + 30.2 + 33.2 + 39.2) / 15 ≈ 27.78
The MAD (mean absolute deviation) is approximately 27.78.
Second, let's find the IQR:
1. Sort the systolic blood pressure readings in increasing order:
70, 79, 82, 90, 96, 110, 120, 134, 139, 142, 148, 150, 152, 155, 161
2. Calculate the 1st quartile (Q1):
Q1 = (n + 1) / 4th item = (15 + 1) / 4 = 16 / 4 = 4th item = 90
3. Calculate the 3rd quartile (Q3):
Q3 = 3 * (n + 1) / 4th item = 3 * (15 + 1) / 4 = 48 / 4 = 12th item = 150
4. Calculate the IQR (interquartile range):
IQR = Q3 - Q1 = 150 - 90 = 60
The IQR (interquartile range) is 60.
Now, comparing the MAD and IQR:
The MAD measures the average absolute deviation from the mean, while the IQR measures the range between the 1st and 3rd quartiles. Neither measure provides direct information about the shape of the distribution.
However, if we consider the fact that the IQR captures the middle 50% of the data, it may be a better representation of the shape of the distribution in terms of the spread or variability. Therefore, the best measure of variability that represents the shape of the distribution is the IQR.
Answer: 2 (IQR)