The job coordinator at Martin Luther King Magnet School surveyed 16-year old students on their hourly wages (in $) for their part-time jobs:
7.75, 8.00, 8.25, 8.50, 8.50, 8.75, 9.00, 9.25, 9.50, 10.00, 10.00, 10.00, 10.25, 10.50, 10.75
Which measure of variability—IQR or MAD—best describes the spread of the dataset? Enter 1 for IQR or 2 for MAD.
1
To determine which measure of variability, IQR or MAD, best describes the spread of the dataset, we need to understand the characteristics of both measures.
1. IQR (Interquartile Range): The IQR is a measure of variability that calculates the range between the upper quartile (Q3) and the lower quartile (Q1) of a dataset. It gives us an idea of the spread of the middle 50% of the data.
2. MAD (Mean Absolute Deviation): The MAD is a measure of variability that calculates the average distance between each observation and the mean of the dataset. It provides a measure of the average spread of the data.
Now, let's consider the dataset of hourly wages for the 16-year old students:
7.75, 8.00, 8.25, 8.50, 8.50, 8.75, 9.00, 9.25, 9.50, 10.00, 10.00, 10.00, 10.25, 10.50, 10.75
To calculate the IQR, we first need to find the Q1 and Q3. Q1 is the median of the lower half of the data, while Q3 is the median of the upper half.
In this dataset, the Q1 is the median of the first half of the ordered data:
7.75, 8.00, 8.25, 8.50, 8.50, 8.75
Q1 = (8.25 + 8.50) / 2 = 8.375
The Q3 is the median of the second half of the ordered data:
9.00, 9.25, 9.50, 10.00, 10.00, 10.00
Q3 = (9.50 + 10.00) / 2 = 9.75
Now, we can calculate the IQR:
IQR = Q3 - Q1 = 9.75 - 8.375 ≈ 1.375
To calculate the MAD, we first need to find the mean of the dataset:
Mean = (7.75 + 8.00 + 8.25 + 8.50 + 8.50 + 8.75 + 9.00 + 9.25 + 9.50 + 10.00 + 10.00 + 10.00 + 10.25 + 10.50 + 10.75) / 15
≈ 9.047
Next, we find the absolute deviation (the distance between each observation and the mean):
|7.75 - 9.047|, |8.00 - 9.047|, |8.25 - 9.047|, |8.50 - 9.047|, |8.50 - 9.047|, |8.75 - 9.047|, |9.00 - 9.047|, |9.25 - 9.047|, |9.50 - 9.047|, |10.00 - 9.047|, |10.00 - 9.047|, |10.00 - 9.047|, |10.25 - 9.047|, |10.50 - 9.047|, |10.75 - 9.047|
Finally, we calculate the average of these absolute deviations:
MAD = (|7.75 - 9.047| + |8.00 - 9.047| + ... + |10.75 - 9.047|) / 15
≈ 0.922
Comparing the values, we see that the IQR is approximately 1.375, while the MAD is approximately 0.922.
Since the IQR measures the range between the upper and lower quartiles and provides insight into the spread of the middle 50% of the data, it is more suitable for describing the spread of the dataset in this case.
Therefore, the answer is 1 for IQR.