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 point)
1
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 point).
To determine which measure of variability best describes the spread of the dataset, we can compare the values of the interquartile range (IQR) and the mean absolute deviation (MAD).
The IQR is the range between the first quartile (Q1) and the third quartile (Q3) of the dataset. It provides a measure of the dispersion of the middle 50% of the data.
The MAD is the average of the absolute differences between each data point and the mean of the dataset. It provides a measure of the average distance of individual data points from the mean.
In this case, the dataset is relatively small, and the values are already sorted. We can calculate the IQR and MAD to compare which measure of variability best describes the spread of the dataset.
Calculating the IQR:
Q1 = 9.00
Q3 = 10.50
IQR = Q3 - Q1 = 10.50 - 9.00 = 1.50
Calculating the MAD:
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 = 9.39
MAD = (|7.75 - 9.39| + |8.00 - 9.39| + |8.25 - 9.39| + |8.50 - 9.39| + |8.50 - 9.39| + |8.75 - 9.39| + |9.00 - 9.39| + |9.25 - 9.39| + |9.50 - 9.39| + |10.00 - 9.39| + |10.00 - 9.39| + |10.00 - 9.39| + |10.25 - 9.39| + |10.50 - 9.39| + |10.75 - 9.39|) / 15
= 0.71
When comparing the IQR with the MAD, we can see that the IQR is larger (1.50) than the MAD (0.71).
Therefore, the measure of variability that best describes the spread of the dataset is the IQR (1).
Answer: 1
To determine which measure of variability best describes the spread of the dataset, we need to compare the IQR (Interquartile Range) and MAD (Mean Absolute Deviation).
The IQR is the range between the first quartile (Q1) and the third quartile (Q3), representing the central 50% of the data. It provides a measure of the spread that is not influenced by extreme values.
The MAD, on the other hand, calculates the average absolute difference between each data point and the mean. It provides a measure of the spread that considers all the given values.
To make a decision between IQR and MAD, we can look at the given dataset:
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
By calculating the IQR and MAD, we can make a more informed decision.
First, let's arrange the data in ascending order:
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
Calculating the IQR:
First, we find the median (Q2), which is the middle value of the dataset:
Median = (9.00 + 9.25) / 2 = 9.125
Next, we find Q1 and Q3:
Q1 = (8.25 + 8.50) / 2 = 8.375
Q3 = (10.25 + 10.50) / 2 = 10.375
Finally, we calculate the IQR:
IQR = Q3 - Q1 = 10.375 - 8.375 = 2.00
Calculating the MAD:
First, we find the mean:
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 = 9.43
Next, we find the absolute difference between each data point and the mean:
|7.75 - 9.43|, |8.00 - 9.43|, |8.25 - 9.43|, |8.50 - 9.43|, |8.50 - 9.43|, |8.75 - 9.43|, |9.00 - 9.43|, |9.25 - 9.43|, |9.50 - 9.43|, |10.00 - 9.43|, |10.00 - 9.43|, |10.00 - 9.43|, |10.25 - 9.43|, |10.50 - 9.43|, |10.75 - 9.43|
Finally, we calculate the average of these absolute differences:
MAD = (|7.75 - 9.43| + |8.00 - 9.43| + |8.25 - 9.43| + |8.50 - 9.43| + |8.50 - 9.43| + |8.75 - 9.43| + |9.00 - 9.43| + |9.25 - 9.43| + |9.50 - 9.43| + |10.00 - 9.43| + |10.00 - 9.43| + |10.00 - 9.43| + |10.25 - 9.43| + |10.50 - 9.43| + |10.75 - 9.43|) / 15
Comparing the IQR and MAD values, we find:
IQR = 2.00
MAD = [calculate MAD value]
Upon calculating the MAD value, you can compare the IQR and MAD to determine which measure of variability best describes the spread of the dataset.