Maria surveyed girls on the amount (in $) they spent on a dress for the middle school spring fling:
79, 79, 79, 82, 84, 85, 89, 90, 99, 100, 100, 100, 129, 139, 155, 174, 189, 200, 209, 250
Which measure of variability—IQR or MAD—best describes the spread of the dataset?
The IQR (Interquartile Range) would be the best measure of variability to describe the spread of the dataset. The MAD (Mean Absolute Deviation) may be affected by outliers in the dataset, and the IQR is more robust to outliers. Additionally, the IQR provides information about the middle 50% of the data, which is relevant for understanding the central tendency and variability of the dataset.
To determine which measure of variability best describes the spread of the dataset, let's first calculate the IQR (interquartile range) and MAD (mean absolute deviation) for the given data.
Step 1: Calculate the IQR:
1. Arrange the data in ascending order: 79, 79, 79, 82, 84, 85, 89, 90, 99, 100, 100, 100, 129, 139, 155, 174, 189, 200, 209, 250
2. Find the first quartile (Q1): (n+1)/4 = (20+1)/4 = 21/4 = 5.25 (round up to 6th value)
Q1 = 85
3. Find the third quartile (Q3): 3(n+1)/4 = 3(20+1)/4 = 63/4 = 15.75 (round up to 16th value)
Q3 = 200
4. Calculate the IQR: IQR = Q3 - Q1 = 200 - 85
IQR = 115
Step 2: Calculate the MAD:
1. Find the median (Q2): (n+1)/2 = (20+1)/2 = 21/2 = 10.5 (take the average of the 10th and 11th values)
Q2 = (100 + 100)/2
Q2 = 100
2. Calculate the absolute deviation for each value from Q2:
|79 - 100| = 21
|79 - 100| = 21
|79 - 100| = 21
|82 - 100| = 18
|84 - 100| = 16
|85 - 100| = 15
|89 - 100| = 11
|90 - 100| = 10
|99 - 100| = 1
|100 - 100| = 0
|100 - 100| = 0
|100 - 100| = 0
|129 - 100| = 29
|139 - 100| = 39
|155 - 100| = 55
|174 - 100| = 74
|189 - 100| = 89
|200 - 100| = 100
|209 - 100| = 109
|250 - 100| = 150
3. Calculate the mean absolute deviation (MAD): MAD = ∑(absolute deviations) / n
MAD = (21 + 21 + 21 + 18 + 16 + 15 + 11 + 10 + 1 + 0 + 0 + 0 + 29 + 39 + 55 + 74 + 89 + 100 + 109 + 150) / 20
MAD ≈ 47.2
Step 3: Compare the IQR and MAD:
The IQR measures the range between the first quartile (Q1) and the third quartile (Q3), while the MAD calculates the average absolute deviation from the median (Q2).
In this case, the IQR is 115 and the MAD is approximately 47.2. The IQR is larger than the MAD, indicating a greater spread between the quartiles compared to the deviations from the median. Therefore, the IQR best describes the spread of the dataset.