To determine whether IQR or MAD best describes the spread of the dataset, we first need to analyze the data, particularly looking at its distribution and potential outliers.
Step 1: Analyze the dataset
The dataset is: 79, 79, 79, 82, 84, 85, 89, 90, 99, 100, 100, 100, 129, 139, 155, 174, 189, 200, 209, 250
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Calculate the median (Q2):
- For 20 numbers, the median is the average of the 10th and 11th values (100 and 100) = 100.
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Calculate the first quartile (Q1) and third quartile (Q3):
- Q1 (the median of the first half: 79, 79, 79, 82, 84, 85, 89, 90, 99, 100) is the average of 85 and 89 = 87.
- Q3 (the median of the second half: 100, 100, 129, 139, 155, 174, 189, 200, 209, 250) is the average of 174 and 189 = 181.5.
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Calculate IQR (Interquartile Range):
- IQR = Q3 - Q1 = 181.5 - 87 = 94.5
Step 2: Identify potential outliers
Using the IQR to determine outliers:
- Calculate lower and upper bounds:
- Lower Bound = Q1 - 1.5 * IQR = 87 - 1.5 * 94.5 = -11.25 (no negatives applicable)
- Upper Bound = Q3 + 1.5 * IQR = 181.5 + 1.5 * 94.5 = 279.75
Since the maximum value of the dataset is 250, which is less than the upper bound, but 250 is the highest significant value. This suggests there may be an outlier consideration.
Step 3: Use of MAD (Mean Absolute Deviation)
Mean Absolute Deviation is less influenced by outliers than IQR. However, given that the dataset is skewed to the right (one high outlier of 250), this suggests the standard deviation would be affected, making IQR a better measure for this data set.
Conclusion
The best measure of variability for this dataset where there is a notable skew and potential outliers is IQR; the data distribution is skewed.
Therefore, the correct response is:
IQR; the data distribution is skewed.