Which measure of variability—range or IQR—best describes the spread of the dataset?
For a statistics assignment, Shayna randomly surveyed students on how many hours per week they spend playing online games:
1.0 2.0 2.0 7.5 10.0 10.0 10.0 10.0 11.5 17.0 17.0 18.0
In this case, the IQR (Interquartile Range) is a better measure of variability to describe the spread of the dataset. The IQR is less sensitive to extreme values and outliers compared to the range. It provides a measure of the dispersion that captures the spread of the middle 50% of the data, which is often more representative of the overall spread.
To determine which measure of variability, range or IQR, best describes the spread of the dataset, we can calculate both measures and compare them.
1. Range: The range is the difference between the maximum and minimum values in the dataset. For this dataset, the maximum value is 18.0 and the minimum value is 1.0. Therefore, the range is 18.0 - 1.0 = 17.0.
2. IQR (Interquartile Range): The IQR is a measure of the spread of the central 50% of the data. To calculate the IQR, we need to find the 25th percentile (lower quartile, Q1) and the 75th percentile (upper quartile, Q3). Firstly, let's order the dataset in ascending order:
1.0 2.0 2.0 7.5 10.0 10.0 10.0 10.0 11.5 17.0 17.0 18.0
The median is the middle value, or the average of the two middle values if there is an even number. In this case, the median is (10.0 + 10.0) / 2 = 10.0.
To find the quartiles, we need to divide the dataset into two halves. The lower half contains 1.0, 2.0, 2.0, 7.5, 10.0, and 10.0. The upper half contains 10.0, 10.0, 11.5, 17.0, 17.0, and 18.0.
Q1 is the median of the lower half, which is (2.0 + 7.5) / 2 = 4.75.
Q3 is the median of the upper half, which is (11.5 + 17.0) / 2 = 14.25.
Finally, the IQR is Q3 - Q1 = 14.25 - 4.75 = 9.5.
Comparing the range and IQR, we can see that the range is 17.0, while the IQR is 9.5. The range provides information about the overall spread of the dataset, including the extreme values, while the IQR focuses on the spread of the middle 50% of the data, excluding outliers. Therefore, if we want to describe the spread of the dataset, including extreme values, the range would be more appropriate. However, if we want to focus on the spread of the central values, excluding outliers, the IQR would be a better choice.