Determine whether the range or interquartile range best describes the spread of a dataset.
The times (in seconds) for a high school boys’ 100-meter race:
11.0 11.2 11.7 12.2 12.4 12.6 12.8 12.9 13.1 13.3 13.8 14.2 14.3 14.3 16.5 17.6 18.0
In this case, the interquartile range would best describe the spread of the dataset. The interquartile range is a measure of variability that considers only the middle 50% of the data. It is calculated by finding the difference between the first quartile (Q1) and the third quartile (Q3).
To determine whether the range or interquartile range best describes the spread of a dataset, we can calculate both and compare their values.
1. Range:
The range is the difference between the largest and smallest values in a dataset. In this case, the largest value is 18.0, and the smallest value is 11.0. Therefore, the range = 18.0 - 11.0 = 7.0.
2. Interquartile Range (IQR):
The interquartile range is the difference between the first quartile (Q1) and the third quartile (Q3). It gives us the spread of the middle 50% of the data.
To calculate the quartiles, we first need to sort the dataset in ascending order:
11.0 11.2 11.7 12.2 12.4 12.6 12.8 12.9 13.1 13.3 13.8 14.2 14.3 14.3 16.5 17.6 18.0
The median (Q2) is the middle value of the dataset. In this case, we have an odd number of values, so the median is the 9th value, which is 12.9.
Q1 is the median of the lower half of the dataset, excluding the median itself. In this case, we have 8 values in the lower half, so the median of the lower half is the average of the 4th and 5th values: (11.7 + 12.2) / 2 = 11.95.
Q3 is the median of the upper half of the dataset, excluding the median itself. In this case, we have 8 values in the upper half, so the median of the upper half is also the average of the 4th and 5th values from the upper half of the dataset: (14.2 + 14.3) / 2 = 14.25.
Now we can calculate the IQR: IQR = Q3 - Q1 = 14.25 - 11.95 = 2.3.
Comparing the range and interquartile range values:
- Range: 7.0
- Interquartile Range: 2.3
In this case, the interquartile range (IQR) best describes the spread of the dataset because it focuses on the range of the middle 50% of the data, which is less sensitive to extreme values compared to the range.