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
The interquartile range would be a better descriptor of the spread in this dataset since there are a few outliers that could significantly affect the range.
Well, let's examine the dataset for the boys' 100-meter race times. It seems like we have a range of values from 11.0 seconds to 18.0 seconds.
Now, the range indicates the difference between the highest and lowest values in a dataset. Taking the difference of 18.0 - 11.0, we get a range of 7.0 seconds.
On the other hand, the interquartile range (IQR) describes the spread between the first quartile (Q1) and the third quartile (Q3) of a dataset. To determine the IQR, we need to sort the data in increasing 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
Now, finding the median, let's split this dataset into two halves.
First half: 11.0, 11.2, 11.7, 12.2, 12.4, 12.6, 12.8, 12.9
Second half: 13.1, 13.3, 13.8, 14.2, 14.3, 14.3, 16.5, 17.6, 18.0
The first quartile (Q1) will be the median of the first half, which is 12.8 seconds. The third quartile (Q3) will be the median of the second half, which is 14.3 seconds.
To calculate the IQR, we subtract Q1 from Q3: 14.3 - 12.8, resulting in an IQR of 1.5 seconds.
Considering both the range and the IQR, it depends on your preference. If you want to know the spread of the entire dataset, including the outliers (if any), you can use the range. However, if you want to analyze the spread of the central 50% of the dataset, disregarding the extremes, the interquartile range (IQR) would be more appropriate.
So, the choice between range and interquartile range would depend on the type of analysis you are conducting. But hey, remember that no matter which one you choose, racing against clowns would still be much more entertaining.
To determine whether the range or interquartile range best describes the spread of a dataset, let's calculate both measures for the given dataset of times:
Step 1: Calculate the Range
The range is the difference between the maximum and minimum values in a dataset.
Minimum value = 11.0 seconds
Maximum value = 18.0 seconds
Range = Maximum value - Minimum value
Range = 18.0 - 11.0 = 7.0 seconds
Step 2: Calculate the Interquartile Range (IQR)
The interquartile range is the range of the middle 50% of values in a dataset.
Step 2.1: 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
Step 2.2: Find the first quartile (Q1) and third quartile (Q3):
The first quartile (Q1) is the median of the lower half of the data.
The third quartile (Q3) is the median of the upper half of the data.
Q1 = Median of the lower half = Median of 11.0, 11.2, 11.7, 12.2, 12.4, 12.6, 12.8 = 12.4
Q3 = Median of the upper half = Median of 13.1, 13.3, 13.8, 14.2, 14.3, 14.3, 16.5, 17.6, 18.0 = 14.3
Step 2.3: Calculate the IQR
IQR = Q3 - Q1
IQR = 14.3 - 12.4 = 1.9 seconds
In the given dataset, the range is 7.0 seconds, and the interquartile range (IQR) is 1.9 seconds.
Based on these calculations, it is generally better to use the interquartile range (IQR) to describe the spread of the dataset, as it is less affected by outliers. The IQR focuses on the middle 50% of the data, providing a more representative measure of the spread.
To determine whether the range or interquartile range best describes the spread of the given dataset, let's first understand what range and interquartile range represent.
The range of a dataset is the difference between the maximum and minimum values. It provides a measure of the total spread of the data. To find the range, we subtract the minimum value from the maximum value.
The interquartile range (IQR) is a measure of variability that describes the range between the 25th and 75th percentiles of the data. It is calculated by finding the difference between the third quartile (Q3) and the first quartile (Q1). The first quartile (Q1) is the median of the lower half of the data, and the third quartile (Q3) is the median of the upper half.
Let's find the range and interquartile range for the given dataset.
First, we 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
Now we can find the range by subtracting the minimum value from the maximum value:
Range = 18.0 - 11.0 = 7.0
To find the interquartile range (IQR), we need to find the first quartile (Q1) and the third quartile (Q3).
Q1 is the median of the lower half of the data. Since we have 17 data points, the lower half is the first 8 values. The median of this lower half is the average of the 4th and 5th values:
Q1 = (12.2 + 12.4) / 2 = 12.3
Q3 is the median of the upper half of the data. The upper half starts from the 9th value and continues to the end. Again, the median is the average of the 4th and 5th values in this upper half:
Q3 = (14.2 + 14.3) / 2 = 14.25
Finally, we can calculate the IQR by subtracting Q1 from Q3:
IQR = 14.25 - 12.3 = 1.95
Now, to determine whether the range or interquartile range best describes the spread of the dataset, we need to consider the presence of outliers.
If the dataset contains outliers, the range can be heavily influenced and become a poor representation of the spread. In such cases, the interquartile range is preferred because it is less affected by outliers.
In the given dataset, the times range from 11.0 to 18.0, resulting in a range of 7.0. The interquartile range, however, is only 1.95.
Since this dataset does not appear to have any extreme outliers, both the range and interquartile range can provide meaningful information about the spread.
However, if you want a measure of spread that is less affected by outliers, the interquartile range (IQR) is generally preferred.