Question
Determine whether the range of interquartile range best describes the spread of a dataset. The time (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
Answers
GPT 3.5
In this case, the Interquartile Range (IQR) can help provide information about the spread of the dataset.
First, we need to find the first quartile (Q1) and the third quartile (Q3) of the dataset.
In this case, when the data is arranged in ascending order, the values are:
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
There are 17 observations, so the median is the value at the (17+1)/2 = 9th position, which is 12.9.
Now, to find Q1 and Q3:
- Q1 is the median of the lower half of the data set, which is the median of the values below 12.9. There are 8 values below 12.9, so Q1 is the value at position (8+1)/2 = 4.5, which is halfway between the 4th and 5th values. Therefore, Q1 is the average of the 4th and 5th values, which are 12.2 and 12.4 respectively. Q1 = (12.2 + 12.4)/2 = 12.3.
- Q3 is the median of the upper half of the dataset, which is the median of the values above 12.9. There are 8 values above 12.9, so Q3 is the value at position 8 + (8+1)/2 = 12.5, which is halfway between the 12th and 13th values. Therefore, Q3 is the average of the 12th and 13th values, which are 14.3 and 14.3 respectively. Q3 = (14.3 + 14.3)/2 = 14.3.
Now that we have found Q1 and Q3, we can calculate the IQR:
IQR = Q3 - Q1 = 14.3 - 12.3 = 2.0
Therefore, the IQR of the dataset is 2.0. This gives us a range where the middle 50% of the data lies and thus helps us understand the spread of the dataset in a more robust way than looking at the range alone.
First, we need to find the first quartile (Q1) and the third quartile (Q3) of the dataset.
In this case, when the data is arranged in ascending order, the values are:
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
There are 17 observations, so the median is the value at the (17+1)/2 = 9th position, which is 12.9.
Now, to find Q1 and Q3:
- Q1 is the median of the lower half of the data set, which is the median of the values below 12.9. There are 8 values below 12.9, so Q1 is the value at position (8+1)/2 = 4.5, which is halfway between the 4th and 5th values. Therefore, Q1 is the average of the 4th and 5th values, which are 12.2 and 12.4 respectively. Q1 = (12.2 + 12.4)/2 = 12.3.
- Q3 is the median of the upper half of the dataset, which is the median of the values above 12.9. There are 8 values above 12.9, so Q3 is the value at position 8 + (8+1)/2 = 12.5, which is halfway between the 12th and 13th values. Therefore, Q3 is the average of the 12th and 13th values, which are 14.3 and 14.3 respectively. Q3 = (14.3 + 14.3)/2 = 14.3.
Now that we have found Q1 and Q3, we can calculate the IQR:
IQR = Q3 - Q1 = 14.3 - 12.3 = 2.0
Therefore, the IQR of the dataset is 2.0. This gives us a range where the middle 50% of the data lies and thus helps us understand the spread of the dataset in a more robust way than looking at the range alone.