The shoe sizes for a varsity field hockey team are:
5.5, 5.5, 6.0, 6.5, 6.5, 7.0, 7.0, 7.5, 7.5, 7.5, 8.0, 8.0, 8.0, 8.5, 8.5, 9.0
Which of the following is the IQR and explains how this measure of variability describes how the values of a dataset vary with a single number?
To find the IQR (interquartile range) of the given dataset, we need to first find the first quartile (Q1) and the third quartile (Q3).
1) Arrange the data in ascending order:
5.5, 5.5, 6.0, 6.5, 6.5, 7.0, 7.0, 7.5, 7.5, 7.5, 8.0, 8.0, 8.0, 8.5, 8.5, 9.0
2) Find Q1:
To find Q1, we need to locate the median of the lower half of the data, which is the value separating the first 25% of the data from the remaining 75%. In this case, the lower half consists of the first 8 values:
5.5, 5.5, 6.0, 6.5, 6.5, 7.0, 7.0, 7.5
The median of this lower half is the average of the numbers in the middle: (6.5 + 7.0) / 2 = 6.75.
Therefore, Q1 = 6.75.
3) Find Q3:
Similarly, to find Q3, we need to locate the median of the upper half of the data, which is the value separating the first 75% of the data from the remaining 25%. In this case, the upper half consists of the last 8 values:
7.5, 7.5, 7.5, 8.0, 8.0, 8.0, 8.5, 8.5, 9.0
The median of this upper half is the average of the numbers in the middle: (8.0 + 8.5) / 2 = 8.25.
Therefore, Q3 = 8.25.
4) Calculate the IQR:
The IQR is found by subtracting Q1 from Q3:
IQR = Q3 - Q1 = 8.25 - 6.75 = 1.5.
The IQR, 1.5, represents the range between the first quartile (Q1) and the third quartile (Q3). It describes how the middle 50% of the data is spread out. In this case, it means that half of the shoe sizes fall within a range of 1.5 units, ranging from 6.75 to 8.25. This measure of variability is useful because it summarizes the spread of the middle portion of the dataset into a single number, allowing for a concise understanding of how values are distributed within that range.
To find the IQR (Interquartile Range) for the given shoe sizes, we need to first determine the first quartile (Q1) and the third quartile (Q3) of the dataset.
To do this, we will arrange the values in ascending order:
5.5, 5.5, 6.0, 6.5, 6.5, 7.0, 7.0, 7.5, 7.5, 7.5, 8.0, 8.0, 8.0, 8.5, 8.5, 9.0
The dataset consists of 16 values, so the median value would be the average of the two middle terms: (7.5 + 7.5) / 2 = 7.5
Next, we need to find Q1 and Q3. Since there are an even number of values in our dataset, we will split the data in half at the median, and work with each half separately.
The lower half of the dataset is: 5.5, 5.5, 6.0, 6.5, 6.5, 7.0, 7.0
The upper half of the dataset is: 8.0, 8.0, 8.0, 8.5, 8.5, 9.0
To find Q1 (the first quartile), we need to find the median of the lower half:
(6.0 + 6.5) / 2 = 6.25
To find Q3 (the third quartile), we need to find the median of the upper half:
(8.0 + 8.5) / 2 = 8.25
Now, we can calculate the IQR by subtracting Q1 from Q3:
IQR = Q3 - Q1 = 8.25 - 6.25 = 2.0
The IQR for the given shoe sizes is 2.0.
The IQR is a measure of variability that describes how the values of a dataset, in this case, the shoe sizes, vary within a single number. It represents the range covered by the middle 50% of the data. In other words, it provides information about the spread of the dataset around the median. In this case, an IQR of 2.0 means that most of the shoe sizes fall within 2 sizes of the median (7.5), with the lower 25% ranging from 5.5 to 6.25 and the upper 25% ranging from 8.25 to 9.0.