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.
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?
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