Calculating Interquartile Range (IQR) for a Data Set
The interquartile range (IQR) is the difference between the first quartile (Q1) and the third quartile (Q3) in a set of data. It measures the middle 50% of the data or the spread of the middle half of the data.
First, we need to find Q1 and Q3 for the given data set:
\[ 1, 2, 4, 4, 4, 4, 4, 8, 8 \]
Since there are 9 data points, the median (second quartile, Q2) will be the middle value, which is the fifth value when the data is ordered - in this case, 4.
To find Q1, we take the lower half of the data not including the median:
\[ 1, 2, 4, 4 \]
The lower half has an even number of data points (4), so Q1 will be the average of the middle two numbers in this subset:
\[ Q1 = \frac{(2 + 4)}{2} = \frac{6}{2} = 3 \]
Next, find Q3 by taking the upper half of the data not including the median:
\[ 4, 4, 8, 8 \]
The upper half also has an even number of data points (4), so Q3 will be the average of the middle two numbers in this subset:
\[ Q3 = \frac{(4 + 8)}{2} = \frac{12}{2} = 6 \]
Now let's calculate the IQR:
\[ IQR = Q3 - Q1 = 6 - 3 = 3 \]
The interquartile range for the given data set is 3.
