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Using the following set of data, calculate the lower quartile, the upper quartile, and the interquartile range.
20, 22, 25, 28, 29, 30, 32, 33, 34
Be sure to show your work for finding:
the lower quartile
the upper quartile
the interquartile range
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To calculate the lower quartile, upper quartile, and the interquartile range, we need to first understand what quartiles are.
Quartiles divide a set of data into four equal parts. The lower quartile (Q1) marks the boundary between the lowest 25% of the data and the remaining 75% of the data. The upper quartile (Q3) marks the boundary between the lowest 75% of the data and the highest 25% of the data. The interquartile range (IQR) represents the range of values that lie between the lower quartile and the upper quartile.
Now let's find the lower quartile, upper quartile, and interquartile range for the given set of data: 20, 22, 25, 28, 29, 30, 32, 33, 34.
Step 1: Arrange the data in ascending order:
20, 22, 25, 28, 29, 30, 32, 33, 34
Step 2: Find the median of the data set:
The median is the middle value of the data set. Since we have an odd number of data points (9), the median is the value in the middle position, which is the 5th value: 29.
Step 3: Find the lower quartile (Q1):
Since Q1 divides the lower 25% of the data, we need to find the median of the lower half of the data set. The lower half of the data set consists of the first four values: 20, 22, 25, 28. Arrange these values in ascending order and find the median, which is the 2nd value: 22. So, Q1 = 22.
Step 4: Find the upper quartile (Q3):
Since Q3 divides the upper 25% of the data, we need to find the median of the upper half of the data set. The upper half of the data set consists of the last four values: 30, 32, 33, 34. Arrange these values in ascending order and find the median, which is the 2nd value: 32. So, Q3 = 32.
Step 5: Calculate the interquartile range (IQR):
IQR is simply the difference between the upper quartile (Q3) and the lower quartile (Q1). So, IQR = Q3 - Q1 = 32 - 22 = 10.
Therefore, the lower quartile (Q1) is 22, the upper quartile (Q3) is 32, and the interquartile range (IQR) is 10.