To calculate the range and the interquartile range (IQR) for the given dataset, we can follow these steps:
Dataset: 21, 24, 25, 27, 30, 32, 33, 34, 34, 36, 41, 44
Step 1: Calculate the Range
The range is calculated using the formula:
\[ \text{Range} = \text{Maximum Value} - \text{Minimum Value} \]
- Maximum Value = 44
- Minimum Value = 21
\[ \text{Range} = 44 - 21 = 23 \]
Step 2: Calculate the IQR
The interquartile range (IQR) is calculated as follows:
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Find the median (Q2): Since there are 12 data points, the median will be the average of the 6th and 7th values.
The sorted dataset is: 21, 24, 25, 27, 30, 32, 33, 34, 34, 36, 41, 44.
The 6th and 7th values are 32 and 33.
\[ Q2 = \frac{32 + 33}{2} = \frac{65}{2} = 32.5 \]
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Find Q1 (the first quartile): Q1 is the median of the first half of the dataset (lower half):
The lower half is: 21, 24, 25, 27, 30, 32
The median (Q1) of this group is the average of the 3rd and 4th values (25 and 27):
\[ Q1 = \frac{25 + 27}{2} = \frac{52}{2} = 26 \]
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Find Q3 (the third quartile): Q3 is the median of the second half of the dataset (upper half):
The upper half is: 33, 34, 34, 36, 41, 44
The median (Q3) of this group is the average of the 3rd and 4th values (34 and 36):
\[ Q3 = \frac{34 + 36}{2} = \frac{70}{2} = 35 \]
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Calculate IQR: The IQR is then calculated as:
\[ \text{IQR} = Q3 - Q1 \] \[ \text{IQR} = 35 - 26 = 9 \]
Summary of the Measures of Variability
- Range: 23
- IQR: 9