To find the interquartile range (IQR) of the given data set, we first need to identify the first (Q1) and third (Q3) quartiles. The IQR is defined as:
\[ \text{IQR} = Q3 - Q1 \]
Steps to calculate IQR:
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Sort the Data: The provided data set is already sorted in ascending order.
\[ 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 \]
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Find the Median (Q2):
- The median is the middle value of the data set. For 21 data points (which is an odd number), the median is the value at position \( (n + 1) / 2 \).
- Median position: \( (21 + 1) / 2 = 11 \)
The median (Q2) is the 11th value:
\[ Q2 = 50 \]
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Find Q1 (first quartile):
- Q1 is the median of the lower half of the data (the first 10 values).
- Lower half: \( 0, 5, 10, 15, 20, 25, 30, 35, 40, 45 \)
- This also has an even number of values (10), so the Q1 is the average of the 5th and 6th values:
\[ Q1 = (20 + 25) / 2 = 22.5 \]
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Find Q3 (third quartile):
- Q3 is the median of the upper half of the data (the last 10 values).
- Upper half: \( 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 \)
- This also has an even number of values. Q3 is the average of the 5th and 6th values:
\[ Q3 = (75 + 80) / 2 = 77.5 \]
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Calculate the IQR: Now, substitute Q1 and Q3 into the IQR formula:
\[ \text{IQR} = Q3 - Q1 = 77.5 - 22.5 = 55 \]
Therefore, the interquartile range (IQR) of the data set is:
\[ \text{IQR} = 55 \]
Adjusting the Data Set:
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To make the IQR smaller:
- Decrease the spread of the middle 50% of the data. For example, you could bring the upper data points closer to the lower ones or eliminate some of the outliers.
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To make the IQR larger:
- Increase the spread of the middle 50% of the data. For example, you could increase the values of some high data points or add higher values far from the current data set.
Feel free to enter "55" for your response to the IQR.