To find out how the inclusion of today's temperature (67º) affects the interquartile range (IQR), we first need to determine the IQR of the original dataset and then the IQR with the new value included.
Original dataset:
{71, 74, 81, 78, 67, 69, 64}
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Sort the original dataset:
{64, 67, 69, 71, 74, 78, 81} -
Find the quartiles:
- The median (Q2) is the middle value. With 7 values, the median is the 4th value (71).
- To find Q1 (the median of the lower half: {64, 67, 69}), we take the median of these values, which is 67.
- To find Q3 (the median of the upper half: {74, 78, 81}), we take the median of these values, which is 78.
- Calculate the interquartile range (IQR):
IQR = Q3 - Q1 = 78 - 67 = 11.
Now, including today's temperature (67º):
New dataset: {64, 67, 67, 69, 71, 74, 78, 81}
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Sort the new dataset:
{64, 67, 67, 69, 71, 74, 78, 81} -
Find the quartiles for the new dataset:
- The number of values is now 8, so the median (Q2) is the average of the 4th and 5th values: (69 + 71) / 2 = 70.
- To find Q1, we look at the lower half: {64, 67, 67, 69}. The median of this is (67 + 67) / 2 = 67.
- To find Q3, we look at the upper half: {71, 74, 78, 81}. The median of this is (74 + 78) / 2 = 76.
- Calculate the new interquartile range (IQR):
IQR = Q3 - Q1 = 76 - 67 = 9.
Conclusion:
The original IQR was 11, and the new IQR is 9. Thus, including today's temperature of 67º decreases the IQR from 11 to 9.
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