A meteorologist recorded the following temperatures: {71,74,81,78,67,69,64}. Today's temperature is 67º. If today's temperature is included in the dataset, how does it change the interquartile range?
The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1) of a dataset.
First, we need to find the Q1 and Q3 values of the dataset {71,74,81,78,67,69,64}.
To find Q1:
1. Arrange the data in ascending order: {64,67,69,71,74,78,81}
2. Find the median of the lower half of the data: Q1 = (69+71) / 2 = 70
To find Q3:
1. Find the median of the upper half of the data: Q3 = (74+78) / 2 = 76
Now, calculate the IQR before including today's temperature:
IQR = Q3 - Q1 = 76 - 70 = 6
If today's temperature of 67º is included in the dataset, the new dataset will be {71,74,81,78,67,69,64,67}.
To find the new Q1 and Q3 values:
To find Q1:
1. Arrange the new data in ascending order: {64,67,67,69,71,74,78,81}
2. Find the median of the lower half of the new data: Q1 = (67+69) / 2 = 68
To find Q3:
1. Find the median of the upper half of the new data: Q3 = (74+78) / 2 = 76
Now, calculate the new IQR after including today's temperature:
IQR = Q3 - Q1 = 76 - 68 = 8
Therefore, including today's temperature of 67º in the dataset increases the interquartile range from 6 to 8.