Question

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?

Answer 1

To find the new interquartile range (IQR) after including today's temperature of 67º into the dataset, we first need to incorporate it into the original set of temperatures.

The original temperatures are:
\[
\{71, 74, 81, 78, 67, 69, 64\}
\]
Including today’s temperature:
\[
\{71, 74, 81, 78, 67, 69, 64, 67\}
\]

Now, we will arrange this dataset in ascending order:
\[
\{64, 67, 67, 69, 71, 74, 78, 81\}
\]

Next, we will find the first quartile (Q1) and the third quartile (Q3).

1. **Finding Q1**: The first quartile is the median of the first half of the data. The first half of our ordered data is:
\[
\{64, 67, 67, 69\}
\]
The median of this set (Q1) is:
\[
\text{Q1} = \frac{67 + 67}{2} = 67
\]

2. **Finding Q3**: The third quartile is the median of the second half of the data. The second half is:
\[
\{71, 74, 78, 81\}
\]
The median of this set (Q3) is:
\[
\text{Q3} = \frac{74 + 78}{2} = 76
\]

3. **Calculating IQR**: The interquartile range (IQR) is found by subtracting Q1 from Q3:
\[
\text{IQR} = Q3 - Q1 = 76 - 67 = 9
\]

We also need to find the original IQR before including today's temperature. The original temperatures in ascending order are:
\[
\{64, 67, 69, 71, 74, 78, 81\}
\]
- For the original dataset:
- The first half is:
\[
\{64, 67, 69\}
\]
So, Q1 is:
\[
\text{Q1} = 67
\]

- The second half is:
\[
\{71, 74, 78, 81\}
\]
So, Q3 is:
\[
\text{Q3} = \frac{74 + 78}{2} = 76
\]

- Original IQR:
\[
\text{IQR}_{\text{original}} = Q3 - Q1 = 76 - 67 = 9
\]

4. **Conclusion**: The inclusion of today's temperature (67º) into the dataset does not change the IQR because the quartiles remain the same.

**Therefore, the IQR remains 9 after including today's temperature.**