To determine which pairs of numbers will increase the standard deviation of the set `{9, 12, 15, 18, 21}`, we will analyze each pair step by step.
### Step 1: Calculate the Mean and Variance of the Original Set
1. **Calculate the Mean:**
\[
\text{Mean} = \frac{9 + 12 + 15 + 18 + 21}{5} = \frac{75}{5} = 15
\]
2. **Calculate the Variance:**
\[
\text{Variance} = \frac{(9-15)^2 + (12-15)^2 + (15-15)^2 + (18-15)^2 + (21-15)^2}{5}
\]
\[
= \frac{(-6)^2 + (-3)^2 + (0)^2 + (3)^2 + (6)^2}{5}
\]
\[
= \frac{36 + 9 + 0 + 9 + 36}{5} = \frac{90}{5} = 18
\]
3. **Calculate the Standard Deviation:**
\[
\text{Standard Deviation} = \sqrt{18} = 3\sqrt{2}
\]
### Step 2: Analyze Each Pair of Numbers
#### Pair I: (14, 16)
1. **New Set:**
\[
\{9, 12, 14, 15, 16, 18, 21\}
\]
2. **Calculate the New Mean:**
\[
\text{New Mean} = \frac{9 + 12 + 14 + 15 + 16 + 18 + 21}{7} = \frac{105}{7} = 15
\]
3. **Calculate the New Variance:**
\[
\text{New Variance} = \frac{(9-15)^2 + (12-15)^2 + (14-15)^2 + (15-15)^2 + (16-15)^2 + (18-15)^2 + (21-15)^2}{7}
\]
\[
= \frac{36 + 9 + 1 + 0 + 1 + 9 + 36}{7} = \frac{92}{7} \approx 13.14
\]
4. **New Standard Deviation:**
\[
\text{New Standard Deviation} = \sqrt{\frac{92}{7}} \approx 3.62
\]
- The standard deviation has decreased.
#### Pair II: (9, 21)
1. **New Set:**
\[
\{9, 9, 12, 15, 18, 21, 21\}
\]
2. **Calculate the New Mean:**
\[
\text{New Mean} = \frac{9 + 9 + 12 + 15 + 18 + 21 + 21}{7} = \frac{105}{7} = 15
\]
3. **Calculate the New Variance:**
\[
\text{New Variance} = \frac{(9-15)^2 + (9-15)^2 + (12-15)^2 + (15-15)^2 + (18-15)^2 + (21-15)^2 + (21-15)^2}{7}
\]
\[
= \frac{36 + 36 + 9 + 0 + 9 + 36 + 36}{7} = \frac{162}{7} \approx 23.14
\]
4. **New Standard Deviation:**
\[
\text{New Standard Deviation} = \sqrt{\frac{162}{7}} \approx 4.80
\]
- The standard deviation has increased.
#### Pair III: (15, 100)
1. **New Set:**
\[
\{9, 12, 15, 15, 18, 21, 100\}
\]
2. **Calculate the New Mean:**
\[
\text{New Mean} = \frac{9 + 12 + 15 + 15 + 18 + 21 + 100}{7} = \frac{190}{7} \approx 27.14
\]
3. **Calculate the New Variance:**
\[
\text{New Variance} = \frac{(9-27.14)^2 + (12-27.14)^2 + (15-27.14)^2 + (15-27.14)^2 + (18-27.14)^2 + (21-27.14)^2 + (100-27.14)^2}{7}
\]
- The calculations will show a significant increase in variance due to the outlier (100).
4. **New Standard Deviation:**
- The standard deviation will definitely increase significantly due to the outlier.
### Conclusion
- **Pair I (14, 16)**: Does not increase standard deviation.
- **Pair II (9, 21)**: Increases standard deviation.
- **Pair III (15, 100)**: Increases standard deviation.
### Final Answer
The pairs that increase the standard deviation of the set are **II and III**.
