Of the following lists of numbers , which has the largest standard deviation ?

Options

A. 2,7,15
B. 3,7,14
C. 5,7,12
D. 10,11,12

Explanation

To determine which of the given lists of numbers has the largest standard deviation, we will follow these steps for each list:

1. **Calculate the Mean (μ)**: The mean is calculated by summing all the numbers in the list and dividing by the total count of numbers.

2. **Calculate the Squared Differences**: For each number in the list, subtract the mean and square the result.

3. **Sum the Squared Differences**: Add all the squared differences together.

4. **Calculate the Variance**: Divide the sum of squared differences by the number of values in the list.

5. **Calculate the Standard Deviation**: Take the square root of the variance.

Now, let's apply these steps to each of the four sets of numbers.

### Set 1: {2, 7, 15}
1. **Calculate the Mean (μ)**:
\[ \mu = \frac{2 + 7 + 15}{3} = \frac{24}{3} = 8 \]

2. **Calculate the Squared Differences**:
- \( (2 - 8)^2 = (-6)^2 = 36 \)
- \( (7 - 8)^2 = (-1)^2 = 1 \)
- \( (15 - 8)^2 = (7)^2 = 49 \)

3. **Sum the Squared Differences**:
\[ 36 + 1 + 49 = 86 \]

4. **Calculate the Variance**:
\[ \text{Variance} = \frac{86}{3} \approx 28.67 \]

5. **Calculate the Standard Deviation**:
\[ \sigma = \sqrt{28.67} \approx 5.36 \]

### Set 2: {3, 7, 14}
1. **Calculate the Mean (μ)**:
\[ \mu = \frac{3 + 7 + 14}{3} = \frac{24}{3} = 8 \]

2. **Calculate the Squared Differences**:
- \( (3 - 8)^2 = (-5)^2 = 25 \)
- \( (7 - 8)^2 = (-1)^2 = 1 \)
- \( (14 - 8)^2 = (6)^2 = 36 \)

3. **Sum the Squared Differences**:
\[ 25 + 1 + 36 = 62 \]

4. **Calculate the Variance**:
\[ \text{Variance} = \frac{62}{3} \approx 20.67 \]

5. **Calculate the Standard Deviation**:
\[ \sigma = \sqrt{20.67} \approx 4.55 \]

### Set 3: {5, 7, 12}
1. **Calculate the Mean (μ)**:
\[ \mu = \frac{5 + 7 + 12}{3} = \frac{24}{3} = 8 \]

2. **Calculate the Squared Differences**:
- \( (5 - 8)^2 = (-3)^2 = 9 \)
- \( (7 - 8)^2 = (-1)^2 = 1 \)
- \( (12 - 8)^2 = (4)^2 = 16 \)

3. **Sum the Squared Differences**:
\[ 9 + 1 + 16 = 26 \]

4. **Calculate the Variance**:
\[ \text{Variance} = \frac{26}{3} \approx 8.67 \]

5. **Calculate the Standard Deviation**:
\[ \sigma = \sqrt{8.67} \approx 2.94 \]

### Set 4: {10, 11, 12}
1. **Calculate the Mean (μ)**:
\[ \mu = \frac{10 + 11 + 12}{3} = \frac{33}{3} = 11 \]

2. **Calculate the Squared Differences**:
- \( (10 - 11)^2 = (-1)^2 = 1 \)
- \( (11 - 11)^2 = (0)^2 = 0 \)
- \( (12 - 11)^2 = (1)^2 = 1 \)

3. **Sum the Squared Differences**:
\[ 1 + 0 + 1 = 2 \]

4. **Calculate the Variance**:
\[ \text{Variance} = \frac{2}{3} \approx 0.67 \]

5. **Calculate the Standard Deviation**:
\[ \sigma = \sqrt{0.67} \approx 0.82 \]

### Summary of Standard Deviations:
- Set 1: \( \sigma \approx 5.36 \)
- Set 2: \( \sigma \approx 4.55 \)
- Set 3: \( \sigma \approx 2.94 \)
- Set 4: \( \sigma \approx 0.82 \)

### Conclusion:
The set with the largest standard deviation is **Set 1: {2, 7, 15}**.