Question

Use the table to answer the question.

Value −3
−2
−1
0
1
Total Observations 1 2 3 2 1
Calculate the standard deviation of the dataset. Round the answer to the nearest hundredth.

(1 point)
The standard deviation is
.

Answer 1

its not 1.15

Answer 2

To calculate the standard deviation of the dataset, follow these steps:

1. **Find the Mean (Average)**:
First, we need to calculate the mean of the dataset.

The values and their frequencies are:
- Value = -3, Frequency = 1
- Value = -2, Frequency = 2
- Value = -1, Frequency = 3
- Value = 0, Frequency = 2
- Value = 1, Frequency = 1

The total number of observations (N) = 1 + 2 + 3 + 2 + 1 = 9.

Now we find the sum of the values weighted by their frequencies:
\[
\text{Sum} = (-3 \times 1) + (-2 \times 2) + (-1 \times 3) + (0 \times 2) + (1 \times 1)
= -3 - 4 - 3 + 0 + 1 = -9
\]

Now calculate the mean:
\[
\mu = \frac{\text{Sum}}{N} = \frac{-9}{9} = -1
\]

2. **Calculate the Variance**:
The variance is calculated as follows:
\[
\sigma^2 = \frac{\sum (x_i - \mu)^2 \cdot f_i}{N}
\]
Where \(f_i\) is the frequency of each value \(x_i\).

Calculate \((x_i - \mu)^2\) for each value:
- For x = -3: \((-3 - (-1))^2 = (-2)^2 = 4\), Contribution to variance = \(4 \cdot 1 = 4\)
- For x = -2: \((-2 - (-1))^2 = (-1)^2 = 1\), Contribution to variance = \(1 \cdot 2 = 2\)
- For x = -1: \((-1 - (-1))^2 = (0)^2 = 0\), Contribution to variance = \(0 \cdot 3 = 0\)
- For x = 0: \((0 - (-1))^2 = (1)^2 = 1\), Contribution to variance = \(1 \cdot 2 = 2\)
- For x = 1: \((1 - (-1))^2 = (2)^2 = 4\), Contribution to variance = \(4 \cdot 1 = 4\)

Now add these contributions:
\[
\text{Total} = 4 + 2 + 0 + 2 + 4 = 12
\]

Now calculate the variance:
\[
\sigma^2 = \frac{12}{9} = \frac{4}{3} \approx 1.3333
\]

3. **Calculate the Standard Deviation**:
The standard deviation is the square root of the variance:
\[
\sigma = \sqrt{1.3333} \approx 1.1547
\]

4. **Round to the nearest hundredth**:
Finally, rounding this value to the nearest hundredth gives us:
\[
\sigma \approx 1.15
\]

Thus, the standard deviation of the dataset is **1.15**.