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
(its not 1.15)

Answer 1

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

1. **Calculate the Mean**: First, we compute the mean (average) of the dataset.

2. **Calculate the Variance**: Next, we use the mean to calculate the variance.

3. **Calculate the Standard Deviation**: Lastly, the standard deviation is the square root of the variance.

### Step 1: Calculate the Mean

From the given table:

- Values: \(-3\), \(-2\), \(-1\), \(0\), \(1\)
- Total Observations:
- \(1\) observation of \(-3\)
- \(2\) observations of \(-2\)
- \(3\) observations of \(-1\)
- \(2\) observations of \(0\)
- \(1\) observation of \(1\)

Now, we can calculate the total sum and total count:

\[
\text{Total Sum} = (-3 \times 1) + (-2 \times 2) + (-1 \times 3) + (0 \times 2) + (1 \times 1)
\]
\[
= -3 + (-4) + (-3) + 0 + 1 = -9
\]

The total number of observations is:
\[
1 + 2 + 3 + 2 + 1 = 9
\]

Now, we calculate the mean \( \mu \):
\[
\mu = \frac{\text{Total Sum}}{\text{Total Count}} = \frac{-9}{9} = -1
\]

### Step 2: Calculate the Variance

Now we calculate each value's squared deviation from the mean:
- For \(-3\): \((-3 - (-1))^2 = (-2)^2 = 4\)
- For \(-2\): \((-2 - (-1))^2 = (-1)^2 = 1\) (occurs 2 times)
- For \(-1\): \((-1 - (-1))^2 = (0)^2 = 0\) (occurs 3 times)
- For \(0\): \((0 - (-1))^2 = (1)^2 = 1\) (occurs 2 times)
- For \(1\): \((1 - (-1))^2 = (2)^2 = 4\)

Now, we multiply these squared deviations by their respective frequencies and sum them up:

\[
\text{Variance} = \frac{1}{N} \sum (f_i \cdot (x_i - \mu)^2)
\]

Where:
- \(f_i\) is the frequency,
- \(x_i\) is the value,
- \(\mu\) is the mean,
- \(N\) is the total number of observations (which is 9 here).

Calculating the weighted squared deviations:

\[
= (1 \times 4) + (2 \times 1) + (3 \times 0) + (2 \times 1) + (1 \times 4)
\]

Calculating each part:
- \((1 \times 4) = 4\)
- \((2 \times 1) = 2\)
- \((3 \times 0) = 0\)
- \((2 \times 1) = 2\)
- \((1 \times 4) = 4\)

Adding these gives:

\[
4 + 2 + 0 + 2 + 4 = 12
\]

Now we calculate the variance:

\[
\text{Variance} = \frac{12}{9} \approx 1.3333
\]

### Step 3: Calculate the Standard Deviation

Taking the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{1.3333} \approx 1.1547
\]

### Final Answer

Rounded to the nearest hundredth, the standard deviation is:

\[
\text{Standard Deviation} \approx 1.15
\]

However, since you mentioned that it is not 1.15, I will provide a concise answer, but the calculations suggest \( \mathbf{1.15} \) for the standard deviation based on the provided data. Please confirm if there is any error or misinterpretation.