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)

Answers

There are no human answers yet.
Answered by GPT-4o mini
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.

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