To calculate the standard deviation of the dataset, we need to follow these steps:
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Calculate the Mean: First, we compute the mean (average) of the dataset.
-
Calculate the Variance: Next, we use the mean to calculate the variance.
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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.