To calculate the standard deviation of the dataset, we first need to compute the mean, then the variance, and finally the standard deviation.
Step 1: Calculate the Mean
The first step is to calculate the mean of the dataset. We can find this by summing the product of each value and its corresponding number of occurrences (given by "Total Observations"), and then dividing by the total number of observations.
Values: -3, -2, -1, 0, 1
Observations: 1, 2, 3, 2, 1
Calculating the Total Sum:
\[ \text{Total sum} = (-3 \times 1) + (-2 \times 2) + (-1 \times 3) + (0 \times 2) + (1 \times 1) \]
\[ = -3 + (-4) + (-3) + 0 + 1 = -9 \]
Calculating the Total Number of Observations:
Total observations = \(1 + 2 + 3 + 2 + 1 = 9\)
Mean:
\[ \text{Mean} = \frac{\text{Total sum}}{\text{Total observations}} = \frac{-9}{9} = -1 \]
Step 2: Calculate the Variance
Variance is calculated using the formula:
\[ \text{Variance} = \frac{\sum (x_i - \text{mean})^2 \cdot f_i}{N} \]
Where:
- \(x_i\) is each value,
- \(f_i\) is the frequency (number of observations of \(x_i\)),
- \(N\) is the total number of observations
Calculating each term:
-
For \(-3\): \[ (-3 - (-1))^2 \cdot 1 = (-2)^2 \cdot 1 = 4 \cdot 1 = 4 \]
-
For \(-2\): \[ (-2 - (-1))^2 \cdot 2 = (-1)^2 \cdot 2 = 1 \cdot 2 = 2 \]
-
For \(-1\): \[ (-1 - (-1))^2 \cdot 3 = (0)^2 \cdot 3 = 0 \cdot 3 = 0 \]
-
For \(0\): \[ (0 - (-1))^2 \cdot 2 = (1)^2 \cdot 2 = 1 \cdot 2 = 2 \]
-
For \(1\): \[ (1 - (-1))^2 \cdot 1 = (2)^2 \cdot 1 = 4 \cdot 1 = 4 \]
Now summing these up:
\[ \text{Total} = 4 + 2 + 0 + 2 + 4 = 12 \]
Calculating the variance:
\[ \text{Variance} = \frac{12}{9} \approx 1.3333 \]
Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{1.3333} \approx 1.1547 \]
Step 4: Round to the Nearest Hundredth
Rounding \(1.1547\) to the nearest hundredth gives us:
\[ \text{Standard Deviation} \approx 1.15 \]
Final Answer
The standard deviation is 1.15.
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