To calculate the standard deviation of the dataset from the given frequency table, follow these steps:
- Find the mean of the dataset.
First, we'll calculate the total sum of the values and the total number of observations.
From the table:
- Value = -3 → Frequency = 1 → Contribution = -3 * 1 = -3
- Value = -2 → Frequency = 2 → Contribution = -2 * 2 = -4
- Value = -1 → Frequency = 3 → Contribution = -1 * 3 = -3
- Value = 0 → Frequency = 2 → Contribution = 0 * 2 = 0
- Value = 1 → Frequency = 1 → Contribution = 1 * 1 = 1
Now, we can calculate the total sum:
\[ \text{Total sum} = -3 - 4 - 3 + 0 + 1 = -9 \]
Next, calculate the total number of observations:
\[ \text{Total observations} = 1 + 2 + 3 + 2 + 1 = 9 \]
Now, calculate the mean:
\[ \text{Mean} = \frac{\text{Total sum}}{\text{Total observations}} = \frac{-9}{9} = -1 \]
- Calculate the variance.
Using the formula for variance, we need the squared deviations from the mean:
\[ \text{Variance} = \frac{\sum (x - \text{mean})^2 \cdot f}{N} \]
Where \( f \) is the frequency of occurrences of each \( x \), and \( N \) is the total number of observations.
Calculating each squared deviation:
- For \( x = -3 \): \((-3 - (-1))^2 \cdot 1 = ( -2)^2 \cdot 1 = 4 \cdot 1 = 4\)
- For \( x = -2 \): \((-2 - (-1))^2 \cdot 2 = (-1)^2 \cdot 2 = 1 \cdot 2 = 2\)
- For \( x = -1 \): \((-1 - (-1))^2 \cdot 3 = (0)^2 \cdot 3 = 0 \cdot 3 = 0\)
- For \( x = 0 \): \((0 - (-1))^2 \cdot 2 = (1)^2 \cdot 2 = 1 \cdot 2 = 2\)
- For \( x = 1 \): \((1 - (-1))^2 \cdot 1 = (2)^2 \cdot 1 = 4 \cdot 1 = 4\)
Now sum these values:
\[ \sum (x - \text{mean})^2 \cdot f = 4 + 2 + 0 + 2 + 4 = 12 \]
Now, calculate the variance:
\[ \text{Variance} = \frac{12}{9} \approx 1.33 \]
- Calculate the standard deviation.
The standard deviation is the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{1.33} \approx 1.15 \]
Since the problem states that it is not approximately 1.15, let's double-check the inputs or interpretation, although the calculated standard deviation is correct based on the calculations shown.
Final Answer:
**The standard deviation is approximately ** 1.15, rounding to the nearest hundredth gives 1.15. If there is any discrepancy on the expected answer, please double-check the dataset or context of the problem given.