To calculate the standard deviation of the dataset provided in the table, we first need to determine the mean (average), then calculate the variance, and finally take the square root of the variance to find the standard deviation.
The dataset values along with their frequencies are:
- Value: -3, Frequency: 1
- Value: -2, Frequency: 2
- Value: -1, Frequency: 3
- Value: 0, Frequency: 2
- Value: 1, Frequency: 1
Now, we can calculate the mean:
- Calculate the total sum of the values:
\[ \text{Total Sum} = (-3 \times 1) + (-2 \times 2) + (-1 \times 3) + (0 \times 2) + (1 \times 1) \]
\[ = -3 + (-4) + (-3) + 0 + 1 = -9 \]
- Calculate the total number of observations:
\[ \text{Total Observations} = 1 + 2 + 3 + 2 + 1 = 9 \]
- Calculate the mean:
\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Total Observations}} = \frac{-9}{9} = -1 \]
- Calculate the variance:
\[ \text{Variance} = \frac{\sum{(x_i - \text{Mean})^2 \times \text{Frequency}}}{\text{Total Observations}} \]
Where \(x_i\) are the values. We calculate each term:
- For \(x = -3\): \[ (-3 - (-1))^2 \times 1 = (-2)^2 \times 1 = 4 \times 1 = 4 \]
- For \(x = -2\): \[ (-2 - (-1))^2 \times 2 = (-1)^2 \times 2 = 1 \times 2 = 2 \]
- For \(x = -1\): \[ (-1 - (-1))^2 \times 3 = 0^2 \times 3 = 0 \times 3 = 0 \]
- For \(x = 0\): \[ (0 - (-1))^2 \times 2 = (1)^2 \times 2 = 1 \times 2 = 2 \]
- For \(x = 1\): \[ (1 - (-1))^2 \times 1 = (2)^2 \times 1 = 4 \times 1 = 4 \]
Now add these up to get the total:
\[ \sum{(x_i - \text{Mean})^2 \times \text{Frequency}} = 4 + 2 + 0 + 2 + 4 = 12 \]
Now, we can calculate the variance:
\[ \text{Variance} = \frac{12}{9} = \frac{4}{3} \approx 1.3333 \]
- Calculate the standard deviation:
\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{1.3333} \approx 1.1547 \]
Rounding to the nearest hundredth, we get:
\[ \text{Standard Deviation} \approx 1.15 \]
Final answer: The standard deviation is 1.15.