To calculate the standard deviation of the dataset given the values and their total observations, we can follow these steps:
-
List the values and their frequencies:
- Value -3: 1 observation
- Value -2: 2 observations
- Value -1: 3 observations
- Value 0: 2 observations
- Value 1: 1 observation
-
Calculate the mean (\( \mu \)) of the dataset: \[ \text{Mean} = \frac{\sum (x_i \cdot f_i)}{N} \] where \( x_i \) is the value and \( f_i \) is the frequency, and \( N \) is the total number of observations.
-
Let's first calculate the sum of all values weighted by their frequencies: \[ \sum (x_i \cdot f_i) = (-3 \cdot 1) + (-2 \cdot 2) + (-1 \cdot 3) + (0 \cdot 2) + (1 \cdot 1) \] \[ = -3 - 4 - 3 + 0 + 1 = -9 \]
-
The total number of observations \( N \) is: \[ N = 1 + 2 + 3 + 2 + 1 = 9 \]
-
So the mean is: \[ \mu = \frac{-9}{9} = -1 \]
-
-
Calculate the variance (\( \sigma^2 \)): \[ \sigma^2 = \frac{\sum f_i (x_i - \mu)^2}{N} \]
-
We will calculate \( (x_i - \mu)^2 \) for each value:
- For -3: \( (-3 - (-1))^2 = (-3 + 1)^2 = (-2)^2 = 4 \)
- For -2: \( (-2 - (-1))^2 = (-2 + 1)^2 = (-1)^2 = 1 \)
- For -1: \( (-1 - (-1))^2 = 0^2 = 0 \)
- For 0: \( (0 - (-1))^2 = (0 + 1)^2 = 1 \)
- For 1: \( (1 - (-1))^2 = (1 + 1)^2 = 2^2 = 4 \)
-
Now calculate \( f_i (x_i - \mu)^2 \): \[ \sum f_i (x_i - \mu)^2 = (1 \cdot 4) + (2 \cdot 1) + (3 \cdot 0) + (2 \cdot 1) + (1 \cdot 4) \] \[ = 4 + 2 + 0 + 2 + 4 = 12 \]
-
Therefore, the variance is: \[ \sigma^2 = \frac{12}{9} = \frac{4}{3} \approx 1.3333 \]
-
-
Calculate the standard deviation (\( \sigma \)): \[ \sigma = \sqrt{\sigma^2} = \sqrt{\frac{4}{3}} \approx 1.1547 \]
-
Round to the nearest hundredth: \[ \sigma \approx 1.15 \]
Thus, the standard deviation of the dataset is 1.15.