To calculate the standard deviation of the given data, we will first need to find the mean (average) of the values and then use it to find the variance and subsequently the standard deviation.
Step 1: Calculate the Mean
The values and their corresponding frequencies are:
- Value -3 has 1 observation.
- Value -2 has 2 observations.
- Value -1 has 3 observations.
- Value 0 has 2 observations.
- Value 1 has 1 observation.
To find the mean, we first calculate the total sum of the values multiplied by their frequencies and divide by the total number of observations.
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:
\[ \text{Total Observations} = 1 + 2 + 3 + 2 + 1 = 9 \]
Now, calculate the mean:
\[ \text{Mean} = \frac{\text{Total Sum}}{\text{Total Observations}} = \frac{-9}{9} = -1 \]
Step 2: Calculate the Variance
The variance is calculated using the formula:
\[ \text{Variance} = \frac{\sum (x_i - \text{mean})^2 \cdot f_i}{N} \]
Where:
- \(x_i\) = value
- \(f_i\) = frequency of each value
- \(N\) = total number of observations
Now, we calculate \( (x_i - \text{mean})^2 \cdot f_i \) for each value:
-
For \(x_1 = -3\): \[ (-3 - (-1))^2 \cdot 1 = (-3 + 1)^2 \cdot 1 = (-2)^2 \cdot 1 = 4 \cdot 1 = 4 \]
-
For \(x_2 = -2\): \[ (-2 - (-1))^2 \cdot 2 = (-2 + 1)^2 \cdot 2 = (-1)^2 \cdot 2 = 1 \cdot 2 = 2 \]
-
For \(x_3 = -1\): \[ (-1 - (-1))^2 \cdot 3 = (0)^2 \cdot 3 = 0 \cdot 3 = 0 \]
-
For \(x_4 = 0\): \[ (0 - (-1))^2 \cdot 2 = (0 + 1)^2 \cdot 2 = (1)^2 \cdot 2 = 1 \cdot 2 = 2 \]
-
For \(x_5 = 1\): \[ (1 - (-1))^2 \cdot 1 = (1 + 1)^2 \cdot 1 = (2)^2 \cdot 1 = 4 \cdot 1 = 4 \]
Adding all these values together for the variance:
\[ \sum (x_i - \text{mean})^2 \cdot f_i = 4 + 2 + 0 + 2 + 4 = 12 \]
Now we calculate the variance:
\[ \text{Variance} = \frac{12}{9} \approx 1.3333 \]
Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{1.3333} \approx 1.1547 \]
Final Answer
Calculating to the nearest hundredth:
The standard deviation is approximately 1.15.