To calculate the standard deviation of the given data set, we first need to determine the frequency of each value based on the "Total Observations."
We have the following data:
- Value 4: 1 observation
- Value 5: 3 observations
- Value 6: 1 observation
Now, we can list the data set as:
- 4, 5, 5, 5, 6
Step 1: Calculate the Mean (Average)
First, we need to calculate the mean of the data:
\[ \text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of observations}} = \frac{4 + 5 + 5 + 5 + 6}{5} = \frac{25}{5} = 5 \]
Step 2: Calculate the Variance
Next, we calculate the variance using the formula:
\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N} \]
Where \( x_i \) represents each value, and \( N \) is the number of observations.
Now, we find \( (x_i - \text{Mean})^2 \) for each value:
-
For \( x_1 = 4 \): \[ (4 - 5)^2 = (-1)^2 = 1 \]
-
For \( x_2 = 5 \): \[ (5 - 5)^2 = 0^2 = 0 \]
-
For \( x_3 = 5 \): \[ (5 - 5)^2 = 0^2 = 0 \]
-
For \( x_4 = 5 \): \[ (5 - 5)^2 = 0^2 = 0 \]
-
For \( x_5 = 6 \): \[ (6 - 5)^2 = 1^2 = 1 \]
Now we sum these values:
\[ \sum (x_i - \text{Mean})^2 = 1 + 0 + 0 + 0 + 1 = 2 \]
Now divide by the number of observations \( N = 5 \):
\[ \text{Variance} = \frac{2}{5} = 0.4 \]
Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.4} \approx 0.6325 \]
Step 4: Round the Answer
Rounding to the nearest hundredth gives us:
\[ \text{Standard Deviation} \approx 0.63 \]
Based on the provided options, none exactly match 0.63, but since you asked for the closest among the given options:
- 0.71 (Response)
- 0.25 (Response)
- 0.50 (Response)
- 5.00 (Response)
The closest option would be 0.71.