To calculate the standard deviation of the given data set, we first need to understand the data in the table:
- Value 4: 1 observation
- Value 5: 3 observations
- Value 6: 1 observation
This means the data set can be expanded as follows:
\[ 4, 5, 5, 5, 6 \]
Next, we will find the mean (average) of the data set.
Step 1: Calculate the Mean
\[ \text{Mean} = \frac{\text{Sum of observations}}{\text{Total number of observations}} = \frac{4 + 5 + 5 + 5 + 6}{5} = \frac{25}{5} = 5 \]
Step 2: Calculate the Variance
Variance is calculated as follows:
\[ \text{Variance} = \frac{\sum (x_i - \text{mean})^2}{N} \] Where \( N \) is the number of observations and \( x_i \) are the values.
Now we calculate each squared deviation from the mean:
- For \( 4 \): \( (4 - 5)^2 = (-1)^2 = 1 \)
- For \( 5 \): \( (5 - 5)^2 = (0)^2 = 0 \) (This occurs 3 times)
- For \( 6 \): \( (6 - 5)^2 = (1)^2 = 1 \)
Now we sum these squared deviations:
\[ 1 + 0 + 0 + 0 + 1 = 2 \]
Next, we divide by the number of observations (which is 5):
\[ \text{Variance} = \frac{2}{5} = 0.4 \]
Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance:
\[ \text{Standard Deviation} = \sqrt{0.4} \approx 0.632455532 \approx 0.63 \]
Rounding
Rounding to the nearest hundredth, we get:
\[ \text{Standard Deviation} \approx 0.63 \]
Conclusion
The closest answer to the calculated standard deviation is not presented in your options. However, the correct value of the standard deviation rounded to the nearest hundredth is 0.63. Please check the response options again.