To calculate the standard deviation of the data set, we first need to determine the mean, variance, and then take the square root of the variance.
Step 1: Calculate the Mean
We have the following values and their respective observations:
- Value 4: 1 observation
- Value 5: 3 observations
- Value 6: 1 observation
First, calculate the total number of observations: \[ \text{Total Observations} = 1 + 3 + 1 = 5 \]
Now calculate the sum of the values: \[ \text{Sum} = (4 \times 1) + (5 \times 3) + (6 \times 1) = 4 + 15 + 6 = 25 \]
Next, find the mean (average): \[ \text{Mean} = \frac{\text{Sum}}{\text{Total Observations}} = \frac{25}{5} = 5 \]
Step 2: Calculate the Variance
Next, we calculate the variance using the formula: \[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2 \times \text{frequency}}{N} \]
Where \( x_i \) is each value, and \( N \) is the total number of observations (5 in this case).
Now, compute the squared differences and multiply by the frequency:
- For \( x_1 = 4 \): \[ (4 - 5)^2 \times 1 = (−1)^2 \times 1 = 1 \]
- For \( x_2 = 5 \): \[ (5 - 5)^2 \times 3 = (0)^2 \times 3 = 0 \]
- For \( x_3 = 6 \): \[ (6 - 5)^2 \times 1 = (1)^2 \times 1 = 1 \]
Now sum these values: \[ \text{Total} = 1 + 0 + 1 = 2 \]
Finally, calculate the variance: \[ \text{Variance} = \frac{2}{5} = 0.4 \]
Step 3: Calculate the Standard Deviation
The standard deviation is the square root of the variance: \[ \text{Standard Deviation} = \sqrt{0.4} \approx 0.6325 \]
Rounding to the nearest hundredth gives us: \[ \text{Standard Deviation} \approx 0.63 \]
Answer
0.63