To calculate the standard deviation of the given data set, we can follow these steps:
- Calculate the mean.
- Calculate the variance.
- Take the square root of the variance to find the standard deviation.
Step 1: Calculate the Mean
First, we will calculate the total number of observations and the sum of the values.
The values and their total observations are:
- Value 4: 1 observation
- Value 5: 3 observations
- Value 6: 1 observation
Now we calculate the total number of observations: \[ \text{Total Observations} = 1 + 3 + 1 = 5 \]
Next, we find the sum of the values: \[ \text{Sum} = (4 \times 1) + (5 \times 3) + (6 \times 1) = 4 + 15 + 6 = 25 \]
Now, we calculate the mean: \[ \text{Mean} = \frac{\text{Sum}}{\text{Total Observations}} = \frac{25}{5} = 5 \]
Step 2: Calculate the Variance
Now, we need to calculate the variance. The variance is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2 \cdot f_i}{N} \] where \(x_i\) is each value, \(\mu\) is the mean, \(f_i\) is the frequency of each value, and \(N\) is the total number of observations.
Now, we calculate each term:
- For \(x_1 = 4\) with \(f_1 = 1\): \((4 - 5)^2 \cdot 1 = (−1)^2 \cdot 1 = 1\)
- For \(x_2 = 5\) with \(f_2 = 3\): \((5 - 5)^2 \cdot 3 = (0)^2 \cdot 3 = 0\)
- For \(x_3 = 6\) with \(f_3 = 1\): \((6 - 5)^2 \cdot 1 = (1)^2 \cdot 1 = 1\)
Now, sum these values: \[ \sum (x_i - \mu)^2 \cdot f_i = 1 + 0 + 1 = 2 \] Now we plug into the variance formula: \[ \sigma^2 = \frac{2}{5} = 0.4 \]
Step 3: Calculate the Standard Deviation
Now, we take the square root of the variance: \[ \sigma = \sqrt{0.4} \approx 0.632455532 \] Rounding this to the nearest hundredth gives: \[ \sigma \approx 0.63 \]
However, let's recheck the multiple choice answers you provided:
- \(0.25\)
- \(5.00\)
- \(0.71\)
- \(0.50\)
None of these options directly match our calculation of \(0.63\), but it looks like the closest option available is \(0.71\).
Thus, the best response is \(0.71\).