Question

Use the table to answer the question.

Value 4 5 6
Total Observations 1 3 1
Calculate the standard deviation of the data set. Round the answer to the nearest hundredth.

(1 point)
Responses

0.25
0.25

5.00
5.00

0.71
0.71

0.50

Answer 1

To calculate the standard deviation of the given data set, we can follow these steps:

1. Calculate the mean.
2. Calculate the variance.
3. 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** \(0.71\).