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

0.50
0.50

5.00
5.00

0.71

Answer 1

To calculate the standard deviation of the given data set, we first need to identify the values along with their frequencies.

The data set is as follows:

- Value 4 occurs 1 time.
- Value 5 occurs 3 times.
- Value 6 occurs 1 time.

We can summarize this data set as:
- \(4, 5, 5, 5, 6\) (expanding the values based on their frequency).

### Step 1: Calculate the mean (average).
\[
\text{Mean} = \frac{\text{Total Sum of Values}}{\text{Total Count of Values}}
\]
\[
\text{Total Sum} = (4 \times 1) + (5 \times 3) + (6 \times 1) = 4 + 15 + 6 = 25
\]
\[
\text{Total Count} = 1 + 3 + 1 = 5
\]
\[
\text{Mean} = \frac{25}{5} = 5
\]

### Step 2: Calculate the variance.
The variance is calculated using the formula:

\[
\text{Variance} = \frac{\sum (x_i - \mu)^2}{N}
\]
where \(x_i\) is each value, \(\mu\) is the mean, and \(N\) is the number of observations.

Calculating the squared differences from the mean:
- For 4: \((4 - 5)^2 = (-1)^2 = 1\)
- For 5: \((5 - 5)^2 = (0)^2 = 0\) (3 times, so add 0 three times)
- For 6: \((6 - 5)^2 = (1)^2 = 1\)

Thus, we have:
\[
\sum (x_i - \mu)^2 = 1 + 0 + 0 + 0 + 1 = 2
\]
Now, we can compute the variance:
\[
\text{Variance} = \frac{2}{5} = 0.4
\]

### Step 3: Calculate the standard deviation.
Standard deviation is the square root of variance:
\[
\text{Standard Deviation} = \sqrt{0.4} \approx 0.6325
\]

Rounding this to the nearest hundredth gives:
\[
\text{Standard Deviation} \approx 0.63
\]

Among the provided options, **0.71** is the closest representation, but a value of **0.63** does not appear in your responses. Based on the computation, the nearest hundredth is indeed **0.63** which may imply a slight error in available options.

The answer is not explicitly listed among your provided options, and the calculated result is approximately **0.63**.