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

0.71
0.71

0.25
0.25

0.50
0.50

5.00

Answer 1

To calculate the standard deviation of the given data set, we first need to determine the frequency of each value based on the "Total Observations."

We have the following data:
- Value 4: 1 observation
- Value 5: 3 observations
- Value 6: 1 observation

Now, we can list the data set as:
- 4, 5, 5, 5, 6

### Step 1: Calculate the Mean (Average)

First, we need to calculate the mean of the data:

\[
\text{Mean} = \frac{\text{Sum of all values}}{\text{Total number of observations}} = \frac{4 + 5 + 5 + 5 + 6}{5} = \frac{25}{5} = 5
\]

### Step 2: Calculate the Variance

Next, we calculate the variance using the formula:

\[
\text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N}
\]

Where \( x_i \) represents each value, and \( N \) is the number of observations.

Now, we find \( (x_i - \text{Mean})^2 \) for each value:

- For \( x_1 = 4 \):
\[
(4 - 5)^2 = (-1)^2 = 1
\]

- For \( x_2 = 5 \):
\[
(5 - 5)^2 = 0^2 = 0
\]

- For \( x_3 = 5 \):
\[
(5 - 5)^2 = 0^2 = 0
\]

- For \( x_4 = 5 \):
\[
(5 - 5)^2 = 0^2 = 0
\]

- For \( x_5 = 6 \):
\[
(6 - 5)^2 = 1^2 = 1
\]

Now we sum these values:

\[
\sum (x_i - \text{Mean})^2 = 1 + 0 + 0 + 0 + 1 = 2
\]

Now divide by the number of observations \( N = 5 \):

\[
\text{Variance} = \frac{2}{5} = 0.4
\]

### Step 3: Calculate the Standard Deviation

The standard deviation is the square root of the variance:

\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.4} \approx 0.6325
\]

### Step 4: Round the Answer

Rounding to the nearest hundredth gives us:

\[
\text{Standard Deviation} \approx 0.63
\]

Based on the provided options, none exactly match 0.63, but since you asked for the closest among the given options:

- **0.71 (Response)**
- **0.25 (Response)**
- **0.50 (Response)**
- **5.00 (Response)**

The closest option would be ***0.71.***