Question

value: 40 50 60
total observations: 1 2 1


Calculate the standard deviation of the dataset. Round the answer to the nearest
hundredth.

(1 point)
Responses

0.71
0.71

7.07
7.07

0.82
0.82

8.16

Answer 1

To calculate the standard deviation of the dataset with the given values and observations, we will follow these steps:

1. Calculate the mean (average).
2. Calculate the variance.
3. Take the square root of the variance to find the standard deviation.

The given values and their corresponding observations are:
- Value 40 → 1 observation
- Value 50 → 2 observations
- Value 60 → 1 observation

### Step 1: Calculate the Mean

First, we find the total number of observations (N):
\[
N = 1 + 2 + 1 = 4
\]

Next, calculate the sum of the weighted values:
\[
\text{Sum} = (40 \times 1) + (50 \times 2) + (60 \times 1) = 40 + 100 + 60 = 200
\]

Now, we can find the mean \(\mu\):
\[
\mu = \frac{\text{Sum}}{N} = \frac{200}{4} = 50
\]

### Step 2: Calculate the Variance

Now we will calculate the variance \(\sigma^2\):
\[
\sigma^2 = \frac{\sum (x_i - \mu)^2 \cdot f_i}{N}
\]
where \(x_i\) are the values, \(\mu\) is the mean, and \(f_i\) is the frequency (number of observations).

Calculating each term:
- For value 40:
\[
(40 - 50)^2 \times 1 = (-10)^2 \times 1 = 100
\]
- For value 50:
\[
(50 - 50)^2 \times 2 = (0)^2 \times 2 = 0
\]
- For value 60:
\[
(60 - 50)^2 \times 1 = (10)^2 \times 1 = 100
\]

Now sum these up:
\[
\sum (x_i - \mu)^2 \cdot f_i = 100 + 0 + 100 = 200
\]

Now we can calculate the variance:
\[
\sigma^2 = \frac{200}{4} = 50
\]

### Step 3: Calculate the Standard Deviation

Finally, we take the square root of the variance:
\[
\sigma = \sqrt{50} \approx 7.07
\]

Rounding to the nearest hundredth, the standard deviation is approximately:
\[
7.07
\]

### Answer

The correct answer is **7.07**.