Question

Value −3
−2
−1
0 1
Total Observations 1 2 3 2 1
Calculate the standard deviation of the dataset. Round the answer to the nearest hundredth.

(1 point)
The standard deviation is
.

Answer 1

To calculate the standard deviation of the dataset given the values and their total observations, we can follow these steps:

1. **List the values and their frequencies**:
- Value -3: 1 observation
- Value -2: 2 observations
- Value -1: 3 observations
- Value 0: 2 observations
- Value 1: 1 observation

2. **Calculate the mean (\( \mu \)) of the dataset**:
\[
\text{Mean} = \frac{\sum (x_i \cdot f_i)}{N}
\]
where \( x_i \) is the value and \( f_i \) is the frequency, and \( N \) is the total number of observations.

- Let's first calculate the sum of all values weighted by their frequencies:
\[
\sum (x_i \cdot f_i) = (-3 \cdot 1) + (-2 \cdot 2) + (-1 \cdot 3) + (0 \cdot 2) + (1 \cdot 1)
\]
\[
= -3 - 4 - 3 + 0 + 1 = -9
\]

- The total number of observations \( N \) is:
\[
N = 1 + 2 + 3 + 2 + 1 = 9
\]

- So the mean is:
\[
\mu = \frac{-9}{9} = -1
\]

3. **Calculate the variance (\( \sigma^2 \))**:
\[
\sigma^2 = \frac{\sum f_i (x_i - \mu)^2}{N}
\]
- We will calculate \( (x_i - \mu)^2 \) for each value:
- For -3: \( (-3 - (-1))^2 = (-3 + 1)^2 = (-2)^2 = 4 \)
- For -2: \( (-2 - (-1))^2 = (-2 + 1)^2 = (-1)^2 = 1 \)
- For -1: \( (-1 - (-1))^2 = 0^2 = 0 \)
- For 0: \( (0 - (-1))^2 = (0 + 1)^2 = 1 \)
- For 1: \( (1 - (-1))^2 = (1 + 1)^2 = 2^2 = 4 \)

- Now calculate \( f_i (x_i - \mu)^2 \):
\[
\sum f_i (x_i - \mu)^2 = (1 \cdot 4) + (2 \cdot 1) + (3 \cdot 0) + (2 \cdot 1) + (1 \cdot 4)
\]
\[
= 4 + 2 + 0 + 2 + 4 = 12
\]

- Therefore, the variance is:
\[
\sigma^2 = \frac{12}{9} = \frac{4}{3} \approx 1.3333
\]

4. **Calculate the standard deviation (\( \sigma \))**:
\[
\sigma = \sqrt{\sigma^2} = \sqrt{\frac{4}{3}} \approx 1.1547
\]

5. **Round to the nearest hundredth**:
\[
\sigma \approx 1.15
\]

Thus, the standard deviation of the dataset is **1.15**.