Questions: Consider the following data: x -4 -3 -2 -1 0 P(X=x) 0.3 0.1 0.2 0.1 0.3 Step 3 of 5: Find the standard deviation, Round your answer to one decimal place.

Solution Steps

The mean \( \mu \) of the distribution is calculated using the formula:

\[ \mu = \sum (x \cdot P(X=x)) = -4 \times 0.3 + -3 \times 0.1 + -2 \times 0.2 + -1 \times 0.1 + 0 \times 0.3 \]

Calculating this gives:

\[ \mu = -1.2 - 0.3 - 0.4 - 0.1 + 0 = -2.0 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \sum ((x - \mu)^2 \cdot P(X=x)) \]

Substituting the values, we have:

\[ \sigma^2 = (-4 - -2.0)^2 \times 0.3 + (-3 - -2.0)^2 \times 0.1 + (-2 - -2.0)^2 \times 0.2 + (-1 - -2.0)^2 \times 0.1 + (0 - -2.0)^2 \times 0.3 \]

Calculating each term:

\[ = (2^2 \times 0.3) + (1^2 \times 0.1) + (0^2 \times 0.2) + (1^2 \times 0.1) + (2^2 \times 0.3) \] \[ = (4 \times 0.3) + (1 \times 0.1) + (0 \times 0.2) + (1 \times 0.1) + (4 \times 0.3) \] \[ = 1.2 + 0.1 + 0 + 0.1 + 1.2 = 2.6 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{2.6} \approx 1.6 \]

Final Answer