Calculate Variance & Standard Deviation of Normally Distributed Series - Step-by-Step

To calculate the variance and standard deviation, we can use the following formulas:

Variance = Σ[(x - E(R))^2 * P(x)] Standard Deviation = √Variance

Where: x = return E(R) = expected return P(x) = probability of return x

Let's calculate the variance first:

Variance = (6 - 9)^2 * 0.06 + (7 - 9)^2 * 0.10 + (8 - 9)^2 * 0.17 + (9 - 9)^2 * 0.34 + (10 - 9)^2 * 0.17 + (11 - 9)^2 * 0.10 + (12 - 9)^2 * 0.06

Variance = 3.24 * 0.06 + 4.41 * 0.10 + 1.69 * 0.17 + 0 * 0.34 + 1 * 0.17 + 4.41 * 0.10 + 9 * 0.06

Variance = 0.1944 + 0.441 + 0.2873 + 0 + 0.17 + 0.441 + 0.54

Variance = 2.0737

Now, let's calculate the standard deviation:

Standard Deviation = √Variance

Standard Deviation = √2.0737

Standard Deviation ≈ 1.439 (rounded to 3 decimal places)

Therefore, the variance of the series is 2.0737 and the standard deviation is approximately 1.439.