How to Find the Standard Deviation of a Probability Distribution

The standard deviation is a key measure of variability in a probability distribution. To find it, follow these steps:

1. Determine the mean (μ): Begin by calculating the mean of the probability distribution. For a discrete distribution, this is done by summing the products of each outcome and its probability:
   μ = Σ (xᵢ * P(xᵢ)), where xᵢ represents each value and P(xᵢ) its probability.
2. Calculate the variance (σ²): The variance is the average of the squared differences from the mean. For a discrete probability distribution, it’s calculated as:
   σ² = Σ [ (xᵢ – μ)² * P(xᵢ) ]. This measures how spread out the values are around the mean.
3. Take the square root: Finally, the standard deviation (σ) is the square root of the variance:
   σ = √σ². This will give you a sense of the average distance of each outcome from the mean.

Understanding the standard deviation helps in assessing the risk and variability associated with random variables in probability distributions.