E(X) = \sum x \cdot P(X=x)

Var(X) = \sum (x - E(X))^2 \cdot P(X=x)

SD(X) = \sqrt{Var(X)}

Multiply each value of X by its probability and sum the results.

Standard deviation is the square root of the variance.

It's the long-run average outcome if you repeat an experiment many times. It doesn't have to be a possible value of X.

Variance quantifies the spread of data points around the mean. A higher variance indicates greater variability.

Standard deviation measures the typical deviation of values from the mean. It's in the same units as the original data, making it easier to interpret than variance.

Probabilities act as weights, influencing how much each outcome contributes to the overall average. Outcomes with higher probabilities have a greater impact on the mean.

Squaring makes all deviations positive (so positive and negative deviations don't cancel out) and emphasizes larger deviations.

Variance: Average squared distance from the mean, units are squared. | Standard Deviation: Square root of variance, units are the same as the data.

Parameter: Describes a population. | Random Variable: Assigns numerical values to outcomes of random phenomena.

Discrete: Countable values (e.g., number of texts). | Continuous: Any value within a range (e.g., height).

Variance: Requires summing the squared differences from the mean, weighted by probabilities. | Standard Deviation: Simply taking the square root of the calculated variance.

Variance: Indicates the average squared deviation from the mean; harder to directly interpret. | Standard Deviation: Indicates the typical deviation from the mean in original units; easier to interpret in context.