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The expected value, often denoted as \( E(X) \), of a probability distribution is essentially a weighted average of all possible outcomes. It provides us with a central measure, giving us an idea of what the average result would be if we could repeat the random experiment infinite times.
This is calculated by multiplying each possible outcome by its probability and then summing all these products. It's like figuring out a destination, even if the routes there vary. In our context, two distributions may have the same expected value. Picture it as both paths leading to the same endpoint, even if they take different twists and turns to get there.
Understanding the expected value helps us anticipate the average outcome, but it doesn't tell us about the variability or possible deviation in outcomes from this central point.

While the expected value gives us a point to focus on, the standard deviation paints a richer picture by illustrating how far the values lie from this average point. It's a measure of dispersion in a probability distribution. Calculated as the square root of the variance, the standard deviation is expressed in the same units as the data, making it very intuitive to interpret.
When analyzing distributions, a smaller standard deviation indicates that the data points cluster tightly around the mean, while a larger one suggests that the data is more spread out. Even if two distributions share the same expected value, they may have vastly different standard deviations. Imagine two archers aiming for the center of a target: one archer may consistently hit closer to the bullseye (lower standard deviation), while the other might have a broader spread of arrows (higher standard deviation), though both have the same average score.

Variance is an essential concept underpinning standard deviation and offers insights into the distribution's variability. Given by the symbol \( \sigma^2 \), it measures how much the values in the distribution spread out from the expected value (or mean).
Calculating variance involves determining the average squared deviation from the mean. This squaring step is key because it amplifies larger deviations, providing a more pronounced picture of the distribution's dispersion.
While variance itself is not expressed in the original data units due to the squaring, it remains a fundamental step in calculating the standard deviation. In context, two distributions can have identical expected values and still have differing variances, highlighting disparities in how the data is spread. Variance helps identify these differences, even when the central tendencies remain aligned.