The Mean Is the Balance Point, the Variance Is the Spread
Last lesson we said a distribution is the whole shape of how probability is spread. But carrying a whole distribution around is heavy. In practice we summarize a distribution with just two numbers: roughly where it's gathered, and how widely it's spread. These two are the expected value and the variance. The expected value E[X] is the center of the distribution — more precisely, its center of mass. View the distribution as the weights of bars, and the expected value is exactly the fulcrum where those bars balance. It pins "on average, values come out near here" to a single point. The variance is how far, on average, the values scatter from that center — the size of the spread. For even at the same center, whether things huddle close together or spread out wide is a completely different situation. In this lesson you'll handle the expected value as a balance point and the variance as the average of squared distances, learning in your hands what the two summarizing numbers of a distribution mean.
First, feel the expected value as a scale. On a beam I've placed a few weights of different masses — the big ones are heavy, the small ones light. The fulcrum where this beam balances exactly, that point is the center of mass. Drag the weights around. The fulcrum instantly follows to rebalance. You can see it lean toward the heavy weights. Send a light weight far away and the fulcrum barely moves, but shift a heavy weight a little and it follows a lot. This is exactly the expected value. In probability, a weight's "mass" is the probability of that value, and a weight's "position" is the value itself. The more probable a value, the harder it pulls the average toward it. So the expected value is "the center measured with probability as weight," the single point where the distribution balances.
How do you compute that balance point? A weighted average. Not the plain average where you just add the values and divide by the count. You multiply each value by the probability of that value and add them up. As a formula, E[X] = Σ x·p(x). Take a die: a fair die has each of 1 through 6 with equal probability 16, so the average is 3.5. But with a "loaded die," weighted to one side, the story changes. Drag the bar heights to raise the probability of a particular face. Once that face comes up often, the mean E[X] gets pulled toward it, because a more probable value contributes more to the sum. Each value gets a "say" equal to its probability and joins a tug-of-war over the position of the mean. Turn block 1's center of mass into numbers and it is exactly this weighted sum: mass = probability, position = value, balance point = Σ x·p(x).
Once you know the center, you measure the spread. That's the variance. The key question is "how far, on average, do the values sit from the mean?" You can't just average the distances. Values above the mean are positive and those below are negative, so they cancel to zero. So you square the distance. Squaring kills the sign and makes everything positive. Drag the points apart. For each point a square is drawn whose side is the distance from that point to the mean. The area of that square is exactly the distance squared. The variance is the average of all these square areas. When the points huddle near the mean the squares are small and the variance is small; spread them wide and the squares grow and the variance shoots up. Because you squared the distance, a point farther out is punished far more heavily. So the variance is a strict measure of spread, sensitive to extreme values.
Variance has one inconvenience. Because you squared the distance, the units get squared too. If you measured height in cm, the units of variance are cm² — an area. "The spread of heights is 25 square centimeters" just doesn't land. So you undo the square with a square root. That's the standard deviation. Standard deviation = √variance, and the units return to cm, the same units as X. "The spread of heights is 5 cm" is immediately understood. On the normal curve, drag σ. The standard deviation σ is the natural "width" of the bell — the distance from the central peak out to where the shoulders bend. When σ is small the curve turns narrow and sharp; when large, wide and flat. To sum up: the variance σ² is a quantity in squared units that is nice to compute with, while the standard deviation σ is the width we use in everyday life, letting us feel how spread out things really are in the original units.
Finally, see how these two numbers change when you act on the values. Turning X into aX+b is very common, as in a unit conversion or a calibration — Celsius to Fahrenheit, meters to feet. Move a and b with the sliders. First, b adds the same value to everyone, so it slides the whole distribution sideways with its shape intact. The mean therefore moves by b, but the spread doesn't change at all, because everything moved together so the distances between them stay the same. The a, on the other hand, stretches or shrinks. The mean becomes a times as large, and the width of the curve stretches by a as well. So the standard deviation scales by |a|, and the variance, being a squared-distance quantity, scales by a². As formulas, E[aX+b]=aE[X]+b and Var(aX+b)=a²Var(X). Adding moves only the center and leaves the spread; multiplying stretches both center and spread, but the variance by the square. Thanks to this rule, when you change units or standardize data, you can work out in your head right away how the statistics will follow.