LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A normal distribution is the bell curve used to describe a random variable by two numbers: a mean, the average, and a variance, the spread. The square root of the variance is the standard deviation. About 68 percent of outcomes fall within one standard deviation of the mean, about 95 percent within two, and about 99.7 percent within three. A single exact value, on a continuous curve, has probability zero. Probabilities attach to ranges.
Why it matters
The curve is a model of where outcomes pile up. The center is the mean, usually written with the Greek letter mu. The width is the variance, sigma squared, or its square root sigma. Saying a variable is normal is saying those two numbers are enough to draw the whole bell. The empirical rule then turns the width into rough chances: one, two, and three standard deviations, at about 68, 95, and 99.7 percent.
The idea in plain words
Place the mean at the middle of the bell. Add and subtract one standard deviation and you have an interval that, under the model, catches about 68 percent of draws. Two standard deviations catch about 95 percent. Three catch about 99.7 percent. Farther from the center, the curve gets lower. Those thin ends are the tails. Outcomes there are possible and uncommon under this model.
The lecture’s numerical sketch sets a return’s mean at 0 and its variance at 0.2 squared, so the standard deviation is 0.2. Inside one standard deviation the model puts the return between negative 0.2 and positive 0.2, about 68 percent of the time. The lecture treats 0.2 as a very wide annual move, a gain or a loss of that size, and uses it only to make the interval easy to see. If the true returns are not normal, that 68 percent is a distorted picture of the chance.
The expectation of this sketched return is 0, because that is the mean that was assumed. The variance is 0.2 squared. Those two sentences are the whole model. They do not estimate a mean from data. They assume one.
The density, written as a lowercase f, gives the height of the curve at a point. For a normal law it is 1 over sigma times the square root of 2 pi, times an exponential that falls as the point moves away from the mean, in units of sigma. The height is a likelihood for values near that point, not the probability of the point. On a continuous line the chance of any one number is zero: one outcome over an infinite range of outcomes. Dividing by an arbitrarily large stand-in for infinity is the lecture’s quick reason. Whether a probability-zero event can still occur is a side question the lecture declines to chase.
The cumulative distribution function, the capital F, is the area under the density from negative infinity up to a chosen point. That area is a probability: the chance the outcome lands at or below the point. You rarely integrate it by hand. Numerical routines in Python, spreadsheets, and elsewhere return the area or the density height.
How to use it
Name mu and sigma before you talk about chances. Then read intervals off the empirical rule instead of quoting a single point. “Within one standard deviation” is a statement the model can answer. “Exactly 0.1” is not a probability, though the density at 0.1 is a well-defined height. Plug the point into the formula if you need that height, and use the cumulative function if you need the chance of landing below the point.
What usually goes wrong
The common error is to treat the bell as a fact about prices because it is easy to draw. The lecture’s own return example is labeled as a poor fit to real returns, even while the arithmetic goes through. Publishing the 68 percent band without that label overstates how often quiet years happen and understates the tails.
Questions people ask
What two numbers define the normal curve?
The mean and the variance. Sigma, the spread’s square root, is the unit the empirical rule adds to the mean and subtracts from it.
What are the 68, 95, and 99.7 percent bands?
Under a normal model, they are the chances of landing within one, two, and three standard deviations of the mean. In the return sketch, one standard deviation is 0.2, so the inner band runs from negative 0.2 to 0.2.
Why is the probability of one exact return zero?
The curve is continuous from minus infinity to infinity. One point is one outcome in an unbounded set of outcomes, so its probability is zero. Use an interval, through the cumulative function, when you need a chance.