Normal Probabilities and Standardizing

Every normal distribution has its own mean and standard deviation, which means there are infinitely many normal curves, each with values measured on a different scale. Standardizing fixes this by converting any value x into a z-score: z = (x - mean) / standard deviation. Subtracting the mean re-centers the distribution at 0. Dividing by the standard deviation rescales it so one unit equals one standard deviation. The result is always the same curve, the standard normal distribution, with mean 0 and SD 1. Because every normal variable can be squashed onto this same scale, a single table of areas under the standard normal curve works for any normal problem you will ever meet.

Once you have a z-score, you look up the area to its left in the standard normal table, and that area is the probability of getting a value less than x. For a value above x, subtract that area from 1. For a value between two numbers, standardize both, look up both areas, and subtract the smaller from the larger. This connects directly to why normal models matter at all in this unit: sample proportions and sample means later turn out to follow approximately normal distributions, and the exact same standardizing move is what lets you compute probabilities for those sampling distributions too. Standardizing here is the technique you will reuse constantly for the rest of the course.