Distributions
Values spread out in a pattern. When can one smooth curve stand in for that pattern, and what does it let you work out?
Where you are 8 inside
What this area is about
A distribution is the pattern of how values are spread. This area replaces a histogram with a smooth model, above all the normal curve N(\mu, \sigma), and reads shares of values as areas under it. Those areas hold only when the data, or the statistic, really is close to normal, so checking that is part of the work.
How these fit together
Definitions. Density curves are smooth outlines with total area 1, where area is proportion. The normal distribution is the symmetric bell pinned down by \mu and \sigma, and the standard normal is the one with \mu = 0 and \sigma = 1.
Procedures. z-scores and the empirical rule measure distance from the mean in standard deviations. Normal calculations go from a value to a share and back. Checking whether data look normal decides whether a normal model is fair.
Statistics as variables. Sampling distributions describe how \bar{x} or \hat{p} varies over random samples, and the central limit theorem says when the shape for \bar{x} is close to normal even though the population is not.
A path through it
The normal distribution and density curves come first, in either order. z-scores follow the normal distribution; the standard normal needs all three; normal calculations come last of these. Checking normality can come any time after the normal distribution, and sampling distributions need it. The central limit theorem and sampling distributions interleave: the theorem comes after sampling variability and before the sampling distribution of \bar{x}.
Next: confidence intervals and hypothesis testing.