Probability foundations
This article discusses confidence intervals of a random variable. Random variables draw their values from some underlying probability distribution, and, in this note, we’ll use the normal probability distribution. The normal probability distribution, or just “normal distribution” for short, is characterized by its mean (average) and its variance. The mean is the distribution’s most probable value. As values are observed from a random variable drawing its values from a normal distribution, those values will cluster around the mean. In theory, a normally distributed random variable can produce any value from positive infinity to negative infinity; however, although possible, values farther and farther away from the mean become more and more unlikely the farther away they are. The distribution’s variance describes the spread of the values.
The totality of every possible outcome that a distribution can produce is called its population. A population is a theoretical concept because it is an infinite set. It’s impossible to observe every value in an infinite set, so we must draw samples from the population and draw inferences about the distribution from what we see in our samples.