Random Variable vs Statistic

A random variable is a rule that attaches a number to every outcome of a chance process. The rule itself is not random. The randomness sits in which outcome occurs, and once the outcome is known the variable has one definite value. Notation carries that distinction: a capital letter, usually $X$, names the rule, and a lowercase $x$ names one value the rule can produce. That is why probabilities are written $P(X = x)$, read as the probability that $X$ takes the particular value $x$.

Flip a fair coin twice. The sample space is HH, HT, TH, TT, each outcome with probability 0.25. Let $X$ count the heads. Then HH maps to 2, HT and TH both map to 1, and TT maps to 0. Four outcomes collapse into three values, and the probabilities follow the map: $P(X = 1) = 0.50$, while $P(X = 0)$ and $P(X = 2)$ are each 0.25.

"Blood type is a random variable" is the error worth naming. The output has to be a number, because everything done with a random variable afterwards is arithmetic. Blood type is a categorical outcome, so it does not qualify. Build a number out of it and you are fine: the count of type O donors among 20 arrivals is a random variable, and counts can be averaged. The related slip is writing $P(x = 2)$ in lowercase, which asks about a value instead of about the variable.

Random variables split into two kinds according to the values they can take. Discrete ones have values you can list, and continuous ones take any value in an interval and get their probabilities from area under a density curve. That split decides everything downstream, including whether you add probabilities or integrate areas to answer a question.

Random variables are introduced at topic 2.8 of Unit 2, and their summaries, the mean $\mu_X$ (mu-X) and the standard deviation $\sigma_X$ (sigma-X), at topic 2.9, Parameters of Random Variables.

A statistic is a number computed from sample data alone. Because its value depends on which individuals were drawn, it changes from one sample to the next, and that movement is what separates it from a parameter, which stays put. Statistics take Roman letters or hats: $\bar{x}$ (x-bar) for the sample mean, $s$ for the sample standard deviation, $\hat{p}$ (p-hat) for the sample proportion, $r$ for the sample correlation. Each is paired with the parameter it estimates, $\mu$, $\sigma$, $p$, and $\rho$ in turn.

Suppose 60 percent of a large population would vote yes, so $p = 0.60$. Draw a random sample of 80 and find 52 yes votes, giving $\hat{p} = \frac{52}{80} = 0.65$. Draw a second sample of 80 and find 44, giving $\hat{p} = \frac{44}{80} = 0.55$. Two different statistics, one unchanged parameter, and no mistake in either sample. That spread is sampling variability, and the sampling distribution of $\hat{p}$ describes it.

"The sample proportion is 0.65, so the population proportion is 0.65" is the error to name out loud. A statistic is an estimate and it is almost never exactly right, so the honest version attaches an interval or a standard error to that 0.65. The written form of the same mistake is putting $p = 0.65$ where $\hat{p} = 0.65$ belongs. That collapse wrecks everything downstream, because a test compares an observed $\hat{p}$ against a hypothesized $p$, and no comparison is left once both symbols name one quantity.

A statistic has two lives. Before the sample is drawn it is a random variable with a distribution of its own, which is what makes phrases like the mean and standard deviation of $\hat{p}$ meaningful. Once the sample is in hand it is a single number.

Two boundaries. A value computed from an entire population is not a statistic, however it was calculated, and a statistic from a badly chosen sample is still a statistic, just a poor estimator. Topic 3.1 grades estimators on exactly those two axes: whether they are centered on the parameter, and how much they vary.