# Random variable vs Statistic? [duplicate]

What’s the difference between a random variable and a statistic?

It seems that formally, a random variable is simply any real-valued function (and its domain is a set that we call a “sample space”).

But isn’t the exact same also true of a statistic? That is, isn’t a statistic also simply any real-valued function (whose domain is a set called a “sample space”)?

Example. We roll a die. The sample space is $$\Omega=\{1,2,3,4,5,6\}$$.

Let $$T$$ be double the value of the die roll: Formally, $$T:\Omega\rightarrow \mathbb R$$ is the function defined by $$T(x)=2x$$.

Is $$T$$ a random variable, a statistic, or both?

1. A precise definition of a random variable
2. A precise definition of a statistic
3. An answer to the question in my example: “Is $$T$$ a random variable, a statistic, or both?”

A statistic is a function defined over one or more random variables.

So yes, a statistic is a random variable, and follows a distribution.

Another answer gave the example of the mean of a bunch of iid normal random variables.
$$X_1,…,X_n\sim N(\mu,\sigma^2)$$

The mean is a statistic because it is a function defined over random variables
$$\bar{X}= g(X_1, X_2 … X_n) = \frac{1}{n} \sum_{i=1}^n X_i$$

There is one condition however, which is that a statistic cannot explicitly depend on unknown parameters. Take the following definition of $$g$$ :

$$g(X_1) = \frac{X_1 – \mu}{\sigma}$$

While $$g$$ here is a function of a random variable, and it follows a standard normal distribution, it’s not a statistic (unless $$\mu$$ and $$\sigma$$ are known).

For a more detailed explanation see pg. 122 of this.