A statisticMathworldMathworldPlanetmath, or sample statistic, S is simply a function, usually real-valued, of a set of (sample) data or observations 𝑿=(X1,X2,,Xn): S=S(𝑿). More formally, let Ω be the sample space of the data 𝑿, then S is a function from Ω to some set T, usually a subset of k. The data 𝑿 is usually considered as a vector of iid random variablesMathworldPlanetmath Xi.


  1. 1.

    100 light bulbs out of 1,000,000 are tested for their functionality. Then the number n, of defective light bulbs in the 100 samples is a statistic. To see this, define, for each i from 1 to 100,

    xi={1if the event Xi={the ith light bulb is defective}0otherwise.

    Then n=i=1100xi, a function of the data. Similarly, the number of operating light bulbs is also a statistic if we switch the 1 and 0 in the above definitions for the xi’s. If we make all xi=1, then n is just the count of the observations, one of the simplest forms of sample statistics. If we make all xi=0, then n=0 is a statistic that is not at all useful.

  2. 2.

    Let w1,w2,,w20 be the weights of 20 students from a particular college. Then the averageMathworldPlanetmath weight defined by


    is a statistic. It is commonly called the sample mean. It is often used to estimate E[X], the expectation of a particular random variable, which, in this case, is the weight of a student in the college. Of course, other averages, such as medians, mode, trimmed mean, are also examples of (sample) statistics.

  3. 3.

    Using the same example as in (2), we can define


    This is also a statistic, for, after some substitution and rewriting,


    which is a function explicitly in terms of the wi’s. This statistic is known as the sample variance, which is a common estimate of Var[X], the varianceMathworldPlanetmath of the random variable X. Again, in this example, the X is the weight of a student in the college.

  4. 4.

    Again, borrowing from the same example above, we can simply order the weights of the 20 students in an ascending order, so we get a vector of 20 real numbers (w(1),w(2),,w(20)). This is also a statistic, called an order statisticMathworldPlanetmath. It is not real-valued and its range is a subset of 20.

  5. 5.

    Given a set of numeric observations X1,X2,,Xn, without knowing the distributionPlanetmathPlanetmath of these observations, one can define what is known as the empirical distribution function F^, which is a real-valued function, based on the observations. This is an example of a statistic whose range is a function space.


  • Any function of a statistic is again a statistic.

  • Since the underlying data is assumed to be random, a statistic is necessarily a random variable.

  • Although mostly real-valued, a statistic can be vector-valued, or even function-valued as we have seen in earlier examples.

Title statistic
Canonical name Statistic
Date of creation 2013-03-22 14:46:18
Last modified on 2013-03-22 14:46:18
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 62A01
Defines sample mean
Defines sample variance