Statistics theory

Statistics refers primarily to a branch of mathematics that specializes in enumeration, or counted, data and their relation to measured data. It may also refer to a fact of classification, which is the chief source of all statistics, and has a relationship to psychometric applications in the social sciences.

An individual statistic refers to a derived numerical value, such as a mean, a coefficient of correlation, or some other single concept of descriptive statistics . It may also refer to an idea associated with an average, such as a median, or standard deviation, or some value computed from a set of data.

More precisely, in mathematical statistics, and in general usage, a statistic is defined as any measurable function of a data sample . A data sample is described by instances of a random variable of interest, such as a height, weight, polling results, test performance, etc., obtained by random sampling of a population.

Illustration of concept

Suppose one wishes to embark on a quantitative study of the height of adult males in some country C. How would one go about doing this and how can the data be summarized? In statistics, the approach taken is to assume/model the quantity of in interest, i.e., "height of adult men from the country C" as a random variable X, say, taking on values in [0,5] (measured in metres) and distributed according to some unknown probability distribution F on [0,5] . One important theme studied in statistics is to develop theoretically sound methods (firmly grounded in probability theory) to learn something about the postulated random variable X and also its distribution F by collecting samples, for this particular example, of the height of a number of men randomly drawn from the adult male population of C. This is referred to as statistical inference.

Suppose that N men labeled $M_{1},M_{2},\ldots ,M_{N}$ have been randomly drawn whose heights are $x_{1},x_{2},\ldots ,x_{N}$, respectively. An important yet subtle point to note here is that, due to random sampling, the data sample $x_{1},x_{2},\ldots ,x_{N}$ obtained is actually an instance or realization of a sequence of independent random variables $X_{1},X_{2},\ldots ,X_{N}$ with each random variable $X_{i}$ being distributed identically according to the distribution of $X$ (that is, each $X_{i}$ has the distribution F). Such a sequence $X_{1},X_{2},\ldots ,X_{N}$ is referred to in statistics as independent and identically distributed (i.i.d) random variables. To further clarify this point, suppose that there are two other investigators, Tim and Allen, who are also interested in the same quantitative study and they in turn also randomly sample N adult males from the population of C. Let Tim's height data sample be $y_{1},y_{2},\ldots ,y_{N}$ and Allen's be $z_{1},z_{2},\ldots ,z_{N}$, then both samples are also realizations of the i.i.d sequence $X_{1},X_{2},\ldots ,X_{N}$, just as the first sample $x_{1},x_{2},\ldots ,x_{N}$ was.

From a data sample $x_{1},x_{2},\ldots ,x_{N}$ one may define a statistic T as $T=f(x_{1},x_{2},\ldots ,x_{N})$ for some real-valued function f which is measurable (here with respect to the Borel sets of $\mathbb {R} ^{N}$). Two examples of commonly used statistics are:

1. $T\,=\,{\bar {x}}\,=\,{\frac {x_{1}+x_{2}+\ldots +x_{N}}{N}}$. This statistic is known as the sample mean
2. $T\,=\,\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2}/N$. This statistic is known as the sample variance. Often the alternative definition $T\,=\,{\frac {1}{N-1}}\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2}$ of sample variance is preferred because it is an unbiased estimator of the variance of X, while the former is a biased estimator.

See also

• Coefficient of correlation
• Data
• Descriptive statistics
• Kappa statistic
• Mathematics
• Mean
• Median
• Mode
• Psychometrics
• Recursive partitioning
• Sensitivity and specificity
• Set
• Standard deviation