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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. ^[1]

More precisely, in mathematical statistics, and in general usage, a statistic is defined as any measurable function of a data sample ^[2]. 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.

Statistics in action

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^[3] Fon [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 by simple random sampling (this means that each men in the population is equally likely to be selected in the sampling process) 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 $\scriptstyle x_{1},x_{2},\ldots ,x_{N}$ obtained is actually an instance or realization of a sequence of independent random variables $\scriptstyle X_{1},X_{2},\ldots ,X_{N}$ with each random variable $X_{i}$ being distributed identically according to the distribution of $X$ (that is, each $\scriptstyle X_{i}$ has the distribution F). Such a sequence $\scriptstyle X_{1},X_{2},\ldots ,X_{N}$ 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 $\scriptstyle y_{1},y_{2},\ldots ,y_{N}$ and Allen's be $\scriptstyle z_{1},z_{2},\ldots ,z_{N}$ , then both samples are also realizations of the i.i.d sequence $\scriptstyle X_{1},X_{2},\ldots ,X_{N}$ , just as the first sample $\scriptstyle x_{1},x_{2},\ldots ,x_{N}$ was.

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

$T\,=\,{\bar {x}}\,=\,{\frac {x_{1}+x_{2}+\ldots +x_{N}}{N}}$ . This statistic is known as the sample mean
$T\,=\,\sum _{i=1}^{N}(x_{i}-{\bar {x}})^{2}/N$ . 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.

References

↑ Guilford, J.P., Fruchter, B. (1978). Fundamental statistics in psychology and education. New York: McGraw-Hill.
↑ Shao, J. (2003). Mathematical Statistics (2 ed.). ser. Springer Texts in Statistics, New York: Springer-Verlag, p. 100.
↑ This is the case in non-parametric statistics. On the other hand, in parametric statistics the underlying distribution is assumed to be of some particular type, say a normal or exponential distribution, but with unknown parameters that are to be estimated.

[1] Guilford, J.P., Fruchter, B. (1978). Fundamental statistics in psychology and education. New York: McGraw-Hill.

[2] Shao, J. (2003). Mathematical Statistics (2 ed.). ser. Springer Texts in Statistics, New York: Springer-Verlag, p. 100.

[3] This is the case in non-parametric statistics. On the other hand, in parametric statistics the underlying distribution is assumed to be of some particular type, say a normal or exponential distribution, but with unknown parameters that are to be estimated.

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@@ Line 6: / Line 6: @@
 More precisely, in [[mathematical statistics]], and in general usage, a statistic is defined as any [[measurable  function]] of a data sample <ref>Shao, J. (2003). ''Mathematical Statistics'' (2 ed.). ser. Springer Texts in Statistics, New York: Springer-Verlag, p. 100.</ref>.  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==
+==Statistics in action==
-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''<ref>This is the case in [[non-parametric statistics]]. On the other hand, in [[parametric statistics]] the underlying distribution is assumed to be of some particular type, say a normal or exponential distribution, but with unknown parameters that are to be estimated.</ref> [[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 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]]<ref>This is the case in [[non-parametric statistics]]. On the other hand, in [[parametric statistics]] the underlying distribution is assumed to be of some particular type, say a normal or exponential distribution, but with unknown parameters that are to be estimated.</ref> ''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 <math> M_1,M_2,\ldots,M_N</math> have been randomly drawn whose heights are <math> x_1,x_2,\ldots,x_N</math>, respectively. An important yet subtle point to note here is that, due to random sampling, the data sample <math> x_1,x_2,\ldots,x_N</math> obtained is actually an ''instance'' or ''realization'' of a sequence of ''independent'' random variables <math> X_1,X_2,\ldots,X_N</math> with each random variable <math> X_i</math> being distributed ''identically'' according to the distribution of <math>X</math> (that is, each <math> X_i</math> has the distribution ''F''). Such a sequence <math> X_1,X_2,\ldots,X_N</math> 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 <math> y_1,y_2,\ldots,y_N</math> and Allen's be <math> z_1,z_2,\ldots,z_N</math>, then both samples are also realizations of the i.i.d sequence <math> X_1,X_2,\ldots,X_N</math>, just as the first sample <math> x_1,x_2,\ldots,x_N</math> was.
+Suppose that ''N'' men labeled <math> M_1,M_2,\ldots,M_N</math> have been randomly drawn by [[simple random sampling]] (this means that each men in the population is equally likely to be selected in the sampling process) whose heights are <math> x_1,x_2,\ldots,x_N</math>, respectively. An important yet subtle point to note here is that, due to random sampling, the data sample <math>\scriptstyle x_1,x_2,\ldots,x_N</math> obtained is actually an ''instance'' or ''realization'' of a sequence of ''independent'' random variables <math> \scriptstyle X_1,X_2,\ldots,X_N</math> with each random variable <math> X_i</math> being distributed ''identically'' according to the distribution of <math>X</math> (that is, each <math>\scriptstyle X_i</math> has the distribution ''F''). Such a sequence <math> \scriptstyle X_1,X_2,\ldots,X_N</math> 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 <math>\scriptstyle y_1,y_2,\ldots,y_N</math> and Allen's be <math>\scriptstyle z_1,z_2,\ldots,z_N</math>, then both samples are also realizations of the i.i.d sequence <math>\scriptstyle X_1,X_2,\ldots,X_N</math>, just as the first sample <math>\scriptstyle x_1,x_2,\ldots,x_N</math> was.
-From a data sample <math> x_1,x_2,\ldots,x_N</math> one may define a statistic ''T'' as <math> T=f(x_1,x_2,\ldots,x_N)</math> for some real-valued function ''f'' which is [[measurable function|measurable]] (here with respect to the [[Borel set]]s of <math> \mathbb{R}^N</math>). Two examples of commonly used statistics are:
+From a data sample <math>\scriptstyle x_1,x_2,\ldots,x_N</math> one may define a statistic ''T'' as <math>\scriptstyle T=f(x_1,x_2,\ldots,x_N)</math> for some real-valued function ''f'' which is [[measurable function|measurable]] (here with respect to the [[Borel set]]s of <math>\scriptstyle \mathbb{R}^N</math>). Two examples of commonly used statistics are:
 #<math> T\,=\,\bar{x}\,=\,\frac{x_1+x_2+\ldots+x_N}{N}</math>. This statistic is known as the ''sample mean''

Statistics theory: Difference between revisions

Revision as of 02:43, 24 May 2008

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Statistics theory: Difference between revisions

Revision as of 02:43, 24 May 2008

Statistics in action

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References

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