Statistics theory

Statistics is a mathematical approach to describe something, predict events, or analyze the relationship between things. Statistical analysis can, for example, describe the average income of a population, test whether two groups have the same average income, or analyze factors that might explain the income level for a particular group.

Statistics uses mathematics to describe, predict, explain or analyze things. Use of mathematics makes it possible to test relationships between two or more groups or to test how observations compare to a prediction. Some of the statistical concepts include mean (average), standard deviation (how concentrated or spread out things are), and correlation (how related two different variables are). These concepts are further explained in this article.

A current example of the use of statistics is trying to figure out how to get the economy growing again. We build a 'model' of the economy, adding in or taking out many different factors or variables, and seeing which variable, or set of variables, results in growth. The model can be used to "predict" growth. However, building a model is not a simple task. Different people make different assumptions about the model and about the different variables to be used in the model. Some may believe, for example, that job creation will result in increased income and consequently increased spending. Thus, they will argue, the role of the government is to create jobs. On the other hand, others may believe that reducing taxes will result in more wealth going to people, who will then spend more. Thus, they believe, the job of the government is to reduce taxes. The point is that statistics is a tool to help build models, analyze relationships, and often predict outcomes. However, behind most statistical analysis is a set of assumptions and beliefs that people bring to the modeling, and those assumptions and beliefs are, very often, not statistical. The first step in using statistics, then, is to make clear the assumptions and beliefs that are being used, so that others can understand the starting points of the analysis.

A second major point to keep in mind is that the usefulness of statistical analysis depends crucially upon the validity of the methods by which statistics are collected, and serious errors have resulted from lack of professional attention to that factor.

The theory of 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.

Simple illustration

Suppose one wishes to study the height of adult males in some country C. How should one go about doing this and how can the data be summarized? In statistics, the approach taken is to model the quantity of 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.

Suppose that N men labeled $\scriptstyle M_{1},M_{2},\ldots ,M_{N}$ have been randomly drawn by simple random sampling (this means that each man in the population is equally likely to be selected in the sampling process) whose heights are $\scriptstyle x_{1},x_{2},\ldots ,x_{N}$, 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:

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.

Transforming data

Statisticians may transform data by taking the logarithm, square root, reciprocal, or other function if the data does not fit a normal distribution. Data needs to be transformed back to its original form in order to present confidence intervals.

Summary statistics

• Descriptive statistics

Measurements of central tendency

• Mean In general understanding, the mean is the average. Suppose there are five people. The people have, respectively, 1 TV in their house, 4 TVs, 2 TVs, no TVs and 3 TVs. You would say that the 'mean' number of TVs in the house is 2 (1+4+2+0+3)/5=2.
• Median The median is the point at which half are above and half are below. In the above example, the median is also 2, because in this group of 5 people, 2 people have more than 2 TVs in their house and 2 people have fewer than 2 TVs in their house.

Measurements of variation

• Standard deviation (SD) is a measure of variation or scatter. The standard deviation does not change with sample size.
• Variance is the square of the standard deviation:

$s^{2}$

• Standard error of the mean (SEM) measures the how accurately you know the mean of a population and is always smaller than the SD. The SEM becomes smaller as the sample size increases. The sample standard devision (S) and SEM are related by:

$SE_{\bar {x}}\ ={\frac {s}{\sqrt {n}}}$

• 95% confidence interval is + 1.96 * standard error.

Problems in reporting of statistics

In medicine, common problems in the reporting and usage of statistics have been inventoried. These problems tend to exaggerated treatment differences.