In applying statistics to a scientific, industrial, or societal problem, one begins with a process or population to be studied. This might be a population of people in a country, of crystal grains in a rock, or of goods manufactured by a particular factory during a given period. It may instead be a process observed at various times; data collected about this kind of "population" constitute what is called a time series.
For practical reasons, rather than compiling data about an entire population, one usually instead studies a chosen subset of the population, called a sample. Data are collected about the sample in an observational or experimental setting. The data are then subjected to statistical analysis, which serves two related purposes: description and inference.
Descriptive statistics can be used to summarize the data, either numerically or graphically, to describe the sample. Basic examples of numerical descriptors include the mean and standard deviation. Graphical summarizations include various kinds of charts and graphs.
Inferential statistics is used to model patterns in the data, accounting for randomness and drawing inferences about the larger population. These inferences may take the form of answers to yes/no questions (hypothesis testing), estimates of numerical characteristics (estimation), forecasting of future observations, descriptions of association (correlation), or modeling of relationships (regression). Other modeling techniques include ANOVA, time series, and data mining.
The concept of correlation is particularly noteworthy. Statistical analysis of a data set may reveal that two variables (that is, two properties of the population under consideration) tend to vary together, as if they are connected. For example, a study of annual income and age of death among people might find that poor people tend to have shorter lives than affluent people. The two variables are said to be correlated. However, one cannot immediately infer the existence of a causal relationship between the two variables; see correlation does not imply causation. The correlated phenomena could be caused by a third, previously unconsidered phenomenon, called a lurking variable.
If the sample is representative of the population, then inferences and conclusions made from the sample can be extended to the population as a whole. A major problem lies in determining the extent to which the chosen sample is representative. Statistics offers methods to estimate and correct for randomness in the sample and in the data collection procedure, as well as methods for designing robust experiments in the first place; see experimental design.
The fundamental mathematical concept employed in understanding such randomness is probability. Mathematical statistics (also called statistical theory) is the branch of applied mathematics that uses probability theory and analysis to examine the theoretical basis of statistics.
The use of any statistical method is valid only when the system or population under consideration satisfies the basic mathematical assumptions of the method. Misuse of statistics can produce subtle but serious errors in description and interpretation — subtle in that even experienced professionals sometimes make such errors, and serious in that they may affect social policy, medical practice and the reliability of structures such as bridges and nuclear power plants.
Even when statistics is correctly applied, the results can be difficult to interpret for a non-expert. For example, the statistical significance of a trend in the data — which measures the extent to which the trend could be caused by random variation in the sample — may not agree with one's intuitive sense of its significance. The set of basic statistical skills (and skepticism) needed by people to deal with information in their everyday lives is referred to as statistical literacy.
Statistical methods
[edit] Experimental and observational studies
A common goal for a statistical research project is to investigate causality, and in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on response or dependent variables. There are two major types of causal statistical studies, experimental studies and observational studies. In both types of studies, the effect of differences of an independent variable (or variables) on the behavior of the dependent variable are observed. The difference between the two types is in how the study is actually conducted. Each can be very effective.
The researchers were interested in whether increased illumination would increase the productivity of the assembly line workers. The researchers first measured productivity in the plant then modified the illumination in an area of the plant to see if changes in illumination would affect productivity. As it turns out, productivity improved under all the experimental conditions (see Hawthorne effect). However, the study is today heavily criticized for errors in experimental procedures, specifically the lack of a control group and blindedness.
An example of an observational study is a study which explores the correlation between smoking and lung cancer. This type of study typically uses a survey to collect observations about the area of interest and then perform statistical analysis. In this case, the researchers would collect observations of both smokers and non-smokers and then look at the number of cases of lung cancer in each group.
The basic steps for an experiment are to:
plan the research including determining information sources, research subject selection, and ethical considerations for the proposed research and method,
design the experiment concentrating on the system model and the interaction of independent and dependent variables,
summarize a collection of observations to feature their commonality by suppressing details (descriptive statistics),
reach consensus about what the observations tell us about the world we observe (statistical inference),
document and present the results of the study.
Levels of measurement
See: Stanley Stevens' "Scales of measurement" (1946): nominal, ordinal, interval, ratio
There are four types of measurements or measurement scales used in statistics. The four types or levels of measurement (nominal, ordinal, interval, and ratio) have different degrees of usefulness in statistical research. Ratio measurements, where both a zero value and distances between different measurements are defined, provide the greatest flexibility in statistical methods that can be used for analysing the data. Interval measurements have meaningful distances between measurements but no meaningful zero value (such as IQ measurements or temperature measurements in degrees Celsius). Ordinal measurements have imprecise differences between consecutive values but a meaningful order to those values. Nominal measurements have no meaningful rank order among values.
[edit] Statistical techniques
Some well known statistical tests and procedures for research observations are:
Student's t-test
chi-square test
Analysis of variance (ANOVA)
Mann-Whitney U
Regression analysis
Correlation
Fisher's Least Significant Difference test
Pearson product-moment correlation coefficient
Spearman's rank correlation coefficient
[edit] Specialized disciplines
Some fields of inquiry use applied statistics so extensively that they have specialized terminology. These disciplines include:
Actuarial science
Biostatistics
Business statistics
Data mining (applying statistics and pattern recognition to discover knowledge from data)
Economic statistics (Econometrics)
Energy statistics
Engineering statistics
Epidemiology
Geography and Geographic Information Systems, more specifically in Spatial analysis
Demography
Psychological statistics
Quality
Social statistics (for all the social sciences)
Statistical literacy
Statistical surveys
Process analysis and chemometrics (for analysis of data from analytical chemistry and chemical engineering)
Reliability engineering
Image processing
Statistics in various sports, particularly baseball and cricket
Statistics form a key basis tool in business and manufacturing as well. It is used to understand measurement systems variability, control processes (as in statistical process control or SPC), for summarizing data, and to make data-driven decisions. In these roles it is a key tool, and perhaps the only reliable tool Statistical computing
The rapid and sustained increases in computing power starting from the second half of the 20th century have had a substantial impact on the practice of statistical science. Early statistical models were almost always from the class of linear models, but powerful computers, coupled with suitable numerical algorithms, caused a resurgence of interest in nonlinear models (especially neural networks and decision trees) and the creation of new types, such as generalised linear models and multilevel models.
Increased computing power has also led to the growing popularity of computationally-intensive methods based on resampling, such as permutation tests and the bootstrap, while techniques such as Gibbs sampling have made Bayesian methods more feasible. The computer revolution has implications for the future of statistics, with a new emphasis on "experimental" and "empirical" statistics. A large number of both general and special purpose statistical packages are now available to practitioners.
This may be the result of outright fraud or of subtle and unintentional bias on the part of the researcher. Thus, Harvard President Lawrence Lowell wrote in 1909 that statistics, "like veal pies, are good if you know the person that made them, and are sure of the ingredients."
As further studies contradict previously announced results, people may become wary of trusting such studies. One might read a study that says (for example) "doing X reduces high blood pressure", followed by a study that says "doing X does not affect high blood pressure", followed by a study that says "doing X actually worsens high blood pressure". Often the studies were conducted on different groups with different protocols, or a small-sample study that promised intriguing results has not held up to further scrutiny in a large-sample study. However, many readers may not have noticed these distinctions, or the media may have oversimplified this vital contextual information, and the public's distrust of statistics is thereby increased.
However, deeper criticisms come from the fact that the hypothesis testing approach, widely used and in many cases required by law or regulation, forces one hypothesis to be 'favored' (the null hypothesis), and can also seem to exaggerate the importance of minor differences in large studies. A difference that is highly statistically significant can still be of no practical significance.
See also criticism of hypothesis testing and controversy over the null hypothesis.
In the fields of psychology and medicine, especially with regard to the approval of new drug treatments by the Food and Drug Administration, criticism of the hypothesis testing approach has increased in recent years. One response has been a greater emphasis on the p-value over simply reporting whether a hypothesis was rejected at the given level of significance α. Here again, however, this summarises the evidence for an effect but not the size of the effect. One increasingly common approach is to report confidence intervals instead, since these indicate both the size of the effect and the uncertainty surrounding it. This aids in interpreting the results, as the confidence interval for a given α simultaneously indicates both statistical significance and effect size.