# User:Dcljr/Statistics

## I hate the fact that I can't edit the lead section without editing the entire article!

This page contains some ideas and lists related to probability and statistics. It is very incomplete. No attempt has been made (yet) to make it presentable or useful (for example, there are lots of dead links since I've decided to "link everything as I go").

I originally intended it to eventually (possibly) replace the current Statistics article, which I am not at all happy with — and some sections are indeed developing in that direction. Others, however, look more like the beginnings of one or more Wikibooks. I will continue to work on this page until portions of it become suitable for moving to other places in the Wiki(media)verse.

## Preamble

Statistics is a broad mathematical discipline which studies ways to collect, summarize, and draw conclusions from data. It is applicable to a wide variety of academic fields from the physical and social sciences to the humanities, as well as to business, government and industry.

Once data is collected, either through a formal sampling procedure or by recording responses to treatments in an experimental setting (see experimental design), or by repeatedly observing a process over time (see time series), graphical and numerical summaries may be obtained using descriptive statistics. The techniques of descriptive statistics can also be applied to census data, which is collected on entire populations.

If the data represents a sample (a subset of the population of interest), inferential statistics can be used to draw conclusions about the larger (mostly unobserved) population. These inferences, which are based on ideas of randomness and uncertainty quantified through the use of probabilities, may take any of several forms:

1. Answers to essentially yes/no questions (hypothesis testing)
2. Estimates of numerical characteristics (estimation)
3. Predictions of future observations (prediction)
4. Descriptions of association (correlation)
5. Modeling of relationships (regression)

The procedures which enable such inferences to be made are sometimes collectively known as applied statistics. In contrast, statistical theory (or, as an academic subject, sometimes called mathematical statistics) is the subdiscipline of applied mathematics which uses probability theory and mathematical analysis to place statistical practice on a firm theoretical basis. (If applied statistics is what you do in statistics, statistical theory tells you why it works.)

The word statistics is often used to refer to data collected on an entire population rather than a subset of it. A common example is sports "statistics" published in newspapers. More formally, in the U.S., the Bureau of Labor Statistics collects data on employment and general economic conditions; the Census Bureau publishes a large annual volume called the Statistical Abstract of the United States based on census data.

The word statistic (with no final s) is typically defined as a numerical quantity calculated from a set of data; see Statistic for further discussion.

## Etymology

The word statistics comes from the modern Latin phrase statisticum collegium (lecture about state affairs), which gave rise to the Italian word statista (statesman or politician — compare to status) and the German Statistik (originally the analysis of data about the state). It acquired the meaning of the collection and classification of data generally in the early nineteenth century. The collection of data about states and localities continues, largely through national and international statistical services.

## Definitions

Some textbook definitions of statistics and related terms (italics added):

Stephen Bernstein and Ruth Bernstein, Schaum's Outline of Elements of Statistics II: Inferential Statistics (1999)
Statistics is the science that deals with the collection, analysis, and interpretation of numerical information.
In descriptive statistics, techniques are provided for collecting, organizing, summarizing, describing, and representing numerical information.
[Inferential statistics provides] techniques.... for making generalizations and decisions about the entire population from limited and uncertain sample information.
Donald A. Berry, Statistics: A Bayesian Perspective (1996)
Statistical inferences have two characteristics:
1. Experimental or observational evidence is available or can be gathered.
2. Conclusions are uncertain.
John E. Freund, Mathematical Statistics, 2nd edition (1971)
Statistics no longer consists merely of the collection of data and their representation in charts and tables — it is now considered to encompass not only the science of basing inferences on observed data, but the entire problem of making decisions in the face of uncertainty.
Gouri K. Bhattacharyya and Richard A. Johnson, Statistical Concepts and Methods (1977)
Statistics is a body of concepts and methods used to collect and interpret data concerning a particular area of investigation and to draw conclusions in situations where uncertainty and variation are present.
E. L. Lehmann, Theory of Point Estimation (1983)
Statistics is concerned with the collection of data and with their analysis and interpretation.
William H. Beyer (editor), CRC Standard Probability and Statistics Tables and Formulae (1991)
The pursuit of knowledge frequently involves data collection; and those responsible for the collection must appreciate the need for analyzing the data to recover and interpret the information therein. Today, statistics are being accepted as the universal language for the results of experimentation and research and the dissemination of information.
Oscar Kempthorne, The Design and Analysis of Eperiments, reprint edition (1973)
Statistics enters [the scientific method] at two places:
1. The taking of observations
2. The comparison of the observations with the predictions from... theory.
Marvin Lentner and Thomas Bishop, Experimental Design and Analysis (1986)
The information obtained from planned experiments is used inductively. That is, generalizations are made about a population from information contained in a random sample of that particular population. ... [Such] inferences and decisions... are sometimes erroneous. Proper statistical analyses provide the tools for quantifying the chances of obtaining erroneous results.
Robert L. Mason, Richard F. Gunst and James L. Hess, Statistical Design and Analysis of Experiments (1989)
Statistics is the science of problem-solving in the presence of variability.
Statistics is a scientific discipline devoted to the drawing of valid inferences from experimental or observational data.
Stephen K. Campbell, Flaws and Fallacies in Statistical Thinking (1974)
Statistics... is a set of methods for obtaining, organizing, summarizing, presenting, and analyzing numerical facts. Usually these numerical facts represent partial rather than complete knowledge about a situation, as is the case when a sample is used in lieu of a complete census.

## Basic concepts

There are several philosophical approaches to statistics, most of which rely on a few basic concepts.

### Population vs. sample

In statistics, a population is the set of all objects (people, etc.) that one wishes to make conclusions about. In order to do this, one usually selects a sample of objects: a subset of the population. By carefully examining the sample, one may make inferences about the larger population.

For example, if one wishes to determine the average height of adult women aged 20-29 in the U.S., it would be impractical to try to find all such women and ask or measure their heights. However, by taking small but representative sample of such women, one may determine the average height of all young women quite closely. The matter of taking representative samples is the focus of sampling.

### Randomness, probability and uncertainty

The concept of randomness is difficult to define precisely. In general, any outcome of an action, or series of actions, which cannot be predicted beforehand may be described as being random. When statisticians use the word, they generally mean that while the exact outcome cannot be known beforehand, the set of all possible outcomes is, at least in theory, known. A simple example is the outcome of a coin toss: whether the coin will land heads up or tails up is (ideally) unknowable before the toss, but what is known is that the outcome will be one of these two possibilities and not, say, on edge (assuming, of course, the coin cannot stand upright on its edge). The set of all possible outcomes is usually called the sample space.

The probability of an event is also difficult to define precisely but is basically equivalent to the everyday idea of the likelihood or chance of the event happening. An event that can never happen has probability zero; an event that must happen has probability one. (Note that the reverse statements are not necessarily true; see the article on probability for details.) All other events have a probability strictly between zero and one. The greater the probability the more likely the event, and thus the less our uncertainty about whether it will happen; the smaller the probability the greater our uncertainty.

There are two basic interpretations of probability used to assign or compute probabilities in statistics:

• Relative frequency interpretation: The probability of an event is the long-run relative frequency of occurrence of the event. That is, after a long series of trials, the probability of event A is taken to be:
$\mbox{P}(A) = {\mbox{number of trials in which event } A \mbox{ happened} \over \mbox{total number of trials}}$
To make this definition rigorous, the right-hand side of the equation should be preceded by the limit as the number of trials grows to infinity.
• Subjective interpretation: The probability of an event reflects our subjective assessment of the likelihood of the event happening. This idea can be made rigorous by considering, for example, how much one should be willing to pay for the chance to win a given amount of money if the event happens. For more information, see Bayesian probability.

Once a procedure has been chosen for assigning probabilities to events, the probabilistic structure of the phenomenon under consideration can be summarized in one or more probability distributions.

Note that the relative frequency interpretation does not require that a long series of trials actually be conducted. Typically probability calculations are ultimately based upon perceived equally-likely outcomes — as obtained, for example, when one tosses a so-called "fair" coin or rolls or "fair" die. Many frequentist statistical procedures are therefore based on simple random samples, in which every possible sample is as likely as any other.

(Ugh.)

Blah, blah...

## Data collection

### Sampling

Main article: Sampling (statistics)

### Experimental design

Main article: Design of experiments

## Data summary: descriptive statistics

Main article: Descriptive statistics

### Levels of measurement

Main article: Level of measurement
• Qualitative (categorical)
• Nominal
• Ordinal
• Quantitative (numerical)
• Interval
• Ratio

### Graphical summaries

Main article: Statistical plot or List of statistical plots

### Numerical summaries

Main article: Summary statistics

Data mining...

Probability:

## Typical course in mathematical probability

Below are the topics typically (?) covered in a one-year course introducing the mathematical theory of probability to undergraduate students in mathematics and statistics. (Actually, this list contains much more material than is typically covered in one year.)

Topics of a more advanced nature are italicized, including those typically only covered in mathematical statistics or graduate-level probability theory courses (e.g., topics requiring measure theory). See also the #Typical course in mathematical statistics below.

order?

• And so on, and so forth...

## Typical course in mathematical statistics

Would cover many of the topics from the #Typical course in mathematical probability outlined above, plus...

• And so on, and so forth...

## Typical course in applied statistics

Less theoretical than the #Typical course in mathematical statistics outlined above. (Sometimes portions of the following form the basis of a second statistics course for mathematics majors — third in the sequence if probability is the first course).

• And so on, and so forth...

Hmm...

## Terms from categorical data analysis

(By chapter: Agresti, 1990.)

## Statistical software

### Free versions of commercial software

• Gnumeric — not a clone of Excel, but implements many of the same functions (can it use Excel add-ins?)
• R — free version of S
• FIASCO or PSPP — free version of SPSS

### World Wide Web

• StatLib — large repository of statistical software and data sets

## References

• Agresti, Alan (1990). Categorical Data Analysis. NY: John Wiley & Sons. ISBN 0-471-85301-1.
• Casella, George & Berger, Roger L. (1990). Statistical Inference. Pacific Grove, CA: Wadsworth & Brooks/Cole. ISBN 0-534-11958-1.
• DeGroot, Morris (1986). Probability and Statistics (2nd ed.). Reading, Massachusetts: Addison-Wesley. ISBN 0-201-11366-X.
• Kempthorne, Oscar (1973). The Design and Analysis of Experiments. Malabar, FL: Robert E. Krieger Publishing Company. ISBN 0-88275-105-0. [Rpt.; orig. 1952, NY: John Wiley & Sons.]
• Kuehl, Robert O. (1994). Statistical Principles of Research Design and Analysis. Belmont, CA: Duxbury Press. ISBN 0-534-18804-4.
• Lentner, Marvin & Bishop, Thomas (1986). Experimental Design and Analysis. Blacksburg, VA: Valley Book Company. ISBN 0-9616255-0-3.
• Manoukian, Edward B. (1986). Modern Concepts and Theorems of Mathematical Statistics. NY: Springer-Verlag. ISBN 0-387-96186-0.
• Mason, Robert L.; Gunst, Richard F.; and Hess, James L. (1989). Statistical Design and Analysis of Experiments: With Applications to Engineering and Science. NY: John Wiley & Sons. ISBN 0-471-85364-X.
• Ross, Sheldon (1988). A First Course in Probability Theory (3rd ed.). NY: Macmillan. ISBN 0-02-403850-4.

### And eventually...

• Berger, James O. (1985). Statistical Decision Theory and Bayesian Analysis (2nd ed.). NY: Springer-Verlag. ISBN 0-387-96098-8. (Also, Berlin: ISBN 3-540-96098-8.)
• Berry, Donald A. (1996). Statistics: A Bayesian Perspective. Belmont, CA: Duxbury Press. ISBN 0-534-23472-0.
• Feller, William (1950). An Introduction to Probability Theory and Its Applications, Vol. 1. NY: John Wiley & Sons. ISBN unknown. (Current: 3rd ed., 1968, NY: John Wiley & Sons, ISBN 0-471-25708-7.)
• Feller, William (1971). An Introduction to Probability Theory and Its Applications, Vol. 2 (2nd ed.). NY: John Wiley & Sons. ISBN 0-471-25709-5.
• Lehmann E. L. [Eric Leo] (1991). Theory of Point Estimation. Pacific Grove, CA: Wadsworth & Brooks/Cole. ISBN 0-534-15978-8. (Orig. 1983, NY: John Wiley & Sons.)
• Lehmann E. L. [Eric Leo] (1994). Testing Statistical Hypotheses (2nd ed.). NY: Chapman & Hall. ISBN 0-412-05321-7. (Orig. 2nd ed., 1986, NY: John Wiley & Sons.)