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Statistics And Probability Tips
Statistics
Statistics is a broad mathematical discipline which studies waysto collect, summarize and draw conclusions from data
.It is applicable to a wide variety of academic disciplines from thephysical and social sciences to the humanities, as well as to business,government, and industry.
Once data is collected, either through a formal sampling procedureor by recording responses to treatments in an experimental setting(cf experimental design), or by repeatedly observing a process overtime (time series), graphical and numerical summaries may be obtainedusing descriptive statistics.
Patterns in the data are modeled to draw inferences about the largerpopulation, using inferential statistics to account for randomnessand uncertainty in the observations. These inferences may take theform of answers to essentially yes/no questions (hypothesis testing),estimates of numerical characteristics (estimation), prediction offuture observations, descriptions of association (correlation), ormodeling of relationships (regression).
The framework described above is sometimes referred to as appliedstatistics. In contrast, mathematical statistics (or simply statisticaltheory) is the subdiscipline of applied mathematics which uses probabilitytheory and analysis to place statistical practice on a firm theoreticalbasis.
Probability
The word probability derives from the Latin probare (to prove, orto test). Informally, probable is one of several words applied touncertain events or knowledge, being more or less interchangeablewith likely, risky, hazardous, uncertain, and doubtful, dependingon the context. Chance, odds, and bet are other words expressing similarnotions. As with the theory of mechanics which assigns precise definitionsto such everyday terms as work and force, so the theory of probabilityattempts to quantify the notion of probable
Last Updated - 8th December 2005
Statistical methods
Experimental and observational studies
The basic steps for an experiment are to:
Levels of measurement
There are four types of measurements or measurement scales used instatistics. The four types or levels of measurement (ordinal, nominal,interval, and ratio) have different degrees of usefulness in statisticalresearch. Ratio measurement, where both a zero value and distancesbetween different measurements are defined, provide the greatest flexibilityin statistical methods that can be used for analysing the data. Intervalmeasurement, with meaningful distances between measurements but nomeaningful zero value (such as IQ measurements or temperature measurementsin degrees Celsius), is also used in statistical research.
Statistical techniques
Some well known statistical tests and procedures for research observationsare:
The general idea of probability is oftendivided into two related concepts:
It is an open question whether aleatory probability is reducibleto epistemic probability based on our inability to precisely predictevery force that might affect the roll of a die, or whether such uncertaintiesexist in the nature of reality itself, particularly in quantum phenomenagoverned by Heisenberg's uncertainty principle. Although the samemathematical rules apply regardless of which interpretation is chosen,the choice has major implications for the way in which probabilityis used to model the real world.
Formalization of probability
Like other theories, the theory of probability is a representationof probabilistic concepts in formal terms -- that is, in terms thatcan be considered separately from their meaning. These formal termsare manipulated by the rules of mathematics and logic, and any resultsare then interpreted or translated back into the problem domain.
There have been at least two successful attempts to formalize probability,namely the Kolmogorov formulation and the Cox formulation. In Kolmogorov'sformulation, sets are interpreted as events and probability itselfas a measure on a class of sets. In Cox's formulation, probabilityis taken as a primitive (that is, not further analyzed) and the emphasisis on constructing a consistent assignment of probability values topropositions. In both cases, the laws of probability are the same,except for technical details:
Representation and interpretation of probabilityvalues
The probability of an event is generally represented as a real numberbetween 0 and 1, inclusive. An impossible event has a probabilityof exactly 0, and a certain event has a probability of 1, but theconverses are not always true: probability 0 events are not alwaysimpossible, nor probability 1 events certain.
Most probabilities that occur in practice are numbers between 0 and1, indicating the event's position on the continuum between impossibilityand certainty. The closer an event's probability is to 1, the morelikely it is to occur.
For example, if two mutually exclusive events are assumed equallyprobable, such as a flipped coin landing heads-up or tails-up, wecan express the probability of each event as "1 in 2", or,equivalently, "50%" or "1/2".
Probabilities are equivalently expressed as odds, which is the ratioof the probability of one event to the probability of all other events.The odds of heads-up, for the tossed coin, are (1/2)/(1 - 1/2), whichis equal to 1/1. This is expressed as "1 to 1 odds" andoften written "1:1".
Odds a:b for some event are equivalent to probability a/(a+b). Forexample, 1:1 odds are equivalent to probability 1/2, and 3:2 oddsare equivalent to probability 3/5.
There remains the question of exactly what can be assigned probability,and how the numbers so assigned can be used; this is the questionof probability interpretations. There are some who claim that probabilitycan be assigned to any kind of an uncertain logical proposition; thisis the Bayesian interpretation. There are others who argue that probabilityis properly applied only to random events as outcomes of some specifiedrandom experiment, for example sampling from a population; this isthe frequentist interpretation. There are several other interpretationswhich are variations on one or the other of those, or which have lessacceptance at present.
Distributions
A probability distribution is a function that assigns probabilitiesto events or propositions. For any set of events or propositions thereare many ways to assign probabilities, so the choice of one distributionor another is equivalent to making different assumptions about theevents or propositions in question.
There are several equivalent ways to specify a probability distribution.Perhaps the most common is to specify a probability density function.Then the probability of an event or proposition is obtained by integratingthe density function. The distribution function may also be specifieddirectly. In one dimension, the distribution function is called thecumulative distribution function. Probability distributions can alsobe specified via moments or the characteristic function, or in stillother ways.
A distribution is called a discrete distribution if it is definedon a countable, discrete set, such as a subset of the integers. Adistribution is called a continuous distribution if it has a continuousdistribution function, such as a polynomial or exponential function.Most distributions of practical importance are either discrete orcontinuous, but there are examples of distributions which are neither.
Important discrete distributions include the discrete uniform distribution,the Poisson distribution, the binomial distribution, the negativebinomial distribution and the Maxwell-Boltzmann distribution.
Important continuous distributions include the normal distribution,the gamma distribution, the Student's t-distribution, and the exponentialdistribution.
Probability in mathematics
Probability axioms form the basis for mathematical probability theory.Calculation of probabilities can often be determined using combinatoricsor by applying the axioms directly. Probability applications includeeven more than statistics, which is usually based on the idea of probabilitydistributions and the central limit theorem.
To give a mathematical meaning to probability, consider flippinga "fair" coin. Intuitively, the probability that heads willcome up on any given coin toss is "obviously" 50%; but thisstatement alone lacks mathematical rigor - certainly, while we mightexpect that flipping such a coin 10 times will yield 5 heads and 5tails, there is no guarantee that this will occur; it is possiblefor example to flip 10 heads in a row. What then does the number "50%"mean in this context?
One approach is to use the law of large numbers. In this case, weassume that we can perform any number of coin flips, with each coinflip being independent - that is to say, the outcome of each coinflip is unaffected by previous coin flips. If we perform N trials(coin flips), and let NH be the number of times the coin lands heads,then we can, for any N, consider the ratio NH/N.
As N gets larger and larger, we expect that in our example the ratioNH/N will get closer and closer to 1/2. This allows us to "define"the probability Pr(H) of flipping heads as the limit (mathematics),as N approaches infinity, of this sequence of ratios:
In actual practice, of course, we cannot flip a coin an infinitenumber of times; so in general, this formula most accurately appliesto situations in which we have already assigned an a priori probabilityto a particular outcome (in this case, our assumption that the coinwas a "fair" coin). The law of large numbers then says that,given Pr(H), and any arbitrarily small number e, there exists somenumber n such that for all N > n,
In other words, by saying that "the probability of heads is1/2", we mean that, if we flip our coin often enough, eventuallythe number of heads over the number of total flips will become arbitrarilyclose to 1/2; and will then stay at least as close to 1/2 for as longas we keep performing additional coin flips.
Note that a proper definition requires measure theory which providesmeans to cancel out those cases where the above limit does not providethe "right" result or is even undefined by showing thatthose cases have a measure of zero.
The a priori aspect of this approach to probability is sometimestroubling when applied to real world situations. For example, if youflip a coin which keeps coming up heads over and over again, a hundredtimes. You can't decide whether this is just a random event - afterall, it is possible (although unlikely) that a fair coin would givethis result - or whether your assumption that the coin is fair isat fault.
Remarks on probability calculations
The difficulty of probability calculations lie in determining thenumber of possible events, counting the occurrences of each event,counting the total number of possible events. Especially difficultis drawing meaningful conclusions from the probabilities calculated.An amusing probability riddle, the Monty Hall problem demonstratesthe pitfalls nicely.
Applications of probability theory to everydaylife
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A major effect of probability theory on everyday life is in riskassessment and in trade on commodity markets. Governments typicallyapply probability methods in environment regulation where it is called"pathway analysis", and are often measuring well-being usingmethods that are stochastic in nature, and choosing projects to undertakebased on their perceived probable effect on the population as a whole,statistically. It is not correct to say that statistics are involvedin the modelling itself, as typically the assessments of risk areone-time and thus require more fundamental probability models, e.g."the probability of another 9/11". A law of small numberstends to apply to all such choices and perception of the effect ofsuch choices, which makes probability measures a political matter.
A good example is the effect of the perceived probability of anywidespread Middle East conflict on oil prices - which have rippleeffects in the economy as a whole. An assessment by a commodity tradethat a war is more likely vs. less likely sends prices up or down,and signals other traders of that opinion. Accordingly, the probabilitiesare not assessed independently nor necessarily very rationally. Thetheory of behavioral finance emerged to describe the effect of suchgroupthink on pricing, on policy, and on peace and conflict.
It can reasonably be said that the discovery of rigorous methodsto assess and combine probability assessments has had a profound effecton modern society. A good example is the application of game theory,itself based strictly on probability, to the Cold War and the mutualassured destruction doctrine. Accordingly, it may be of some importanceto most citizens to understand how odds and probability assessmentsare made, and how they contribute to reputations and to decisions,especially in a democracy.
Another significant application of probability theory in everydaylife is reliability. Many consumer products, such as automobiles andconsumer electronics, utilize reliability theory in the design ofthe product in order to reduce the probability of failure. The probabilityof failure is also closely associated with the product's warranty.
Specialized disciplines
Some sciences use applied statistics so extensively that they havespecialized terminology. These disciplines include:
Software
One of the most Important Application of Statics and Probabilitywith computers is Simulation
A simulation is an imitation of some real device or state of affairs.Simulation attempts to represent certain features of the behaviorof a physical or abstract system by the behavior of another system.
Simulation is used in many contexts, including the modeling of naturalsystems, and human systems to gain insight into the operation of thosesystems; and simulation in technology and safety engineering wherethe goal is to test some real-world practical scenario. Simulation,using a simulator or otherwise experimenting with a fictitious situationcan show the eventual real effects of some possible conditions.
Physical and Interactive simulation
Simulation in training
Simulation is often used in the training of civilian and militarypersonnel. This usually occurs when it is prohibitively expensiveor simply too dangerous to allow trainees to use the real equipmentin the real world. In such situations they will spend time learningvaluable lessons in a "safe" virtual environment. Oftenthe convenience is to permit mistakes during training for a safety-criticalsystem.
Training simulations typically come in one of four categories:
Medical Simulators
Medical simulators are increasingly being developed and deployedto teach therapeutic and diagnostic procedures as well as medicalconcepts and decision making to personnel in the health professions.Simulators have been developed for training procedures ranging fromthe basics such as blood draw, to laparoscopic surgery and traumacare. Many medical simulators involve a computer connected to a plasticsimulation of the relevant anatomy. In others, computer graphics reproducesall visual components and tool handles reproduce haptic aspects ofthe task. Some contain computer graphics simulations of imagery suchas X-ray or other medical images. Some patient simulators employ alife size mannequin which responds to injected drugs and can be programmedto create simulations of life-threatening emergencies. Some medicalsimulations are disseminated via the web and can be interacted withusing standard web browsers They are currently limited to screenbasedsimulations where users interact with the simulation via standardpointing devices.
Flight simulators
A flight simulator is used to train pilots on the ground. It permitsa pilot to crash his simulated "aircraft" without beinghurt. Flight simulators are often used to train pilots to operateaircraft in extremely hazardous situations, such as landings withno engines, or complete electrical or hydraulic failures. The mostadvanced simulators have high-fidelity visual systems and hydraulicmotion systems. The simulator is normally cheaper to operate thana real trainer aircraft.
Simulation and games
Many video games are also simulators, implemented inexpensively.These are sometimes called "sim games". Such games can simulatevarious aspects of reality, from economics to piloting vehicles, suchas flight simulators.
Engineering simulation
Simulation is an important feature when engineering systems. Forexample in electrical engineering, delay lines may be used to simulatepropagation delay and phase shift caused by an actual transmissionline. Similarly, dummy loads may be used to simulate impedance withoutsimulating propagation, and is used in situations where propagationis unwanted. A simulator may imitate only a few of the operationsand functions of the unit it simulates. Contrast with: emulate.
Most engineering simulations entail mathematical modeling and computerassisted investigation. There are many cases, however, where mathematicalmodeling is not reliable. Simulation of fluid dynamics problems oftenrequire both mathematical and physical simulations. In these casesthe physical models require dynamic similitude.
Computer simulation
Computer simulation, has become a useful part of modeling many naturalsystems in physics, chemistry and biology, and human systems in economicsand social science (the computational sociology) as well as in engineeringto gain insight into the operation of those systems. A good exampleof the usefulness of using computers to simulate can be found in thefield of network traffic simulation. In such simulations the modelbehaviour will change each simulation according to the set of initialparameters assumed for the environment. Computer simulations are oftenconsidered to be human out of the loop simulations.
Traditionally, the formal modeling of systems has been via a mathematicalmodel, which attempts to find analytical solutions to problems whichenables the prediction of the behaviour of the system from a set ofparameters and initial conditions. Computer simulation is often usedas an adjunct to, or substitution for, modeling systems for whichsimple closed form analytic solutions are not possible. There aremany different types of computer simulation, the common feature theyall share is the attempt to generate a sample of representative scenariosfor a model in which a complete enumeration of all possible statesof the model would be prohibitive or impossible.
It is increasingly common to hear simulations of many kinds referredto as "synthetic environments". This label has been adoptedto broaden the definition of "simulation" to encompass virtuallyany computer-based representation.
Simulation in computer science
In computer programming, a simulator is often used to execute a programthat has to run on some inconvenient type of computer. For example,simulators are usually used to debug a microprogram. Since the operationof the computer is simulated, all of the information about the computer'soperation is directly available to the programmer, and the speed andexecution of the simulation can be varied at will.
Simulators may also be used to interpret fault trees, or test VLSIlogic designs before they are constructed. In theoretical computerscience the term simulation represents a relation between state transitionsystems. This is useful in the study of operational semantics.
Simulation in education
Simulations in education are somewhat like training simulations.They focus on specific tasks. In the past,video has been used forteachers and education students to observe, problem solve and roleplay; however, a more recent use of simulations in education includeanimated narrative vignettes (ANV). ANVs are cartoon-like video narrativesof hypothetical and reality based stories involving classroom teachingand learning. ANVs have been used to assess knowledge, problem solvingskills and dispositions of children, pre-service and in-service teachers.
Another form of simulation has been finding favour in business educationin recent years. Business simulations that incorporate a dynamic modelenables experimentation with business strategies in a risk free environmentand provide a useful extension to case study discussions.
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