The analysis of events governed by probability is called statistics. There are several competing interpretations of the actual "meaning" of probabilities.
Overview[ edit ] In applying statistics to a problem, it is common practice to start with a population or process to be studied. Populations can be diverse topics such as "all persons living in a country" or "every atom composing a crystal".
Ideally, statisticians compile data about the entire population an operation called census.
This may be organized by governmental statistical institutes. Descriptive statistics can be used to summarize the population data. Numerical descriptors include mean and standard deviation for continuous data types like incomewhile frequency and percentage are more useful in terms of describing categorical data like race.
When a census is not feasible, a chosen subset of the population called a sample is studied. Once a sample that is representative of the population is determined, data is collected for the sample members in an observational or experimental setting.
Again, descriptive statistics Probability and statistics answers be used to summarize the sample data. However, the drawing of the sample has been subject to an element of randomness, hence the established numerical descriptors from the sample are also due to uncertainty. To still draw meaningful conclusions about the entire population, inferential statistics is needed.
It uses patterns in the sample data to draw inferences about the population represented, accounting for randomness. These inferences may take the form of: Inference can extend to forecastingprediction and estimation of unobserved values either in or associated with the population being studied; it can include extrapolation and interpolation of time series or spatial dataand can also include data mining.
Sampling[ edit ] When full census data cannot be collected, statisticians collect sample data by developing specific experiment designs and survey samples. Statistics itself also provides tools for prediction and forecasting through statistical models. The idea of making inferences based on sampled data began around the mids in connection with estimating populations and developing precursors of life insurance.
Representative sampling assures that inferences and conclusions can safely extend from the sample to the population as a whole. A major problem lies in determining the extent that the sample chosen is actually representative. Statistics offers methods to estimate and correct for any bias within the sample and data collection procedures.
There are also methods of experimental design for experiments that can lessen these issues at the outset of a study, strengthening its capability to discern truths about the population. Sampling theory is part of the mathematical discipline of probability theory.
Probability is used in mathematical statistics to study the sampling distributions of sample statistics and, more generally, the properties of statistical procedures. The use of any statistical method is valid when the system or population under consideration satisfies the assumptions of the method.
The difference in point of view between classic probability theory and sampling theory is, roughly, that probability theory starts from the given parameters of a total population to deduce probabilities that pertain to samples.
Statistical inference, however, moves in the opposite direction— inductively inferring from samples to the parameters of a larger or total population. Experimental and observational studies[ edit ] A common goal for a statistical research project is to investigate causalityand in particular to draw a conclusion on the effect of changes in the values of predictors or independent variables on dependent variables.
There are two major types of causal statistical 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 lies in how the study is actually conducted. Each can be very effective. An experimental study involves taking measurements of the system under study, manipulating the system, and then taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements.
In contrast, an observational study does not involve experimental manipulation.Put statistical theories into practice with PROBABILITY AND STATISTICS FOR ENGINEERING AND THE SCIENCES, 9th Edition.
Always a favorite with statistics students, this calculus-based text offers a comprehensive introduction to probability and statistics while demonstrating how professionals apply concepts, models, and methodologies in today's engineering and scientific careers.
Great textbook for the intro to statistics with mathematical background. Covers a lot of material with reasonable depth. Some of the intro paragraphs to the chapters/sections might seem vague at first, as the authors intended it to be an "interesting example".
The binomial distribution is a discrete probability distribution. It describes the outcome of n independent trials in an experiment.
Each trial is assumed to have only two outcomes, either success or failure. If the probability of a successful trial is p, then the probability of having x successful outcomes in an experiment of n independent trials is as follows. Box-and-whisker plots are a handy way to display data broken into four quartiles, each with an equal number of data values.
The box-and-whisker plot doesn't show frequency, and it doesn't display each individual statistic, but it clearly shows where the middle of the data lies. Statistics and Probability Problems with Answers sample 1.
Problems on statistics and probability are presented. The answers to these problems are at the bottom of the page. Statistics is a branch of mathematics dealing with data collection, organization, analysis, interpretation and presentation.
In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied.
Populations can be diverse topics such as "all people living in a country" or.