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The term "mathematical statistics" designates the mathematical theories of probability and statistical inference, which are used in statistical practice. The relation between statistics and probability theory developed rather late, however.
Statistical inference makes propositions about a population, using data drawn from the population with some form of sampling.Given a hypothesis about a population, for which we wish to draw inferences, statistical inference consists of (first) selecting a statistical model of the process that generates the data and (second) deducing propositions from the model.
Mathematical statistics is the application of probability theory and other mathematical concepts to statistics, as opposed to techniques for collecting statistical data. [1] Specific mathematical techniques that are commonly used in statistics include mathematical analysis , linear algebra , stochastic analysis , differential equations , and ...
Statistical inference uses mathematics to draw conclusions in the presence of uncertainty. This generalizes deterministic reasoning, with the absence of uncertainty as a special case. Statistical inference uses quantitative or qualitative (categorical) data which may be subject to random variations.
He wrote a book entitled Manuscript on Deciphering Cryptographic Messages, containing detailed discussions on statistics and cryptanalysis. [2] [3] [4] Al-Kindi also made the earliest known use of statistical inference. [1] 13th century – An important contribution of Ibn Adlan was on sample size for use of frequency analysis. [1]
Statistics subsequently branched out into various directions, including decision theory, Bayesian statistics, exploratory data analysis, robust statistics, and non-parametric statistics. Neyman-Pearson hypothesis testing made significant contributions to decision theory, which is widely employed, particularly in statistical quality control.
Statistical inference is the process of using data analysis to deduce properties of an underlying probability distribution. [29] Inferential statistical analysis infers properties of a population, for example by testing hypotheses and deriving estimates. It is assumed that the observed data set is sampled from a larger population.
Statistics is the theory and application of mathematics to the scientific method including hypothesis generation, experimental design, sampling, data collection, data summarization, estimation, prediction and inference from those results to the population from which the experimental sample was drawn.