Exploratory statistics Probability calculus Lecture #5 Exploratory statistics Probability calculus

Exploratory analysis First comments on exploratory data analysis. According to Tukey and co - workers - scientists - we have four major themes for exploratory data analyses: Resistant measures. Measures, observations, summaries which prove insensitive to localized 'misbehaviour' of observed sample data. {Examples for location parameter - median}. Re-expression - re-scaling. This includes a possible re-scoring of data to achieve distributions which are more in line of the researchers expectation. Although we will not consider this for the data at hand it seems a very important consideration for social science data where scales are not 'naturally' given and interpretation is often difficult { for example tests which purport to measure knowledge}. Revelation or model proposals. We want to use graphical displays, in addition to a-priori models, to discuss possible population distributions, such graphical displays are to include both univariate, bivariate and successively more 3-dimensional models { Examples: stem-and-leaf, box plots}. Residual examinations. We want to use residuals to see if 'obvious' trends exist indicating that the chosen model leads to biased estimations. According to Tukey a statistician can model himself according to a detective who collects information to create some ‘hunches’, ‘guesses’ to enable planning for some deductive experiments

Historical considerations Probability theory Historical considerations Gambling Pascal Fermat correspondence and dice games 6-49 Theoretical probability Empirical probability Subjective – personal probability Relative frequency approach

Probability theory Introduction to probability theory based on Kolmogorov’s axiomatic development of probability based on 5 axioms. Some set theory concepts: Simple elements and combined sets Venn diagrams Union, intersection, difference of sets (Sum, product, difference) Probability calculus: consider random events A,B,C,…. Based on simple, atomic events E1, E2,… example A = {1,3,5} odd number result for a die A measure 0 ≤ P(A) ≤ 1 is called the probability of the event A to be observed. For a fair die P(A) = 1/2

Probability calculus Mutually exclusive events P(AUB)= P(A) + P(B) Events not mutually exclusive P(AUB)= P(A) + P(B) – P(A∩B) or P(A+B) = P(A) +P(B) – P(AB) Independent events P(A∩B) = P(AB)= P(A) P(B) Conditional events P(A|B) = P(AB)/P(B) iff P(B) ≠0