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Quick guide 10 The statistical problem solving cycle Data are numbers in context and the goal of statistics is to get information from those data usually through problem solving A procedure or paradigm for statistical problem solving and scienti ?c enquir

The statistical problem solving cycle Data are numbers in context and the goal of statistics is to get information from those data usually through problem solving A procedure or paradigm for statistical problem solving and scienti ?c enquiry is illustrated in the diagram The dotted line means that following discussion the problem may need to be re-formulated and at least one more iteration completed Specify the problem and plan Interpret and discuss Collect data Process represent and analyse data Descriptive statistics Given a sample of n observations x x L xn we de ?ne the sample mean to be ? x x x L xn n xi n and the corrected sum of squares by Sxx ? xi ?? x ?? ? xi ?? nx ?? ? xi ?? ? xi n Sxx n is sometimes called the mean squared deviation and an unbiased estimator of the population variance ? is s Sxx n ?? The sample standard deviation is s In calculating s the divisor n- is called the degrees of freedom df Note that s is also sometimes written ? If the sample data are ordered from smallest to largest then the minimum Min is the smallest value lower quartile LQ is the ? n -th value median Med is the middle or the n -th value upper quartile UQ is the n -th value maximum Max is the largest value These ?ve values constitute a ?ve-number summary of the data They can be represented diagrammatically by a box-and-whisker plot commonly called a boxplot Min LQ Med UQ Max Grouped frequency data If the data are given in the form of a grouped frequency distribution where we have ? observations in an interval ? whose mid-point is xi then if ? n x ? ? xi ? ? ? ? xi n and Sxx ? ? xi ?? x ? ? xi ?? ? ? xi n Events probabilities The intersection of two events A and B is A ??B The union of A and B is A ??B A and ?? B are mutually exclusive if they cannot both occur denoted A ??B ? where ? is called the null event For an event A ? P A ? For two events A and B P A ??B P A P B ?? P A ??B If A and B are mutually exclusive then P A ??B P A P B Equally likely outcomes If a complete set of n elementary outcomes are all equally likely to occur then the probability of each elementary outcome is n If an event A consists of m of these n elements then P A m n Independent events A B are independent if and only if P A ??B P A P B Conditional Probability of A given B P A B P A ??B P B if P B ?? Bayes ? Theorem P B A P A B P B P A Theorem of Total Probability The k