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Grouped frequency data The statistical problem solving cycle If the data are gi

Grouped frequency data The statistical problem solving cycle If the data are given in the form of a grouped frequency distribution where we have fi observations in an interval whose mid-point is xi then, if f n = ∑ i 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 scientific 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. Descriptive statistics Given a sample of n observations we define the sample mean to be 1 2 , , , n x x x L x x L + + + = = ∑ 1 2 i n x x x n n and the corrected sum of squares by ( ) ( ) = − ≡ − ≡ −∑ ∑ ∑ ∑ 2 2 2 2 2 i xx i i i x S x x x nx x n . xx S n is sometimes called the mean squared deviation and an unbiased estimator of the population variance is 2 σ 2 ( 1 xx s n = −) S . The sample standard deviation is s. In calculating s2, the divisor (n-1) is called the degrees of freedom (df). Note that s is also sometimes written ˆ σ . If the sample data are ordered from smallest to largest then the: minimum (Min) is the smallest value; lower quartile (LQ) is the ¼(n+1)-th value; median (Med) is the middle [or the ½(n+1)-th] value; upper quartile (UQ) is the ¾(n+1)-th value; maximum (Max) is the largest value. These five values constitute a five-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 i i i i i f x f x x f n = = ∑ ∑ ∑ and ( ) ( ) 2 2 2 i i xx i i i i f x S f x x f x 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, 0≤ P(A )≤ 1. 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 1/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( ) ≠ 0. B Bayes’ Theorem ( | ) ( ) ( | ) ( ) P A B P B P B A P A = . Theorem of Total Probability B B B The k events , 1 2 B 2 ,.., form a partition of the sample space S if B ∪ ∪ k P A 1 3 B ... ∪B = S and no two of the B ’s can occur together. Then ∑ . k ( ) i ) ( | ) ( i i i P A B P B = In this case Bayes Theorem generalizes to ( | ) ( ) P A B P B ( | ) ( | ) ( ) i i i i i i P B A P A B P B = ∑ ( 1 ,2,..., ) i k = . If is the complement of the event , P( )=1 – P( ) ( P A ( | ) P A B ( ) P B ( | P A B B′ B B′ B and ) = + ) ′ is a special ) ( P B ′ case of the theorem of total probability. Also, the complement of the event B is commonly denoted . B Guides to Statistical Information 1 Specify the problem and plan Interpret and discuss Collect data Process, represent and analyse data Probability & Statistics Facts and Formulae Resources to support the learning of mathematics, statistics and OR in higher education. www.mathstore.ac.uk The Statistical Education through Problem Solving (STEPS) glossary www.stats.gla.ac.uk/steps/glossary/ Further reading Kotz, S. and Johnson, L. (1988) Encyclopedia of Statistical Sciences, Vols 1 – 9. New York: John Wiley and Sons. 2 1 Permutations and combinations Variance The Central Limit Theorem The number of ways of selecting r objects out of a total of n, where the order of selection is important, is the number of permutations: ! ! n . ( ) n r P n r = − The variance of a random variable is defined as ( )     If a random sample of size n is taken from any distribution with mean µ and variance σ2, the sampling distribution of the mean will be approximately ~N(µ,σ2/n), where ‘ ~ ’ means ‘is distributed as’. The larger n is, the better the approximation. 2 2 2 ( ) . Var X X X µ µ = − ≡ −         E E Properties: X Var( )≥0 and is equal to 0 only if X is a constant. X The number of ways in which r objects can be selected from n when the order of selection is not important is the number of combinations: ! n  . !( )! n r n C r r n r   = =    −  Var(a X + b) = a2Var( ), where a and b are constants. The standard normal and Student’s t distributions Moment generating functions The moment generating function (mgf) of a random variable is defined as MX(t)= E[exp(t X )] if this exists. X n n C 0 nC must equal 1, so 0!=1 and = 1; = ; + +... + = ; = + . 0 nC 1 n C + n r C r C n n r C − 1 r C − 1 nC 1 n n C − n n C 2n r n n E[ k] can be evaluated as the: (i) coefficient of t r/(r !) in the power expansion of MX(t); Random variables (ii) r-th derivative of MX(t) evaluated at t = 0. Data arise from observations on variables that are measured on different scales. A nominal scale is used for named categories (eg race, gender) and an ordinal scale for data that can be ranked (eg attitudes, position) - no arithmetic operations are valid with either. Interval scale measurements can be added and subtracted only (eg temperature), but with ratio scale measurements (eg age, weight) multiplication and division can be used meaningfully as well. Generally, random variables are either discrete or continuous. Note: all data are discrete because the accuracy of measuring is always limited. X Measures of location The mean or expectation of the random variable X is E[ X ] the long-run average of realisations of X . If a random variable X ~ N(µ,σ2), z = (X-µ)/σ ~ N(0,1), the standard normal distribution. The t distribution with (n-1) degrees of freedom is used in place of z for small samples size n from normal populations when σ2 is unknown. As n increases the distribution of t converges to N(0,1). These distributions are used, for example, for inference about means, differences between means and in regression. The mode is where the pmf or pdf achieves a maximum (if it does so). For a random variable, , X the median is such that P( X ≤ median) = ½, so that 50% of values of X occur above and 50% below the median. Percentiles x A discrete random variable can take one of the values , , x x , the probabilities p 1 2 L i = ( ) i P X x = must satisfy p i i x i ≥ 0 and p1 + p2 + … = 1. The pairs ( , pi) form the probability mass function (pmf) of X. X Fisher’s F distribution p is the 100-p-th percentile of a random variable X if P( X ≤x p) = p. For example, the 5th percentile, x0.05, has 5% of the values smaller than or equal to it. The median is the 50-th percentile, the lower quartile is the 25th percentile, the upper quartile is the 75th percentile. f(F) A continuous random variable takes values from a continuous set of possible values. It has a probability uploads/Geographie/ quick-guide 10 .pdf