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Basic Concepts in Descriptive Statistics Schaum's Outline of Elements of Statistics I: Descriptive Statistics & Probability Chapter 1. Functions. Function (p 10): If two variables are related so that for every permissible specific value x of X there is associated one and only one specific value y of Y, then Y is a function of X. The domain of the function is the set of x values that X can assume, the range is the set of y values associated with the x values, and the rule of association is the function itself. Functions in statistics (p 11-12): independent/dependent variables and cause/effect. In the mathematics function y = f(x), y is said to be the dependent variable and x the independent variable because y depends on x. In the research context the dependent variable is a measurement variable that has values that to some degree depend on the values of a measurement variable associated with the cause. Chapter 2. Measurement scales (p 35-36). Nominal: unique mutually-exclusive categories (=, ), meaning that a measured item is = or some category. For example, fish being shark, flounder, or trout. Ordinal: nominal (=, ) plus ordered (<,>). For example, eggs are small, medium, or large. Interval: ordinal (=, and <,>) plus uniform reference units (+,-). For example, degrees Celsius. Ratio: interval (=, and <,> and +,-) plus absolute zero making ratios meaningful. For example, degrees Kelvin where 300 K is twice as hot as 150 K. Chapter 3. Probabilities for sampling: with and without replacement (p 60). The probability of drawing an ace from a deck of 52 cards is P(ace) = 4/52 = 1/13, and if the sampling is done with replacement, the probability of drawing an ace on a second try is also 1/13. However, if the sampling is without replacement, the probability of drawing the second ace is P(second ace) = 3/51. Chapter 4 and 5. Frequency distributions and graphing frequency distributions. Chapter 6. Measures of central tendency. Mean or average: = x i N (p 140) i Median = value that divides an array of ordered values into two equal parts (p 149) Mode = the measurement that occurs most frequently (p 156) Chapter 7. Measures of dispersion. Variance 2 = ( xi ) 2 (n 1) (p 188, unbiased estimate) Standard Deviation is . i Normal probability density function (bell shaped curve): 68% of the values lie within one from and 95% within 2 from (p 197). Chapter 8. Probability: four interpretations. Basic concepts: experiment = a process that yields a measurement, trial = identical repetition of an experiment, outcome = result of a trial (a measurement), event = a possible specific outcome (p 227). 1. Classical: deals with idealizes situations, like the roll of a perfect die on a flawless surface having equally likely (probabilities of 1/6) outcomes (p 227). 2. Relative frequency: data from experiments are analyzed to obtain the relative frequency of events (p 229). 3. Set theory: the basis for the mathematical theory of probability (p 230). 4. Subjective: in contrast to the objective determination of probabilities above, here the probabilities are determined using “personal judgment” or “educated guesses” (p 237). Chapter 9. Calculating rules and counting rules. P(A B) signifies "the probability of event A or event B." If the events are mutually exclusive, such as, in tossing a die, event A is getting an even number and event B getting a 5, then the probability of A or B is simply 3/6 + 1/6 = 4/6, that is P(A B) = P(A) + P(B) (Special addition rule, p 258) When A and B are not mutually exclusive, P(A B) = P(A) + P(B) – P(A B) (General addition rule, p 264) P(A B) signifies "the probability of event A and event B." In terms of the outcome space, we're thinking about the set-theoretic intersection. Relative to a die, suppose event A is getting a face divisible by 3 (i.e., 3 or 6) and event B is getting an even number. The only outcome satisfying both is a 6. Thus, the probability that a die toss is even and divisible by 3 is 1/6. Conditional probability: P(A|B) signifies "the probability of event A, given the occurrence of event B." Event B is additional information that can narrow the outcome space (the range). P(A|B) = P(A B) / P(B) if P(B) 0 (p 259) P(A B) = P(B) P(A|B) (General multiplication rule, p 261) P(A B) = P(A) P(B) (Special multiplication rule when A and B independent, p 263) Bayes’ Theorem (also known as Bayes’ Law) P(A|B) = P(B|A) P(A) / P(B) (p 270) Chapter 10. Random variables, probability distributions, cumulative distribution functions. A random variable is a function having the sample space as its domain, and an association rule that assigns a real number to each sample point in the sample space. The range is the sample space of numbers defined by the association rule (p 309). Example of a random variable: the experiment is flipping a coin twice, the random variable (a function) is the number of heads on the two flips, the domain S = {HH, HT, TH, TT}, the rule of association is counting the number of heads, and the range S = {0, 1, 2}. A discrete random variable has a sample space that is finite or countably infinite (p 310). A continuous random variable has a sample space that is infinite or not countable (p 310). Understand discrete and continuous probability distributions. Expected value of discrete probability distribution: E(X) = = xf ( x) (p 322) x Variance of discrete probability distribution: Var(X) = = 2 (x ) 2 f ( x) (p 325) xf ( x)dx (p 324) x Expected value of continuous probability distribution: E(X) = = Variance of continuous probability distribution: Var(X) = 2 = ( x ) 2 f ( x)dx (p 328)