Some Basic Concepts Schaum's Outline of Elements of Statistics I: Descriptive Statistics & Probability Chuck Tappert and Allen Stix School of Computer Science and Information Systems Pace University

Chapter 1. Functions Function: 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. domain of the function is the set of x values that X can assume range is the set of y values associated with the x values the rule of association is the function itself

Chapter 1. Functions in statistics Independent/dependent variables and cause/effect In the mathematical 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 Nominal: unique mutually-exclusive categories, meaning that a measured item is equal to some category or not – e.g., fish being shark, flounder, or trout. Ordinal: nominal plus ordered – e.g., eggs are small, medium, or large. Interval: ordinal plus uniform reference units – e.g., degrees Celsius. Ratio: interval plus absolute zero making ratios meaningful – e.g., degrees Kelvin where 300 K is twice as hot as 150 K.

Chapter 3. Probabilities for sampling: with and without replacement The probability of drawing an ace from a deck of 52 cards is P(ace) = 4/52, and if the sampling is done with replacement, the probability of drawing an ace on a second try is also 4/52. 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 Median = value that divides an array of ordered values into two equal parts Mode = the measurement that occurs most frequently

Chapter 7 Measures of dispersion Variance and Standard Deviation Normal probability density function (bell shaped curve): 68% of the values lie within one sigma from the mean, and 95% within two sigma from the mean

Chapter 8 Probability: four interpretations Classical: deals with idealizes situations, like the roll of a perfect die on a flawless surface having equally likely (probabilities of 1/6) outcomes Relative frequency: data from experiments are analyzed to obtain the relative frequency of events Set theory: the basis for the mathematical theory of probability Subjective: in contrast to the objective determination of probabilities above, here the probabilities are determined using “personal judgment” or “educated guesses”

Chapter 9 Calculating rules and counting rules Special addition rule - A and B are mutually exclusive General addition rule - A and B are not mutually exclusive Conditional probability General multiplication rule - A and B not independent Special multiplication rule - A and B independent Bayes’ Theorem (also known as Bayes’ Law)

Chapter 10 Random variables, probability distributions, cumulative distribution functions Random variable – 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, and range is the sample space of numbers defined by the association rule Discrete random variable – sample space is finite or countably infinite Continuous random variable –sample space is infinite or not countable

Chapter 10 (cont) Understand discrete and continuous probability distributions Expected value of discrete probability distribution Variance of discrete probability distribution Expected value of continuous probability distribution Variance of continuous probability distribution