Intro to Parametric Statistics, Assumptions & Degrees of FreedomAssumptions & Degrees of Freedom• Some terms we will need• Normal Distributions•Degrees of freedomg• Z-values of individual cases % & their meaning• 1-sample Z-test vs. t-test – ND & df based on sample size• 2-sample Z-test & t-tests – ND & df based on sample sizes•X2tests – ND & df based on design size• F-tests – ND & df based on design & sample sizes• r – tests – bivariate ND & df based on sample sizeDescriptive Statistics vs. Inferential Statistics vs. Population Parameters• a descriptive statistic is calculated from sample data to describe the value of some attribute of the sample• an inferential statistic is calculated from sample data as a basis to infer the value of some attribute of the population from which the sample is drawn• a parameter is calculated from population data to describe the value of some attribute of the population ft• more often…• inferential statistics from multiple samples converge sufficiently that we begin to refer to that value as the ygpopulation parameter of that attributethe parameter is theoretical/hypothetical with the measurement instrument having been designed/scaled tomeasurement instrument having been designed/scaled to produce that population parameter`Inferential Statistics vs. Inferential Statistical Test• an inferential statistic is calculated from sample data as a basis to infer the value of some attribute of the population from which the sample is drawnthe sample is drawn• an inferential statistical test is calculated from sample data to test some hypothesis about the value of some attribute of the yppopulation from which the sample is drawnStandard DeviationvsStandard Error of the MeanStandard Deviationvs. Standard Error of the Mean• standard deviations tells how much a set of scores (sample or population) varies around the mean of those scorespopulation) varies around the mean of those scores • standard error of the mean tells by how much the mean calculated from a sample (of a given size) is likely to vary from thecalculated from a sample (of a given size) is likely to vary from the true mean of the population from which the sample is drawnFrequency Distribution vs. Sampling Distribution vs. Mathematical SamplingqypgpgDistribution•a frequency distribution tells the frequency/probability of attribute f( f )qy qyp yvalues across a set of scores (sample of population)• what is the freq distnof age for the sample?•a sampling distribution tells the frequency/probability of a•a sampling distribution tells the frequency/probability of a statistical value about an attribute (e.g., mean age), from a set of samples of the same size all drawn from the same population•what is the freq distnof mean age from samples of n=20?what is the freq distof mean age from samples of n 20?• a mathematical sampling distribution tells the expected frequency/probability of statistical values from a set samples of the same size from the same population as defined by thethe same size from the same population, as defined by the mathematical function of that distribution• what is the probability of finding a mean Zage > 1.96?All inferential statistical tests are based on mathematical sampling distributions (e.g., Z, t, X2, F, r)Degrees of FreedomWh ti t t d t t t t h thWhenever we estimate a parameter and want to test whether or not that parameter has some hypothesized value, we will…1. Compute a test statistics (t, r, F, X2, etc.)2. Determine a critical value of that test statistics according to • the df of the distribution and • a specified Type I error rate (e.g., p = .05)3C th t t d th NHST3.Compare the two to do the NHSTSo, for each of these significance tests, we need to know the degrees of freedom!There are several ways of describing df, some more th ti ll t lt ti f imathematically accurate, some are alternative ways of expressing the same thing, and some are just plain confusing.Here’s a reasonably accurate and easy to remember approachHere s a reasonably accurate, and easy to remember approach, that generalizes well across univariate, bivariate and multivariate statistical models…The degrees of freedom for a set of things conveys, “how many pieces of information from a set of things must be know t h ll th i f ti i th t t f thi ”to have all the information in that set of things.”There are several “kinds of dfs” you should know about…1) df of a single group of values df = n - 1)ggpSay you have the 5 data points (n=5) 2 4 6 8 & 10 the df for the sum of the set of numbers tells how many of those values are free to vary without changingthe sumare free to vary without changingthe sum.• in other words, since the sum is 30 if we know any 4 of the numbers, we can determine what the 5thmust be for the group to have that sumhave that sum• e.g., 2 + 4 + 6 + 8 = 20, so the 5thnumber must be 10•eg 4+6+8+10=28 sothe5thnumber must be 2•e.g. 4 + 6 + 8 + 10 = 28, so the 5thnumber must be 2• same for computing the mean, SS, variance or std all df = n – 12) df of nested conditions/groups df = N-k = Σk(n-1)• typical of between groups designs•different cases are (nested) in different conditions•different cases are (nested) in different conditions• for the cases in a given condition, n-1 of the values must be know in order to deduce the sum (mean or std) for the groupdf f ll th i bt i d b i th df f @•df for all the groups is obtained by summing the df fro @ group3) df of crossed conditions/groups df = (k-1)(n-1)tilfithidi•typical of within-groups designs• each case is in every conditionFor @ case – k-1 values are needed to deducek=3G1 G2 G3 SumC1 10 11 34are needed to deduce value of the kth13For @ group – n-1 values are needed to deduce lfththk=3n=4C2 14 9 32C3 18 15 38C43995value of the nth1512The last value for the last case can then be deduced12C4 39Sum 57 47 39 1512case can then be deduced.12So, (3-1)*(4-1) = 6 values are “free”formula for the normal distribution:(x)² / 2²e -( x-)² / 2 ²ƒ(x) = --------------------2π2πFor a given mean () and standard deviation (), plug in g()(), p gany value of x to receive the proportional frequency of that value in that particular normal distribution.Each ND has a different Mean & Std, but all have the same