Formulae - 1 of 16 Updated 23 August 2007 Definitions and Formulae I expect you to memorize terms and formulae marked with an asterisk (*). I suggest that you make flash cards and practice daily. Population* refers to all that there is of a particular thing. The complete set of individuals, objects, or scores that the investigator is interested in studying. Sampling Element* refers to a single member of a population. Sample* refers to a subset of sampling elements selected from a population. Parameter* refers to a numerical characteristic of a population. Statistic* refers to a numerical characteristic of a sample. Also an estimate of a parameter. Sampling Unit* refers to what one observes in a population (also called unit of analysis). Usually, but not always, the sampling element. Sampling Frame* refers to a list of all the sampling units. Descriptive Statistics* refers to techniques used to characterize, summarize, and interpret data obtained from groups (samples or populations). Inferential Statistics* refers to techniques used to evaluate the adequacy of a generalization from obtained sample data to populations. proportion (p)* refers to a decimal fraction ranging from zero to one that shows the part of the total that has a certain characteristic. percentage (%)* refers to the number per hundred that has a certain characteristic N or n* refers to the total number of subjects or cases frequency (f)* refers to the number of subjects or cases having a particular value or range of values for some variable relative frequency (rf)* refers to the proportion of subjects or cases having a particular value or range of values for some variable percentage frequency (%f)* refers to the percentage of subjects or cases having a particular value or range of values for some variable cumulative frequency (cf)* refers to the number of subjects or cases having a particular value or less for some variable (descending), or the number of subjects or cases having a particular value or greater for some variable (ascending)Formulae - 2 of 16 Updated 23 August 2007 Nominal Scale* refers a measurement scale constructed by assigning a numerical value (a label) to a variable based upon • membership in only one of several unordered categories. Ordinal Scale* refers a measurement scale constructed by assigning a numerical value (a rank) to a variable based upon • the order of magnitude along a dimension. • the difference between each pair of ordered adjacent numerals (with respect to change in the magnitude of the variable measured) is not known (unequal intervals). Interval Scale* refers a measurement scale constructed by assigning a numerical value (a rank) to a variable based upon • the order of magnitude along a dimension. • the difference between each pair of ordered adjacent numerals represents equal amounts of change in the magnitude of the variable measured (equal intervals). • the zero point on an interval scale does not represent the absence of the characteristic being measured (an arbitrary zero). Ratio Scale* refers a measurement scale constructed by assigning a numerical value (a rank) to a variable based upon • the order of magnitude along a dimension. • the difference between each pair of ordered adjacent numerals represents equal amounts of change in the magnitude of the variable measured (equal intervals). • the zero point on a ratio scale represents the absence of the characteristic being measured (an absolute zero). Population Sample Parameter Statistic Mean* nΣYµY= * nΣYY = * Sum of Squares* ()2YYµYΣSS −= * ()2YYYΣSS −= *Variance* (Mean Square) nSSσY2Y= * 1-nSSsY2Y= * Standard Deviation* 2YYσσ = * 2YYss = * Standard Error of the Mean* nσσYY= * nssYY= * Note: When we discuss only one variable, the subscript is often omitted, e.g., µ, σ2, σ, s2, and s. However, for the standard error of the mean (Yσ, Ys), the subscript must be included in all cases.Formulae - 3 of 16 Updated 23 August 2007 Percentile Rank ()()icpnYYibLPff−+= where n refers to the sample size. p refers to the percentile expressed as a proportion. YL refers to the true lower limit of the interval containing the score at the percentile. YU refers to the true upper limit of the interval containing the score at the percentile. i refers to the interval width (YU – YL) cfb refers to the cumulative frequency below the interval containing the score at the percentile. fi refers to the frequency within the interval containing the score at the percentile. 1st Quartile (25th percentile) ()()ic.25nYYibL.25ff−+= Median - 2nd Quartile (50th percentile, 5th decile) ()()ic.50nYYibL.50ff−+= 3rd Quartile (75th percentile) ()()ic.75nYYibL.75ff−+= Range* Range = YMax – YMin Interquartile Range* IQR = Y.75 – Y.25 Semi-Interquartile Range* 2IQRSIQR = Standard Score* If Y is normally distributed, YYσµ-Yz = µz = 0 σz = 1 T Score* T = 10z + 50 µT = 50 σT = 10Formulae - 4 of 16 Updated 23 August 2007 Probability of Occurrence of Event A* P(A) P(A) stands for the proportion of times Event A is expected to occur. P(A) takes on decimal fraction values from 0 to 1. Probability of Non-Occurrence* P(~A) = 1 – P(A) P(~A) stands for the proportion of times event A is expected not to occur. ~A stands for “not A” Probability of Both of Two Independent Events Occurring* P(A and B) = P(A) • P(B) P(B) stands for the proportion of times event B is expected to occur. P(A and B) stands for the proportion of times both event A and event B is expected to occur. Independent events are events where the occurrence of one event has no effect on the probability of occurrence of the other event. Probability of At Least One of Two Independent Events Occurring* P(A or B) = P(A) + P(B) – (P(A) • P(B)) P(A or B) stands for the proportion of times event A, or event B, or both event A and event B is expected to occur. Independent Variable* refers to the speculative causal variable. Levels of the Independent Variable* represent the values that the independent variable may assume. If a study involves an evaluation of an intervention, the independent variable might assume the values 0 = no intervention or 1 = intervention. Dependent Variable* refers to the “effect” or outcome variable. Dependent Measure* refers to the procedure used to measure the level of the dependent variable. It also stands for the category, rank, or score values