Single sample testsTwo related samples testsTwo Independent Samples TestsThree or more independent samples testsSTATISTICS REVIEWIntroductionStudents are often intimidated by statistics. This brief overview is intended to place statistics in context and to provide a reference sheet for those who are trying to interpret statistics that they read. It does not attempt to show or to explain the mathematics involved. Although it is helpful if those who use statistics understand the math, the computer age has rendered that understanding unnecessary for many purposes. Practically speaking, students often simply want to know whether a particular result is significant, i.e. how likely it is that the obtained result may be attributable to something other than chance. Computer programs can easily produce numbers that allow such conclusions, if the student knows which tests to use and has an understanding of what thenumbers mean. This summary is intended to help achieve that understanding.Basic ConceptsVariablesMost statistics involve at least two variables: an independent variable and a dependent variable. The independent variable is the one that the researcher focuses on as influencing the dependent variable. The dependent variable “depends” on the independent variable. For example, height “depends” on age: As individuals age, they usually grow taller. One cannot alter age by withholding food and thereby stunting the growth of children, so age cannot logically “depend” on height. As another example, scores on a test “depend” on the amount of knowledge an individual has on the subject matter. Assigning higher test scores would not increase knowledge.AttributesEvery variable has attributes or components that constitute the variable. For example, theattributes of gender include male, female, and transgendered. The attributes of age (in years) include all of the numbers from zero to 122 or so. In research, the list of a variable’s attributes has to be exhaustive, that it, it has to cover all possibilities, and the attributes have to be discrete, that is they cannot overlap with each other. Levels of MeasurementEvery variable fits into one and only one level of measurement: nominal, ordinal, interval, or ratio.Nominal variables are differentiated by names (from Latin nomen, which means name). Examples include ethnicity, language group, and hair color.Ordinal variables are differentiated by their order (from Latin ordo, which means order) in relationship to each other. Examples include class rank, place of finish in a race, and birth order. We know which individual is above or below another individual based on their places in the ordered list, but the place on the ordered list tells nothing about the amount of difference between any two individuals. For example, the second place finisher in a race could have been behind the first place finisher by .01 second or by 100 seconds, but if all we know is the order of finishing, we know nothing about the closenessof their times. Likert scales produce ordinal levels of measurement. Interval variables are differentiated by a measurement that has regular intervals, such as inches, degrees Celsius, and some test scores. With interval measurements, unlike ordinal measurements, one could properly say that one individual was twice as tall as another or that one individual did half as well on a test as another individual.Ratio variables are similar to interval variables, except ratio variables have a true zero. Examples include degrees Kelvin (but not Celsius or Fahrenheit), number of children, number of times married, and number of statistics tests passed.Descriptive and Inferential StatisticsStatistics are divided into two general categories—descriptive and inferential. Descriptivestatistics include mean, median, mode, range, sum of squares (sum of squared deviation scores), variance (also known as mean square, which is short for mean of squared deviation scores), and standard deviation. It is assumed that graduate students have somefamiliarity with each of these statistics.Scores are often converted to z scores. A z score is simply a number that shows how far ascore is from a mean. Therefore, a z score could range from 0 to infinity, but practically speaking, a z score of 4, which means that the raw score is 4 standard deviations from the mean, is so large that the area under the distribution curve beyond the z score is less than 0.0001.Although not a statistic, a regression line is a line through a data plot that best fits the data. The line produces the lowest possible difference between actual values and predicted values.Other descriptive statistics include the error sum of squares (SSE), regression sum of squares (SSR) and total sum of squares (SST). SSE + SSR = SST.Another descriptive statistic is proportion of variance explained (PVE). PVE is a measure of how well the regression line predicts the scores that are actually obtained.A correlation coefficient is a measure of the strength of a relationship between two variables. It is represented by the symbol r, and it can range between –1 and +1. Thenegative and positive signs reflect the direction of the slope of the line that shows the relationship between the two variables. The standard error of estimate is the standard deviation of the prediction errors. It tells how spread out scores are with respect to their predicted values.Inferential statistics involve tests to support conclusions about a population from which a sample is presumed to have been drawn. The tests that follow are considered to be inferential statistics.AssumptionsAll tests are based on the assumption that samples are randomly selected and randomly assigned and that individuals are independent from each other, i.e. that one member’s score does not influence another member’s score.Parametric tests are based on the assumption that populations from which samples are drawn have a normal distribution. Nonparametric tests do not have this assumption.Each test has other assumptions, such as regarding the type of data and the number of data points. The following test descriptions outline those assumptions.Number of SamplesDifferent situations require different testing procedures. The following discussion is organized according to the number of samples that is being evaluated: one sample, two samples, and more than two samples. In each category, both parametric and nonparametric