Descriptive StatisticsFrequency distributionsRegular frequency distributionInterval frequency distributionCumulative frequency distributionGraphs of frequency distributionsShapes of frequency distributionsCentral TendencyModeMedianMeanDisadvantages of the mean. Sometimes, the characteristic that gives the mean an advantage over the mode and the median is also the thing that results in a potential downside. Because every score gets taken into account when computing the mean, scores that are very different from most of the other scores end up having a disproportionately large influence on the mean. In other words, a single or a few outliers on one side of the distribution can pull the mean towards them to the degree that the mean doesn’t look very much like the vast majority of scores in the data set. When this happens the median is often a better choice as a measure of central tendency.The rangeSum of squaresVarianceDegrees of FreedomStandard DeviationThe equation for a standard score is …Percentile scores and the normal curveTransforming percentile scores to standard scoresDescriptive StatisticsDr. Tom PierceDepartment of PsychologyRadford UniversityDescriptive statistics comprise a collection of techniques for understanding what a group of people looks like in terms of the measure or measures you’re interested in. In general, there are four classes of descriptive techniques. First, frequency distributions are used to display information about where the scores in a data set fall along the scale going from the lowest score to the highest score. Second, measures of central tendency, or averages, provide the best single numbers to use in representing all of the scores on a particular measure. Third, measures of variability provide informationabout how spread out a set of scores are. Fourth, the original raw scores one collects are often transformed to other types of scores in order to provide the investigator with different types of information about the research participants in a study. As standard score is a very good example of a transformed score that provides much more information about an individual subject than a raw score can.Frequency distributionsLet’s say that you obtain Beck Depression Inventory scores from each of 400 research participants. The scores on this measure can range anywhere from 1 to 73. Typically, scores fall somewhere between 35 and 55. You’ve got 400 numbers to have to keep trackof here. If someone asks you how the scores in your study came out you could say “well, subject number one had a score of 38, subject two had a 25, subject three had a 46,…”. You get the idea. This is too many number for anyone to able to look at them and beable to get a general ideas about where most of the scores fall on the scale and how spread out the scores are around this point on the scale. The job of a frequency distribution is to take a very large set of numbers and to boil it down to a much smaller set of numbers – a collection of numbers that is small enough for the pathetically limited human mind to keep track of at one time. A good frequency distribution allows the consumer to extract the necessary information about the scores in the data set while working within our cognitive limitations.Regular frequency distributionThe most straight-forward example of a frequency distribution goes like this. Let’s say that you’re given ratings of teaching effectiveness for the students in a large Introduction to Psychology class. There are 400 students in the class. The questionnaire provides students with 15 statements and the student is asked to pick a number between one and five that indicates the degree to which they agree or disagree with each statement. One ofthese statements is “The instructor in this course is outstanding”. A response of “5” indicates that the students agrees with the statement completely. A response of “one” indicates that the student disagrees with the statement completely. A regular frequency Descriptive Statistics – Version 1.4©2008 by Thomas W. PierceRevised 3/17/081distribution will allow the instructor to see how many of the students rated him or her on every possible score ranging from one to five. In other words, how many students gave the instructor a “one”, how many gave them a “two, and so on. You get the idea. This information is often displayed in the form of a table.Table 1.1X f --- --- 5 1504 2003 402 101 0 ----- 400There are two columns of numbers in this table. There is a capital X at the top of the column on the left. Every possible raw score that a subject could provide is contained in this column. A capital X is used to label this column because a capital X is the symbol that is usually used to represent a raw score. The column on the right is labeled with a small-case letter f. The numbers in this column represent the number of times – or the frequency -- that each possible score actually showed up in the data set. The letter f at the top of the column is just short-hand for “frequency”.Thus, this dinky little table contains everything the instructor needs in order to know every score in the data set. Instead of having to keep track of 400 numbers, the instructoronly has to keep track of five – the number of times each possible score appeared in the data set. This table is said to represent a frequency distribution because the it shows us how the scores in the set are distributed as you go from the smallest possible score in the set to the highest possible score. It basically answers the question “where did the scores fall on the scale” This particular example is said to represent a regular frequency distribution because every possible score is displayed in the raw score (capital X) column.Interval frequency distributionA little bit different situation where a frequency distribution might come in handy is in displaying the IQ scores collected from 90 people. In a random sample of people drawn from the population, what would you expect these IQ scores to look like? What is the lowest score you might expect to see in the set? What’s the highest score you might reasonably expect to see in the data set? It turns out that the lowest score in this particular set is 70 and the highest score is 139. Is it reasonable to display these data in a