Descriptive Statistics Tom Pierce 1/7/2003 1Descriptive Statistics Tom Pierce Department of Psychology Radford University Descriptive 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 information about 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 distributions Let’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 track of 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 be able 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 distribution The 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 of these 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 frequencyDescriptive Statistics Tom Pierce 1/7/2003 2distribution 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 1.1 X f --- --- 5 150 4 200 3 40 2 10 1 0 ----- 400 There 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 instructor only 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 distribution A 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 regular frequency distribution? No! So why not?Descriptive Statistics Tom Pierce 1/7/2003 3What would you have to do to generate a regular frequency distribution for these data? You’d start with the highest possible score at the top of the raw score column and in each row below that you’d list the next lowest possible raw scores. Like this… Table 1.2 X --- 139 138 137 136 You get the idea. There are SEVENTY possible raw scores between 70 and 139. That means there would be 70 rows in the table and 70 numbers that you’d have to keep track of. This is too many! The whole idea is to keep the number of values that you have to keep track of to somewhere between five and ten. So what can you do? A regular frequency distribution isn’t very efficient when the number of possible raw scores is greater than ten or twelve. So it’s not a good idea to keep track of how often every possible raw score shows up in the set. You’re best bet is to