Hockenbury Discovering Psychology 5th txtbk

A-6 APPENDIX A Statistics: Understanding Datameasure of variabilityA single number that presents informationabout the spread of scores in a distribution.rangeA measure of variability; the highest score ina distribution minus the lowest score.Although the mean is usually the most representative measure of central tendencybecause each score in a distribution enters into its computation, it is particularlysusceptible to the effect of extreme scores. Any unusually high or lowscore will pull the mean in its direction. Suppose, for example, that in our frequencydistribution for aerobic exercise one exercise zealot worked out 70 hoursper week. The mean number of aerobic exercise hours would jump from 2.33 to4.43. This new mean is deceptively high, given that most of the scores in the distributionare 2 and below. Because of just that one extreme score, the mean hasbecome less representative of the distribution. Frequency tables and graphs areimportant tools for helping us identify extreme scores before we start computingstatistics.Figure A.4 Distributions with DifferentVariability Two distributions with thesame mean can have very different variability,or spread, as shown in these twocurves. Notice how one is more spread outthan the other; its scores are distributedmore widely.MeanMeasures of VariabilityIn addition to identifying the central tendency in a distribution, researchers maywant to know how much the scores in that distribution differ from one another. Arethey grouped closely together or widely spread out? To answer this question, weneed some measure of variability. Figure A.4 shows two distributions with thesame mean but with different variability.A simple way to measure variability is with the range, which is computed by subtractingthe lowest score in the distribution from the highest score. Let’s say thatthere are 15 participants in the traditional diet and exercise group and that theirweights at the beginning of the study varied from a low of 95 pounds to a high of155 pounds. The range of weights in this group would be 155 95 60 pounds.As a measure of variability, the range provides a limited amount of informationbecause it depends on only the two most extreme scores in a distribution (the highestand lowest scores). A more useful measure of variability would give some idea ofthe average amount of variation in a distribution. But variation from what? Themost common way to measure variability is to determine how far scores in a distributionvary from the distribution’s mean. We saw earlier that the mean is usually thebest way to represent the “center” of the distribution, so the mean seems like anappropriate reference point.What if we subtract the mean from each score in a distribution to get a generalidea of how far each score is from the center? When the mean is subtracted from ascore, the result is a deviation from the mean. Scores that are above the mean wouldhave positive deviations, and scores that are below the mean would have negativedeviations. To get an average deviation, we would need to sum the deviations anddivide by the number of deviations that went into the sum. There is a problem withthis procedure, however. If deviations from the mean are added together, the sumwill be 0 because the negative and positive deviations will cancel each other out. Infact, the real definition of the mean is “the only point in a distribution where all thescores’ deviations from it add up to 0.”We need to somehow “get rid of” the negative deviations. In mathematics, sucha problem is solved by squaring. If a negative number is squared, it becomes positive.So instead of simply adding up the deviations and dividing by the numberof scores (N), we first square each deviation, then add together the squareddeviations and divide by N. Finally, we need to compensate for the squaringoperation. To do this, we take the square root of the number just calculated.This leaves us with the standard deviation. The larger the standard deviation,the more spread out are the scores in a distribution.Let’s look at an example to make this clearer. Table A.2 lists the hypotheticalweights of the 15 participants in the traditional group at the beginning ofthe study. The mean, which is the sum of the weights divided by 15, is calculatedto be 124 pounds, as shown at the bottom of the left-hand column. Thefirst step in computing the standard deviation is to subtract the mean from eachscore, which gives that score’s deviation from the mean. These deviations are