Skewness and kurtosis
James Dean Brown
University of Hawai'i at Manoa
[ p. 20 ]The ses can be estimated roughly using the following formula (after Tabachnick & Fidell, 1996): For example, let's say you are using Exceltm and calculate a skewness statistic of -.9814 for a particular test administered to 30 students. An approximate estimate of the ses for this example would be: Since two times the standard error of the skewness is .8944 and the absolute value of the skewness statistic is -.9814, which is greater than .8944, you can assume that the distribution is significantly skewed. Since the sign of the skewness statistic is negative, you know that the distribution is negatively skewed. Alternatively, if the skewness statistic had been positive, you would have known that the distribution was positively skewed. Yet another alternative would be that the skew statistic might fall within the range between - .8944 and + .8944, in which case, you would have to assume that the skewness was within the expected range of chance fluctuations in that statistic, which would further indicate a distribution with no significant skewness problem.
|". . . reporting the median along with the mean in skewed distributions is a generally good idea."|
[ p. 21 ]
[ p. 22 ]Another practical implication should also be noted. If a distribution of test scores is very leptokurtic, that is, very tall, it may indicate a problem with the validity of your decision making processes. For instance, at the University of Hawai'i at Manoa, we give a writing placement test for all incoming native-speaker freshmen (or should that be freshpersons?) that produces scores on a scale of 0-20 (each student's score is based on four raters' scores, which each range from 0-5). Yearly, we test about 3400 students. You can imagine how tall the distribution must look when it is plotted out as a histogram: 20 points wide and hundreds of students high. The decision that we are making is a four way decision about the level of instruction that students should take: remedial writing; regular writing with an extra lab tutorial; regular writing; or honors writing. The problem that arises is that very few points separate these four classifications and that hundreds of students are on the borderline. So a wider distribution would help us to spread the students out and make more responsible decisions especially if the revisions resulted in a more reliable measure with fewer students near each cut point.
|"interpreting . . . [statistics] depends heavily on the type and purpose of the test being analyzed."|
[ p. 23 ]