The Correlation between Mean and Standard Deviation

The particular phenomenon, to be studied in this paper, is, that in concentration tests there exists a positive correlation across subjects between the mean and standard deviation of the reaction times. This phenomenon will be described using the Bourdon-Vos Test developed by Vos (1992). This test is a modification for children of the Bourdon-Wiersma Test (Huiskam & de Mare, 1947). It was originally proposed by Abelson (1911). A similar test was mentioned by Spearman (1927), who called it ''Fours'' (crossing out groups of four dots from among groups of three, four, and five dots). The test consisted of 33 lines, with each line containing, in random order, eight groups of three dots, eight groups of four dots, and eight groups of five dots. The average diameter of the groups of dots was about 5 mm. The examinees marked each group of four dots "as quickly and as accurately" as possible. Two lines on the back of the test sheet illustrated the procedure, with the instructor doing the first line to demonstrate how to perform the task. The examinee completed the second line as a practice. For the application of the previously mentioned inhibition models the task should be over-learned in advance. Note, that this requirement is not met here. This would be the case if the complete test, and not just one line, could have been practiced several times before actually administering the test. In the actual test, at the end of each line, examinees should start immediately with of the next line without a break. The execution time spent on each line was registered in seconds using a stop watch, resulting in a series of 33 consecutive RTs.

The correlation between mean and standard deviation was computed on the basis of the standardization group used by Vos (1992). This group consisted of 732 subjects, ranging from six years old to eighteen years old. (377 boys and 323 girls). The sample is described in more detail in Vos (1992). The number of children age 18 (N = 2) was relatively low in comparison to the other age groups. Therefore, these two subjects were excluded from the analysis. Moreover, 15 datasets were suspicious (last reaction time equal to 99). These datasets were also removed from the sample, leaving 715 children. In van der Ven and Smit (1989) a subsample of 516 cases was used. For each subject the mean and standard deviation across the reaction times 1 to 30 was computed. The reaction times 31 to 33 were disregarded due to an end spurt effect (see van der Ven, Smit and Jansen, 1989). Next the correlations between these two measures across subjects (N=715) was computed. The correlation between mean and standard deviation was equal to 0.804. The correlation between the natural logarithm of the mean and the natural logarithm of the standard deviation was equal to 0.859. The correlation between the standard deviation and the mean deviation was equal to 0.989. The latter correlation shows that it does not make much difference whether one uses the mean deviation or the standard deviation as a measure for concentration. It has been shown in van der Ven and Smit (1989) that the reaction time curves usually show trend: increasing curves as well as decreasing curves, the first resulting from perseveration and the latter from learning (see van der Ven, 1997). The correlation between mean and standard deviation is therefore, partly influenced by the trend in the curves. In order to reduce the effect of trend, it was decided rather arbitrarily to compute the mean and the standard deviation of the reaction times 7 to 30, considering that part of the reaction time series as more or less stationary. The correlation between mean and standard deviation was now equal to 0.778. The correlation between mean deviation and standard deviation was again equal to 0.989. The correlation between the natural logarithm of the mean and the natural logarithm of the standard deviation was equal to 0.836. According to all inhibition models developed thusfar (including the PE model) mean and variance both depend on the real total working time A. The correlation between these variables could therefore be explained by the common variable A. One could use the minimum reaction time as an estimate for A. If A does account for the correlation between the mean and the variance, then the correlation between these variables with the effect of the minimum reaction time partialled out should tend to zero. However, the partial correlations have still large values. The partial correlation between mean and standard deviation, both computed from the reaction times 7 to 30, partialled out for the minimum of the reaction times 1 to 30, was equal to 0.688. The partial correlation between the logarithm of the mean and the logarithm of the standard deviation, partialled out for the logarithm of the minimum reaction time, was equal to 0.746. The effect of the minimum reaction time seems negligible.

These results make very clear, that, if one uses the standard deviation as a measure for concentration, one has to account for the high correlation of the standard deviation with the mean. As long as no explanation is found for this phenomenon, one is not able to make any statement about the specific nature of the standard deviation in comparison to the mean. In the next part of this paper a particular hypothesis regarding this correlation is formulated in terms of the PE model, which is used as a limiting case of the PI model. For reasons of mathematical convenience, from now on the variance will be used as a measure of concentration instead of the standard deviation.