# Abnormal distribution models - Mathematical methods in psychology

## Abnormal distribution models

The law of normal distribution appears in mathematical statistics as a standard law of distribution of a random variable. On the basis of this law, most classical methods of statistical data analysis have been developed. The significance and universality of this law are reflected in the important position of mathematical statistics, which is known as the central limit theorem . According to this provision, the distribution of the mean values ​​of the variable in question in random samples will be approximately normal in form, regardless of the form of its distribution in the general population under the conditions that the sample size is sufficiently large and that the dispersion of the population is limited.

It should be noted that in psychology we often encounter a situation where the normal distribution of the investigated quantities is impossible. Thus, by examining the distribution of the reaction time of the subjects, one hardly expects their normal distribution, since the subject is limited in the ability to accelerate his reactions infinitely much (even at the expense of stimulus anticipations), but at the same time can indefinitely slow them down. Thus, there is an uncontrolled by the experimenter a side factor acting in one direction. As a result, it turns out that the variances of relatively fast and relatively slow answers are equal. Similarly, if we examine any proportion of the subjects' answers, for example, the percentage of problems solved by the subjects, we should be prepared for the variance of these indicators to differ depending on how close these values ​​are to the extreme values. In this case, it may be necessary to evaluate additional distribution parameters. As such, asymmetry and kurtosis are usually used.

Asymmetry and excesses are the central moments of the third and fourth order, respectively. In the case of a normal distribution, they are equal to zero values. The more these parameters differ from the zero ones, the more the distribution of the random variable differs from the normal one. In this case, there may be differences in the direction of both positive and negative values.

Negative asymmetry occurs when the values ​​predominate in the distribution, which are smaller in magnitude; the variance of smaller values ​​of a random variable turns out to be much larger than the variance of large values.

Suppose a researcher has constructed an intelligent test that differentiates people with relatively low intelligence and practically does not distinguish people with a relatively high level of intelligence. Then all those subjects who have relatively higher intellectual abilities, will be on this test to gain approximately the same points. However, subjects who have a relatively low level of intellectual development will differ to a much greater extent. Thus, when applying this test, we must certainly meet with a negative asymmetry in the distribution of empirical scores.

The opposite should occur in the case of positive asymmetry. By the way, the reaction time distribution of the subject, as a rule, has a positive asymmetry, so that relatively long reactions have significantly greater variability (variance) than the more rapid ones. The lack of positive asymmetry in the measurement of the reaction time can actually indicate that the subject predicts his answer by guessing the stimuli before they are presented, so that the very fact of positive or negative asymmetry is by no means indicative of the incorrectness of the experimental procedure, how sometimes it can to be presented.

The excess is also positive and negative. Negative kurtosis indicates that there is a more or less even distribution of the quantities of the random variable of interest to us. In this case, according to empirical data, it is difficult to estimate the mode of distribution, since the frequencies of the appearance of different values ​​of the investigated variable turn out to be approximately the same. Sometimes a negative kurtosis may even indicate the existence of two modes of distribution. The bimodal distribution may indicate that the experimenter actually deals not with one but with two variables describing two different ways of behavior.

For example, if in the study of the reaction time in a group of subjects there is a bimodal distribution, this may indicate that a significant part of the subjects tried to predict the appearance of the stimulus to which they should respond, while the other part performs the experimenter's instruction accurately, target stimulus. If the bimodal distribution is found in the assessments of two different teachers who take the same exam independently of one group of students, this may indicate that the criteria for the quality of the students' answers differ among the two teachers.

A positive kurtosis reflects the fact that random variables are closely grouped around a single value. This can, in particular, indicate that the means used by the experimenter for measuring the investigated characteristic were insufficiently sensitive. In differential psychometrics-psychodiagnostics-a positive kurtosis may indicate, inter alia, that the key used to assess the personality properties of the questionnaire is not correct, or that the subjects, having unraveled the focus of the questionnaire, try to balance their positive and negative responses. As a result, it may turn out that the majority of subjects demonstrate a level close to the average by the measured property.

There are special procedures that allow you to estimate the level of the asymmetry of its distribution from the observed values ​​of the random variable, as well as the excess.

So, the asymmetry value can be estimated by the following formula:

(1.6)

where s is the standard deviation value.

Similarly, the magnitude of the excess is estimated:

(1.7)

However, with manual calculations, these formulas are practically not applied, since they assume rather tedious work, while the estimates themselves are usually needed only as a tool for estimating the normality of the distribution, i.e. play an auxiliary role. Therefore, when evaluating these distribution parameters, it is better to use a computer. The corresponding statistical functions include almost all spreadsheets, as well as specialized statistical packages, almost always include the possibility of calculating kurtosis and asymmetry in their basic descriptive statistics modules. If the calculated values ​​are close to zero, then we are dealing either with a normal distribution, or close to normal. If the values ​​of asymmetry or kurtosis, or both, exceed ± 2, this is most likely indicative of the anormal distribution of the investigated quantity.

Thus, the processing of experimental data begins with the evaluation of the distribution parameters of the random variables under study: mathematical expectation, variance (standard deviation), asymmetry and kurtosis.

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