## Equal interval methods

Among the productive methods of interval scaling include the method of equal sensor distances and the method of consecutive intervals, as well as various versions of the method of fractionation. They involve dividing the given stimulus or the interval between the stimulus into equal parts and comparing these parts. For example, the subject's task with such a scaling can be to select two other stimuli based on an assessment of the subjective difference of the two stimuli, the subjective distance between which will be identical to the distance between the first two stimuli.

It is believed that the method of equal sensory distances originates in J. Plato's studies in the middle of the 50's. XIX century. He offered eight subjects, in the quality of which were professional artists, to choose a shade of gray, which, in their opinion, is at an equal distance from both white and black colors. According to the Weber-Fechner law, this point on the brightness scale would have to correspond to the average geometric value. However, this assumption was not confirmed. It has been shown that measurement results are more likely to be described by the power relationship between the feeling * E * and the stimulus

**S**

In modern psychology, along with the division of the subjective continuum into two parts, variants with a large number of categories are also used. Because this method is also called another method of seemingly equal intervals. This is used as an option for sequential and simultaneous separation.

Suppose, for example, that a subject needs to divide the scale into four equal intervals. The boundaries of these intervals are denoted by the smallest and largest stimulus in the measurement range. When the test subject is sequentially divided, they are first asked to find a stimulus that is perceived as being somewhere between the given edges

range. After that, the subject must find incentives that are in the middle of the ranges between the minimum and average stimulus, and then between the average and maximum stimulus. This procedure is called the method of successive sensory distances.

With simultaneous separation, these three stimuli should be set by the subject at once. This procedure is considered as a variant of the method appearing to be equal intervals.

## Fractionation method and Stevens law

Another fairly common method of scaling intervals was proposed in the works of Stevens. This method, called the method of fractionation, involves dividing a given amount of stimulus into parts (as a rule, division is used in half). The method became widespread after Stevens used it to construct a scale for perceiving loudness of sounds. This scale is called the scale of the soaps.

One example of a study using the fractionation method, where subjects had to be divided into two standard weights, describes T. Engen. This study was aimed at building a weight scale. Following Stevens, it is called the Vega scale. Four groups of subjects, eight in each group, weighed 900, 550.300 and 150 grams. Each group worked with only one weight. The task of the subjects was to find the weight that is perceived by them as exactly half the size. In other words, it was necessary to reduce twice the subjectively perceived weight.

The experimenter gave the examinee the following instruction: "You will be given a container that you can pick up with your preferred hand. First you will be given a standard weight. Then you will be given another container that will be perceived by you as being lighter or heavier than half the standard weight. All I want from you is that you say that I need to add more weight or subtract a little from this weight until it seems to be equal to half the weight of the standard. You can remove or add weight as many times as you want, until you make a final judgment about half the weight. We will repeat this procedure several times.

To change the mass (hereinafter - weights) of the estimated stimulus, the experimenter used a special measuring spoon. It allowed to change the weight of the stimulus in the range from one to ten percent of the standard weight. Each subject made measurements eight times in alternating ascending and descending samples. The average results of work for each group of subjects are presented in Table. 9.1.

Graphically, these results can be represented as shown in Fig. 9.1. We see that between the standard values of weight and weights, which are subjectively perceived as half of the standard, a pronounced linear dependence is revealed. The coefficient of determination between these values is close to 1. However, the observed dependence is not yet psychophysical, since it reflects the relationships between physical quantities, i.e. expresses the relationship stimulus-stimulus & quot ;. Psychophysical dependence should reflect the relationship between the physical magnitude of the stimulus and its subjectively perceived value ("stimulus-sensation").

* Table 9.1 *

** Average weight values, subjectively perceived as half the weight of the standard **

The standard of weight (.9), r |
Subjectively perceived weight as half the standard (5 1/2 ) g |

900 |
541.6 |

550 |
323.3 |

300 |
159.0 |

150 |
93.5 |

* Fig. 9.1. * The result of the division of four reference weights

In order to obtain such a dependence, we will denote the subjectively perceivable weight corresponding to 900 g, as 100. Then the weight, subjectively equal to 50, will obviously correspond to approximately 542 g.

Substituting this value in the linear regression equation, reduced

e on the graph, we get that the weight of about 320 g is perceived as half of 542 g and, respectively, a quarter of 900 g. Consequently, its subjectively perceivable value is 25. Using this logic of reasoning, we find that the weight equal to about 185 g, is subjectively perceived as 1/8 of 900 g, and a weight of about 103 g is as 1/16. Consequently, their subjectively perceived values are 12.5 and 6.25.

By the described method we get the psychophysical dependence "stimulus-sensation", presented in Fig. 9.2. It can be seen that this dependence is slightly different from linear, having a slight positive acceleration.

* Fig. 9.2. *

**Dependence of subjectively perceived weight on its magnitude**

Weber-Fechner's law assumes that the magnitude of the sensation should be proportional to the logarithm of stimulation. In other words, it is expected that a linear dependence should be observed between the logarithms of stimulation and the magnitude of the sensation.

In order to evaluate the correspondence of the data available to us to this classical law, we translate the values of the weights into logarithmic units and postpone the results of the experiment in logarithmic coordinates of physical weight, as shown in Fig. 9.3. As we see, the obtained dependence turns out to be very far from linear. Thus, the results obtained in the Engen experiment cast doubt on the validity of the Weber-Fechner logarithmic law and obviously require another explanation.

* Fig. 9.3. *

**The relationship between the subjectively perceivable weight and the logarithm of its physical magnitude**

As was shown in the studies of Stevens and his followers, the much obtained dependence can be described by means of a power function known as the Stevens power law. Formally, it can be expressed by the following relationship:

(9.1)

Such a relationship also implies a linear relationship, but not between the sensation and the logarithm of the stimulus, but between the logarithm of sensation and the logarithm of stimulation. Unlike Fechner, who advanced the postulate of the equality of barely noticeable differences and equated them to the magnitude of the differential threshold, Stevens suggested that Weber's law should also be transferred to the sensations of subtle differences. In other words, if the assumption that the ratio A * S/S * is constant, then the assumption that the magnitude of the barely noticeable difference should be proportional to the value of the self sensations, i. assuming a constant A

*Assuming the increments of the stimulus and the sense of infinitesimal values (differentials), we get the following differential equation:*

**E/E.**

Integrating the right and left sides of this equation and performing simple transformations, we obtain

(9.2)

where * C - * is an unknown constant. We represent this constant in the form of a logarithm of an unknown number

*Then equation (9.2) can be rewritten as follows:*

**k.**

(9.3)

Now let's postpone the data we have in double logarithmic coordinates (Figure 9.4). As you can see, now the data almost perfectly correspond to the power law and scale of weights, developed by Stevens.

* Fig. 9.4. *

**The ratio between the weight of the stimulus and its sense in the double**

** logarithmic coordinates **

Using the simple linear regression method, we find the values * n * and logk in equation (9.3). They are equal to 1,28 and -4,13, respectively. Substituting these values into equation (9.2), we obtain the following relationship between the values of sensations and stimulation:

Now you can find the value of * to. * It turns out to be approximately 0.016. Thus, the observed relationship between the magnitude of the stimulus and the magnitude of the sensation can be roughly described by the following power ratio:

* E * = 0.0165 -.

(9.4)

Using equation (9.4), we can calculate the values of sensations predicted by this law for the selected stimuli.

In conclusion of this discussion, we note that the described procedure of stimulus fractionation can be applied also in those cases when the number of factors different from 2 is used as the division coefficient. For example, you can ask the subject to divide the standard into three, four, five or even eight parts. However, division in half is used most often, since this division seems to be more common for most subjects.

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