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Scaling

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Scaling seeks to discover how varying the physical parameters of the stimulus affects the psychological parameters. In general, scaling is concerned with the question "How does subjective magnitude (perception) vary with changes in physical magnitude?" e.g. If we double the intensity of a light does it look twice as bright? Investigators have attempted to determine the functional form of this relationship - basically to find functions to describe the relationship between psychological and physical variations in suprathreshold stimuli.

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Some Terminology The type of scaling with which we are concerned here, usually called unidimensional scaling Two different types of dimensionsprothetic and metathetic. Prothetic - dimensions of quantity - how much? Metathetic - dimensions of quality - what kind? Unidimensional scaling is generally concerned with prothetic dimensions

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Indirect Scaling Weber's law This law states that the smallest difference in intensity that can be detected is directly proportional to the background intensity. Ernst Weber (1795-1878)

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Suppose we want to want to find the just noticeable difference (jnd) needed to detect an increment in light intensity We would find that the increment needed was a constant proportion of the background intensity

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Webers Law I/I = K where K is a constant I = small incremental intensity I = background intensity

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But, Weber's does not hold for all values of I. In fact, it's often the case that for very small values of I, Weber's law starts to break down, but if we add Generalize Law to I = K( I + I o ) where I o is the absolute threshold--the smallest intensity of a given physical dimension we can reliably detect. This law holds fairly well over a wide range of background intensities.

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Weber Fraction Rough Index of Sensitivity

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Webers law suggests that standard physical scales are not appropriate for representing perceptual experience The physical difference between 1 and 2 mm in length is the same as between 101 and 102 m But Webers law tells us that while we could readily discriminate the first difference (between 1 and 2 cm), we couldnt the second (between 101 and 102 cm) Fechner wished to construct a psychological scale that would reflect such differences

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Fechner used Webers law to help understand the relationship between changes in physical intensity and the psychological experience of those changes - scaling Fechner argued that our subjective experience of intensity changes is related in a logarithmic fashion to changes in physical intensity Fechner's Law

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Fechner needed a measurement unit that could be used to describe the quantity of the sensory experience. 1) He assumed Weber's Law was correct. ( It does hold over a wide range of stimuli) 2) He assumed that the subjective impression of the difference between 2 stimuli separated by one JND was the same regardless of the absolute magnitude of the 2 stimuli - JND - atom of sensation. 3) Absolute threshold was 0 point. 4) One can add and subtract JND's.

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Fechner was saying that regardless of its size in physical units the JND is the standard unit of sensation magnitude because it is the smallest detectable increment in a sensation and therefore always psychologically the same. Key Assumption

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How does one determine the number of JND's above threshold corresponding to values of the physical stimulus. Empirically: Measure first I above threshold - gives first JND. This is starting point for the measurement of the next I - gives second JND.

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However, Fechner assumed the validity of Weber's Law. Hence, e.g. if I/I=1/5 and the absolute threshold is 10, the first JND is 10*1/5+10=12. Next is 12*1/5+10=14.4 # of JNDs Stim. intensity Log stim. intensity 0 10.0 1.00 1 12.0 1.079 2 14.4 1.158 3 17.28 1.238 etc...

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Fechner's Law: S = k log I S = subjective magnitude k = a constant number appropriate for the modality (like the Weber fraction) I = physical intensity

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Fechners law means that as the magnitude of the physical intensity increases, the magnitude of our subjective intensity increases rapidly at first, but then slows.

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Does doubling intensity double brightness? set k =1 I = 10 ---> S = 1 I = 20 ---> S = 1.3 If I = 100, P = 2

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Problems with Fechner's law: 1) All JNDs are not equal --> e.g. sound 20 JND's above threshold is more than twice as loud as one 10 JND's above. 2) problems with Weber's law at low intensities --> generality of Fechner's law restricted to certain ranges. 3) S.S. Stevens proposed an alternative equation that describes the relationship between sensation and stimulus intensity for a wide range of senses more accurately than Fechner's law.

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Stevens Law Stevens used a technique called magnitude estimation - direct method in that one assign numerals to reflect perception For example, in judging the length of lines first given a standard length (e.g. 10 unit length) Variety of stimuli then presented and task is to assign numerals to these other stimuli in relation to the one with a value of 10. e.g. if you see a line that appears twice as long, you should assign it a number 20. If you see a line that appears 1/5 as long, you should assign it a number 2. S. S. Stevens (1906-1973)

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Brightness - response compression - e.g. doubling intensity causes less than doubling brightness. Shock - response expansion - e.g. doubling intensity cause more than doubling of sensation. Line Length - linear - e.g. doubling line length doubles observer's estimate.

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P = kI n Perceived magnitude, P, equals a constant, k, times the stimulus intensity, I, raised to a power, n. Steven's Power Law

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What about all these curves? Does the relationship between the intensity of a stimulus and our perception of its magnitude follow different rules for different senses?

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No! Replot data On log-log paper gives straight lines Such data are called power functions

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Log P= n Log I+Log k estimate n = y/ x (slope) (intercept) Any systematic deviation of the data points from a straight line on log-log graph indicates that the function is not a power function: (y=ax+b) Take log of power function - P = kI n

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Take out a piece of blank paper and label the lines from A to H Judge the area of each circle relative to the standard Do not measure, simply write down your subjective impression A little demo of Magnitude Estimation

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Standard circle. Call its area 1

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A

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B

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C

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D

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E

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F

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G

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H

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What is the correct law – Fechners or Stevens? Most present day psychophysicists would likely opt for some more direct method of measuring the psychophysical function than the indirect method used by Fechner. Fechners assumptions seem to have been wrong in at least one very fundamental respect – his assumption that jnds of different physical size have the same subjective size. There is still controversy over which formulation may be correct and whether the two 'laws' can be reconciled. It is even possible that both the power function and the logarithmic function are valid under different conditions - or maybe neither is valid! An important feature is shared by both laws - the relationship between stimulus and sensation is a downward curve, so larger increments are needed to have similar effects at higher stimulus levels.

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Appendix to Chapter 1 Mathematics Used in Microeconomics © 2004 Thomson Learning/South-Western.

Appendix to Chapter 1 Mathematics Used in Microeconomics © 2004 Thomson Learning/South-Western.

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