Semester Fall 2010
"Research Methods (STA630)"
Schedule
Opening Date and Time
November 22, 2010 At 12:01 A.M. (Mid-Night)
Closing Date and Time
November 24 , 2010 At 11:59 P.M. (Mid-Night)
Topic/Area for Discussion
"MEASUREMENT OF CONCEPTS"
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Discussion Question
"Which of the scale among ratio and interval scale is better? Justify our answer"
"Which of the scale among ratio and interval scale is better? Justify our answer"
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Solution:
I think Ration Scale is better:
Interval Scale:-Permissible Statistics
mean, standard deviation, correlation, regression, analysis of variance
Ratio Scale:-Permissible Statistics
All statistics permitted for interval scales plus the following: geometric mean, harmonic mean, coefficient of variation, logarithms
All statistics permitted for interval scales plus the following: geometric mean, harmonic mean, coefficient of variation, logarithms
Interval scale
Quantitative attributes are all measurable on interval scales, as any difference between the levels of an attribute can be multiplied by any real number to exceed or equal another difference. A highly familiar example of interval scale measurement is temperature with the Celsius scale. In this particular scale, the unit of measurement is 1/100 of the difference between the melting temperature and the boiling temperature of water at atmospheric pressure. The "zero point" on an interval scale is arbitrary; and negative values can be used. The formal mathematical term is an affine space (in this case an affine line). Variables measured at the interval level are called "interval variables" or sometimes "scaled variables" as they have units of measurement.
Ratios between numbers on the scale are not meaningful, so operations such as multiplication and division cannot be carried out directly. But ratios of differences can be expressed; for example, one difference can be twice another.
The central tendency of a variable measured at the interval level can be represented by its mode, its median, or its arithmetic mean. Statistical dispersion can be measured in most of the usual ways, which just involved differences or averaging, such as range, interquartile range, and standard deviation. Since one cannot divide, one cannot define measures that require a ratio, such as studentized range or coefficient of variation. More subtly, while one can define moments about the origin, only central moments are useful, since the choice of origin is arbitrary and not meaningful. One can define standardized moments, since ratios of differences are meaningful, but one cannot define coefficient of variation, since the mean is a moment about the origin, unlike the standard deviation, which is (the square root of) a central moment.
Ratio measurement
Most measurement in the physical sciences and engineering is done on ratio scales. Mass, length, time, plane angle, energy and electric charge are examples of physical measures that are ratio scales. The scale type takes its name from the fact that measurement is the estimation of the ratio between a magnitude of a continuous quantity and a unit magnitude of the same kind (Michell, 1997, 1999). Informally, the distinguishing feature of a ratio scale is the possession of a non-arbitrary zero value. For example, the Kelvin temperature scale has a non-arbitrary zero point of absolute zero, which is denoted 0K and is equal to -273.15 degrees Celsius. This zero point is non arbitrary as the particles that compose matter at this temperature have zero kinetic energy.
Examples of ratio scale measurement in the behavioral sciences are all but non-existent. Luce (2000) argues that an example of ratio scale measurement in psychology can be found in rank and sign dependent expected utility theory.
All statistical measures can be used for a variable measured at the ratio level, as all necessary mathematical operations are defined. The central tendency of a variable measured at the ratio level can be represented by, in addition to its mode, its median, or its arithmetic mean, also its geometric mean or harmonic mean. In addition to the measures of statistical dispersion defined for interval variables, such as range and standard deviation, for ratio variables one can also define measures that require a ratio, such as studentized range or coefficient of variation.
Interval Scale
An interval scale assumes that the measurements are made in equal units. However, an interval scale does not have to have a true zero. Good examples of interval scales are the Fahrenheit and Celsius temperature scales. A temperature of "zero" does not mean that there is no temperature...it is just an arbitrary zero point. An Interval Scale
Ratio Scale
Ratio scales are similar to interval scales. A ratio scale allows you to compare differences between numbers. For example, if you measured the time it takes 3 people to run a race, their times may be 10 seconds (Racer A), 15 seconds (Racer B) and 20 seconds (Racer C). You can say with accuracy, that it took Racer C twice as long as Racer A. Unlike the interval scale, the ratio scale has a true zero value.
INTERVAL SCALE
A characteristic of data such that the difference between two values measured on the scale has the same substantive meaning/significance irrespective of the common level of the two values being compared. This implies that scores may meaningfully be added or subtracted and that the mean is a representative measure of central tendency. Such data are common in the domain of physical sciences or engineering - e.g. lengths or weights. Also see : MEASUREMENT TYPE, SCALE TYPES, STEVENS' TYPOLOGY.
RATIO SCALE
This is a type of MEASUREMENT SCALE for which it is meaningful to reason in terms of differences in scores (see INTERVAL SCALE) and also in terms of ratios of scores. Such a scale will have a zero point which is meaningful in the sense that it indicates complete absence of the property which the scale measures. The RATIO SCALE may be either unipolar (negative values not meaningful) or bipolar (both positive and negative values meaningful), and either continuous or discrete.