FREQUENCY DISTRIBUTIONS Page 1 of 6 I. Introducton 1. The dea of a frequency dstrbuton for sets of observatons wll be ntroduced, together wth some of the mechancs for constructng dstrbutons of data. Then ths dea of a frequency dstrbuton wll be paralleled by that of probablty dstrbuton. 2. Pror to any concerns wth summarzng the data or wth comparng the obtaned results wth some theoretcal state of affars, the nvestgator must frst measure the phenomena under study. 3. It therefore seems approprate at ths pont to menton the process of measurement and to ndcate the role that measurements consderatons wll play n the remander of ths text. II. Measurement Scales 1. Whenever the scentst makes observatons of any knd, some classfyng and recordng scheme must be used. Any phenomenon or "thng" wll have many dstngushable characterstcs or attrbutes, but the scentst must frst sngle out those propertes relevant to the queston beng studed. 2. Once the scentst has sngled out the property or propertes to be studed and establshed controls for the others, the classfyng scheme s appled to each
FREQUENCY DISTRIBUTIONS Page 2 of 6 observaton. Such a scheme s essental n order to record, categorze, and communcate the thngs observed. 3. At ts very smplest, ths scheme s a rule for arrangng observatons nto equvalence classes, so that observatons fallng nto the same set are thought of as qualtatvely the same and those n dfferent classes as qualtatvely dfferent n some respect. In general, each observaton s placed n one and only one class, makng the classes mutually exclusve and exhaustve. 4. The process of groupng ndvdual observatons nto qualtatve classes s measurement at ts most prmtve level. Sometmes ths s called categorcal or nomnal scalng. The set of equvalence classes s called a nomnal scale. 5. The word measurement s usually reserved, however, to refer to the stuaton n whch each ndvdual s assgned a number; ths number reflects a magntude of some quanttatve property. A defnte rule must exst assocatng one and only one number wth each ndvdual, and the measurement procedure flls the role of ths rule. 6. There are at least three knds of numercal measurement that can be dstngushed: these are often called ordnal scalng, nterval scalng, and rato scalng. 7. Before talkng about these scales we need to dscuss measurng. Ideally, n measurng an object o we should be able to determne ts t(o ) true value drectly. Unfortunately, ths s not always (or even usually) possble.
FREQUENCY DISTRIBUTIONS Page 3 of 6 8. Rather, what we must do s to devse a procedure for parng each o wth another number, say m(o ), that we call ts numercal measurement. The actual procedure we use to assgn the m(o ) value consttutes a measurement rule. 9. However, just any old procedure, regardless of how precse and "scentfc" t appears, wll not do: we want the varous values of m(o ) assgned to the varous possble objects o at least reflect the t(o ) values showng the dfferent degrees of the property. 10. A measurement rule would surely be nonsense f t gave numbers havng no connecton at all wth the true amounts of some property that dfferent objects possess. 11. Even though we may never be able to determne the t(o ) value for any object o exactly, we at least hope to fnd numbers m(o ) that wll be related to these true values n a systematc way. The measurement numbers we obtan must be good reflectons of the true quanttes, so that nformaton about magntudes or amounts of the property can be at least nferred from the value observed. 12. Measurement operatons or procedures dffer n the nformaton that the numercal measurements themselves provde about the true magntudes. Some ways of measurng permt us to make very strong statements about what dfferences or ratos among the true magntudes must be, and thus about the actual dfferences n, or proportonal amounts of, some property that dfferent objects possess.
FREQUENCY DISTRIBUTIONS Page 4 of 6 13. On the other hand, some measurement operatons permt only the roughest nferences to be made about true magntudes from measurement numbers themselves. 14. Now suppose that we have a measurement procedure that gves a number m(o ) to any object o, and also gves a number m(o j) to a dfferent object o j. Then we say that ths s measurement at the ordnal level f the followng statements are true: 1. m(o ) P m(o j) mples that t(o ) P t(o j). 2. m(o ) > m(o j) mples that t(o ) > t(o j). (Hays, 1988, p.68) 15. In other words, when we measure the ordnal level, we can say at least that f two measurements are unequal, the true magntudes are unequal, and f one measurement s larger than another, one magntude exceeds another. However, we really cannot say by how much the object truly dffer on property n queston. 16. Other measurement procedures gve numbers where much stronger statements can be made about the true magntudes from the numercal measurements. Suppose that the followng statement, n addton to statement 1 and 2, 3. For any object o, t(o ) = x f and only f m(o ) = ax + b, where a P 0. (Hays, 1988, p.69)
FREQUENCY DISTRIBUTIONS Page 5 of 6 17. That s, the measurement number m(o) s some lnear functon of the true magntude x (the rule for lnear functon s x multpled by some constant a and added to some constant b). When the statements 1, 2, and 3 are all true, the measurement operaton s called nterval scalng, or measurement at the nterval scale level. 18. When measurement s at the nterval scale level, any of the ordnary operatons of arthmetc may be appled to the dfferences between numercal measurements, and the results nterpreted as a statement about magntudes of the underlyng property. 19. The mportant part s ths nterpretaton of a numercal result as a quanttatve statement about the property shown by the objects. 20. Interval scalng s about the best one can do n most scentfc work, and even ths level of measurement s all too rare n socal and behavoral scences. However, especally n physcal scence, t s sometmes possble to fnd measurement operatons makng the followng statement true: 4. For any object o, t(o ) = x f and only f m(o ) = ax, where a > 0. (Hays, 1988, p. 70)
FREQUENCY DISTRIBUTIONS Page 6 of 6 21. When the measurement operaton defnes a functon such that Statements 1 through 4 are all true, then measurement s sad to be at the rato scale level. For such scales, ratos of numercal measurements can be nterpreted drectly as ratos of magntudes of objects: 22. For example, the usual procedure for fndng the length of objects provdes a rato scale. If one object has a measurement value of 10 ft, and another a value of 20 ft, then t s qute legtmate to say that the second object has twce as much length as the frst. 23. Notce that ths s not a statement one ordnarly makes about the temperatures of object (on an nterval scale): f the frst object has a temperature readng of 10 and the second 20, we do not ordnarly say that the second has twce the temperature of the frst. 24. Only when scalng s at the rato level can the full force of ordnary arthmetc be appled drectly to the measurements themselves, and the results renterpreted as statements about magntudes of objects. 25. When objects are measured on the nterval scale, then dfferences between objects are measured on a rato scale. The true dfference between the rato scale and nterval scale s that nterval scale employ an arbtrary orgn or zero pont where as rato scale employ a fxed zero pont.