Scale Determination

The original scale of item difficulty and student performance is in logarithmic units, based on theories of probability. Each value also has a raw total equivalent on each test form. It is these logarithmic values that are estimated in the field testing and are ultimately expressed on a common scale across test forms. These logarithmic values, and the raw total scores they represent on each test form, have to be converted to a scale in which 65 represents passing and 85 represents passing with distinction, according to the results of the standard setting study. This conversion is accomplished algebraically.

The highest value (perfect score) is assigned 100, and the lowest possible is assigned 0. Passing is assigned 65 and passing with distinction is assigned 85. All values can then be converted to the scale using an algebraic transformation:

scale score = (a)*(x**3) + (b)*(x**2) + (c)*(x) + d.

In this equation, the value of x in each of four equations represents perfect scoring, lowest possible scoring, just passing, or just passing with distinction, respectively. The coefficients can be determined because each equation is set equal to the four known scale values: 0, 65, 85, and 100. These values yield four simultaneous equations. Thereafter, every logarithmic value, which, through equating has the exact same meaning in performance from test form to test form, can be assigned a scale value from 0 to 100 in which 65 is passing. In this way, a logarithmic scale having meaning with reference to achievement of State Learning Standards can be converted to a scale that ranges from 0 to 100 and carries the same meaning with reference to achieving State Learning Standards.

Conclusion

As described above, there is no State manipulation of test results to have a certain level of students pass or fail. Rather, the object is to operationally define what it means to achieve State Learning Standards, and New York State teachers accomplish this through a study. Once the standard setting is completed, it is applied to the scale values of the questions that comprise the test form. These scale values determine scores above which students can answer the questions correctly, and below which students cannot. Standard setting has associated these levels of performance with passing, passing with distinction, etc. Of course, there is much greater detail involved in these processes, and the Office of State Assessment welcomes opportunities to discuss these details and the philosophy of fair and equitable test scoring and scaling with interested audiences.