School Science Lessons
Mathematics and measurement
The theory of rank scaling tables
by Ian W. Wright, Department of Mathematics, The Papua New Guinea University
of Technology, Lae, Papua New Guinea
2022-08-11
Please send comments to: johnelfick@hotmail.com
Preface
The author is grateful to Dr Murli Gupta for his advice, to Dr David Tombs and the University of Technology for assistance with many things, and to Mrs Lenore Adams for typing this manuscript.
The theory of rank scaling tables
by Ian W. Wright, Department of Mathematics, The Papua New Guinea University of Technology, Lae, Papua New Guinea
Introduction and Preliminaries
In calculating students' marks, especially when comparing different subjects, teachers often use so called "T-scores" calculated from each raw score X by T = 50 + 10z, where z = (X - M) / s, with M and s respectively the mean and standard deviation of the class as a whole.
One of the most desirable features of these T-scores is that a student's T-score is virtually unaffected by the degree of difficulty of a test.
With an easy test, students' scores tend to have a high mean and fairly small standard deviation, while with a difficult test the students' marks tend to have a lower mean and larger standard deviation.
From test to test a particular student's T-score usually only shows minor random fluctuations.
One drawback of T-scores is that for a class as small as 25, the labour of calculating X, s and the T-scores is quite considerable especially without an electronic calculator.
For a class of 50, this becomes prohibitive.
Some reliable method of obtaining T-scores without heavy labour is clearly very desirable.
Since the students in each class in Papua New Guinea High Schools are also ranked, it occurred to John Elfick of Goroka Teachers' College, that these ranks might be used in some way to obtain T-scores directly, dispensing with the heavy calculation.
The rank scaling tables used in Papua New Guinea High Schools, Wright and Elfick (1975) operate on this principle, and are based on normal order statistics.
We now explain the relationship between the marks given in your tables and the ordinary T-scores.
If the students' marks X are normally distributed, and we look at many classes of a fixed size n and then calculate the T-score of the student who has rank r in each of these classes, it will be found that these T-scores are distributed about a certain mean value with a certain (small) standard deviation.
For each value of r this distribution is approximately Gaussian.
The rank scaling tables we have prepared give this mean value for all ranks in all class sizes n up to 300.
If we give this mean value as a final score to the student ranked r in a class of n, we believe we will, over time, do justice to all students and perhaps be fairer than some non-T score methods.
However, one feature of T-scores is that an exceptionally good performance is recognized and rewarded.
This is the situation when a top group of students break away from the rest.
It would be a pity if your system for simulating T-scores did not take account of this eventuality.
Accordingly the author has sought a sound method to incorporate this valuable (though subjective) judgment by the teacher in the students' final scores.
An operating scheme devised by the author will now be described.
Operating Procedure for Rank Scaling Tables
1. Mark test papers or add up marks of cumulative assessment.
2. Put marks in rank order, breaking ties by an appropriate method.
3. If the group of students at the top have done particularly well (in the teachers opinion, and by obtaining high scores) a star or other symbol should be placed before the rank number.
4. Look up the Rank Scaling Table for the class of that size.
Give each student the score in the table except that 3 should be added to the scores for the starred group.
[The latter procedure beginning with "Give each student . . ." is not used by J. Elfick.]
The Performance
To see how well the scores determined by your method compare with actual T-scores, we shall have to examine the theoretical distribution of T-scores of students ranked first, second, third, fourth . . . in classes of size 20, 30, 40, 50, 80, 100, 150, 300.
This will give us some insight into the operating procedure.
Later we will compare the results of an actual simulation with these theoretical values to show the performance in practice.
We should remark now that the size of standard deviation of the nth T-score in a class of n is the key to the success of the entire scheme.
Theoretical Distribution of T-Scores
For a class of size n with marks [Xr: 1 < r < n] we calculate M (mean) and Var (X) and this gives the T-score of then nth student as Tr = 10 / √ var (X) (Xr - M) + 50
In the following discussion we shall assume, for numerical convenience, that the normal order statistics are drawn from a normal population with mean 50 and standard deviation 10.
We shall also assume that Var (X) and M for that sample of n are independent of Tr.
Xr = √ Var (X) / 10 (Tr - 50) + M.
so, E(Xr) = E[√ Var (X) / 10] E(Tr -50) + E(M),
which gives, E(Xr) = E(Tr).
So the mean value of the nth T-score is equal to the mean of the nth normal order statistic, which we know how to calculate.
Using conditional expectations we can also obtain a formula for the variance of the nth T-score. If A, B, and Y are independent random variables, we know that:
Var(AY + B) = VarY Var A + (EY)2Var A + (EA)2Var Y + Var B.
Using the known distributions of:
A = √ var X / 10 and B = M (mean)
we obtain:
Var Xr = (1 + 1 / 2n) Var Tr + 1 / 2n (Etr -50)2 + 100 / n
or Var Tr = Var (Xr) - 1 / 2n(ETr - 50)2 - 100 / n / (1 + 1 / 2n)
Our knowledge of the distribution of the order-statistics Xr enables us to evaluate this RHS.
We now give tables of the standard deviation of Tr for various representative class sizes n.