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
Updated: 2011-10-23
Please send comments to: J.Elfick@uq.edu.au
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) judgement 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 rth 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 rth 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 rth T-score is equal to the mean of the rth
normal order statistic, which we know how to calculate.
Using conditional expectations we can also obtain a formula for the
variance of the rth 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.