School Science Lessons
Mathematics and measurement
Rank Scaling Tables
Updated: 2007-12-15
Please send comments to: J.Elfick@uq.edu.au
Rank Scaling Tables
See examples of Rank Scaling Tables
The scaling of test results by using Rank Scaling Tables is the simplest method of converting a set of marks into standard scores. They allow you to read off normal curve scores from rank orders. The scores have a mean of 50 and standard deviation of 10; they are usually called T Scores.
The assumption is that the frequency distribution of test scores or total scores from cumulative assessment is close enough to that of a normal distribution. The distribution of test marks follows the normal curve principle of bunching in the centre and thinning out towards the extremes. If rank values as numbers are used in assessment the tendency towards a normal distribution is ignored.
Rank Scaling assumes that class tests, cumulative assessment and examinations do no more than allow you to place students in order of merit for that class, or school, or population of examination candidates. Testing is a device to enable you to make a sound judgement about who is best, second best, third best in carrying out a task or group of tasks.
If you want to compare and interpret marks gained by a student in different subjects, or if you want to add marks together from different subjects then those marks must be standard scores.
There is an assumption of some form of "innate trait" theory in the argument about difficulty not affecting T-score. This would certainly not have raised eyebrows in the era from which this approach dates.
Comment from a mathematics educator
"Nowadays many within the criterion referencing movement would, I suspect, have reservations about this as there is an implicit assumption that the best on task A will be the best on task B and so on.
This is clearly not necessarily the case, particularly if different assessment tasks sample different attributes. However, provided the assumptions are made explicit, subsequent consistent theory is legitimate.
However this is not to argue that the approach is the best one educationally. This is most evident in C-R assessment when marks are not used, and there is no means (nor meaning) to adding standard
scores. The approach to adjusting for a very good top group of students has an arbitrariness about it that needs elaborating. Why add 3? This needs justifying. I would also suspect that practically in the field, the educational grapevine would quickly start buzzing so that superior top groups would start to appear everywhere. In presenting the paper it would help to distinguish clearly between the theoretical and
the pragmatic when both play a significant role. In summary it seems to me, without getting out my calculator, that there is interesting potential in the approach as a labour saving device for the purposes advanced, the mathematical question. On the other hand I cannot get too enthused about the problem it sets out to make more tractable, the educational question."
Procedure
1. Mark test papers or add up total marks of cumulative assessment.
2. Put marks in rank order (1st, 2nd, 3rd, 4th  nth). No ties!
3. Look up Rank Scaling Tables, T Scores for class of n students
Rank Scaling Table for a class of 28 students.
T scores for class of 28
.
 0 1 2 3 4 5 6 7 8 9
0 .
 70 66 63 61 60 58 57 56 55
10 54 53 52 51 50 50 49 48 47 46
20 45 44 43 42 40 39 37 34 30 .
The best student gains a standard score of 70, the tenth best student gains a standard score of 54, the fourteenth and fifteenth best students gain a standard score of 50 (the average score) the twenty-second best student gains a standard score of 43 and the bottom student gains a standard score of 30.
The theory of rank scaling tables
Dr. I. W. Wright of the Mathematics Department, University of Technology, Lae, Papua New Guinea devised a computer algorithm to calculate these tables.
Comparison of the results of four methods of determining T scores (mean = 50, standard deviation =10)
A comparison of methods of calculating T-scores
A teacher gave a test to a class of 27 students, listed the marks in rank order, then determined T scores by:
Method I Calculating the mean, standard deviation and T scores. Standard score = 10(mark - mean) / standard deviation + 50.
Method II Using the Rank Scaling Tables.
Method III Calculation, using a Percentile scaling method.
Method IV Calculation, using a semi-graphical Sigma scaling method.
The determination of T scores by using these Rank Scaling Tables, see column II, takes only a fraction of the time needed for any of the other three methods. The T scores can be compared below:
T SCORES
Let X = a student's mark and let M = mean of all students' marks
Rank X X-M (X-M)2 Method I Method II Method III Method IV
1 85 30 900 72 70 71 73
2 81 26 676 69 66 67 69
3 78 23 529 67 63 65 67
4 73 18 324 63 61 63 63
5 65 10 100 57 60 59 57
6 65 10 100 57 58 59 57
7 63 8 64 56 57 57 56
8 60 5 25 54 56 55 54
9 59 4 16 53 55 54 53
10 59 4 16 53 54 54 53
11 56 3 9 52 53 52 51
12 54 -1 1 49 52 51 49
13 53 -2 4 48 51 50 49
14 53 -2 4 48 50 50 49
15 52 -3 9 48 49 49 48
16 52 -3 9 48 48 49 48
17 51 -4 16 47 47 48 47
18 50 -5 25 46 46 47 46
19 47 -8 64 44 45 44 44
20 47 -8 64 44 44 44 44
21 46 -9 81 43 43 43 43
22 46 -9 81 43 42 43 43
23 40 -15 225 39 40 33 39
24 40 -15 225 39 39 38 39
25 34 -21 441 35 37 34 34
26 33 -22 484 34 34 33 34
27 31 -24 576 32 30 29 32
Sum 1 473 -10 5 068 1 340 1 350 1 346 1 341
Mean 54.5 -0.37 187.7 49.6 50 49.8 49.7