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Thursday, October 11, 2012

PSLE T-Score: The Role of Standard Deviation

When your child takes the PSLE, he or she will generate 4 raw scores. These are the actual marks he/she gets in the exam. If you imagine all the PSLE raw scores plotted on a graph... there will be 4 bell curves looking like the graph on this site. Some assert here that the 4 subjects' raw score bell curves vary in the size of their bell bottoms in the following manner...

(1) Math - wide bell bottom
(2) Science - less wide bell bottom
(3) English - even less wide bell bottom
(4) Chinese - narrowest bell bottom.

The raw scores for each subject, need to be transformed into a t-score. In essence, each child has 4 t-scores - one for each subject. The cohort average for each subject becomes the t-score of 50. If you score below the cohort average, your t-score is below 50. If you score above the cohort average, your t-score would be above 50. If you score exactly the cohort average for all 4 subjects, your aggregate t-score will be 200 exactly.

EQUAL WEIGHTAGE FOR EACH SUBJECT
Now... let us go back to the notion that the 4 raw score curves have different bell bottom widths. We need to consider this notion in relation to the consideration of EQUAL WEIGHTAGE for each subject.  In order to ensure equal weightage, the transformation from raw scores to t-scores needs to bring all 4 subjects' bell bottom widths into parity, i.e., the transformed curves for each subject all have the same bell bottom width. In this way, every 1 point of t-score in the Mother Tongue curve is equivalent to 1 point of t-score on the Math curve, which is equivalent to 1 point of t-score on the Science curve... add these 1 points together and you get 4 points of equal weightage.

Consider now, that the 4 raw score curves are all either narrower or wider than the t-score curve. Now, if your raw scores bell bottom was narrow... you would have to pull it wider to fit the wider spread of the t-score curve. If your bell bottom were wider... you would have to compress the spread to fit the spread of the t-score curve.This means that every 1 mark increase/decrease in Chinese raw score (see above... narrow curve that is pulled wider) gives you a higher increase/decrease in t-score than every extra mark in Math raw score (wider Math curve compressed).

In layman terms, it means this. If the difference in raw scores between the poorest students in Chinese and the best students in Chinese is small, it means that the bell curve is narrow. If the difference in raw scores between the poorest students in Math and the best students in Math is large, then the bell curve is wide. The width of the bell curve is captured in a statistical device called the Standard Deviation. The Standard Deviation is an integral part of the t-score calculation. The formula for the t-score (taken from here) is as follows...


Note that Z (i.e., the standard deviation) is an integral part of the formula pictured above. A high standard deviation (i.e., wide raw scores bell curve) makes for LOWER t-scores. A low standard deviation (i.e., narrow raw scores bell curve) makes for HIGHER t-scores. If it is true that the raw scores bell curve for Chinese is the narrowest (and Math the widest), then those who score 100 marks in Chinese will have higher Chinese t-scores than the Math t-scores of those who score 100 marks for Math. Add these t-scores up and the student good in Chinese will have the higher aggregate than the one good in Math.

See the table below where I have fitted the t-score formula pictured above to 4 subjects with increasing standard deviations (see column in orange)... and note how the subject t-score decreases correspondingly.



No one really knows 
which of the 4 subjects has 
the narrowest raw score bell curve... and which the widest.
If you have such information, please leave a comment.
I will address the role of the Cohort Average in the T-Score Computation HERE

5 comments:

Celine said...

Hmm. I do not know that anyone who knows is able to tell you about the raw scores vs t-scores. Maybe an Official Secrets Act thing.....

Petunia Lee said...

Celine - Yeeeah... I guess.

Cassandra said...

So sad. My boy's weakest subject happen to be Chinese & strongest is Math. But I think he should be able to score above 50 for Chinese. Base on the bell curve provided by his school last year, Math is indeed widest & Chinese is the narrowest. I believe it applies to all the schools.

Cassandra said...

So sad. My boy's weakest subject happen to be Chinese & strongest is Math. But I think he should be able to score above 50 for Chinese. Base on the bell curve provided by his school last year, Math is indeed widest & Chinese is the narrowest. I believe it applies to all the schools.

Petunia Lee said...

Cassandra - Thanks for your input! :-)