Z-Scores

An Idiot’s Guide to Z-Scores

Published in

On Economics

5 min readJan 2, 2020

These days, there is some controversy around the Z-Score system for university entry. Because, the new cabinet plans to replace this “Old-System” with a “New-System”. Which is supposed to be “school-based” and “scientific” (whatever that means).

The goal of this article is not to compare and contrast these old and new systems. To compare, one must understand first.

I’ve been alarmed and shocked by how so many seem to misunderstand the Z-Score system. Few journalists attempt to explain it. And are happy to represent it as some mysterious enigma. With powers of life and death. Others are happy to quote soundbites from “experts”.

“Prof. Superfluous says, Z-Scores have a high carbon footprint. We hope to phase-out Z-Scores by 2025”.
“According to Dr. Gratuitous, Z-Scores cause pimples. But Salu Sala is well-stocked with preventive medication” [probably meant “Osu Sala”].
“Elton (aged 23) from Kandana complains that Z-Scores are homophobic”.

Ad nauseam.

Hence, in this article, I attempt to clear a bit of this mystery. By explaining Z-Scores.

[I hope to make a comparison of the “Old-System” to the “New-System” in a future article. But let’s first try and understand the “Old-System” first]

The Old-Old System

Consider Alice. At the A. Level exam, she gets:

Chemistry
80Maths
70Physics 
70

Before we adopted Z-Scores (let’s call that the Old-Old-System), raw aggregate marks determined university entry.

For example, Alice’s aggregate is 80 + 70 + 70 = 220. She would have gained entry into any university program, where A) Chemistry, Maths and Physics were sufficient subjects, and B) 220 was a sufficient aggregate.

In this Old-Old-System, any mark from any subject is equal. For example, if Alice got 20 fewer marks for Maths (50) and 20 extra marks for Physics (90), she would have still had the same aggregate of 220. And more importantly, the same chances of university admission.

The Old-System

But is the assumption “any mark from any subject is equal” reasonable? After all, aren’t some subjects “easier” than others? If so, students doing well in “difficult” subjects (which have lower marks) are disadvantaged. Compared to “easier” subjects (which have higher marks).

To analyse this assumption, let’s consider the marks of all the students who took Chemistry, Maths and Physics. While in practice there are 100,000s of students, for simplicity, let’s assume there were only five. Including Alice.

And let’s suppose the marks were as follows, with Alice’s marks in bold.

Chemistry
50, 60, 70, 80, 90Maths
30, 40, 50, 60, 70Physics
10, 30, 50, 70, 90

What can we observe?

It looks like this year’s Maths paper is a lot more “difficult” than the Chemistry paper. And Alice’s 70 for Maths, is the top mark. Even though it is 10 marks less than her 80 for Chemistry.
Similarly, her 70 for Maths looks better than the same 70 for Physics.
While both Physics and Maths have the same “average” mark (50), the marks for Physics seem to have more “variation”. While the marks for Maths are more “closely clustered”. Hence, if Alice was to get 20 marks fewer for Maths, and 20 marks more for Physics, she would have had to do “more worse” for Maths than “more better” for Physics. Because the same 20 marks jump from 70 to 50 for Maths would seem to have more significance than the same jump from 70 to 90 for Physics.

Why are these observations significant?

Alice’s 70 for maths should carry “more weight” in her university admissions than her 80 for Chemistry or 70 for Physics. But it doesn’t. This is unfair.
Alice (who got 70 for Maths and 70 for Physics) is probably better than a student who gets 50 for Maths and 90 for Physics. This is also not considered.

Enter Statistics

All these problems arise from different subjects different in two properties. Statisticians call these “mean” and “standard deviation”. “Mean” is the average mark. “Standard Deviation” is the “amount of deviation or variation” in the marks.

For example, Chemistry, Maths and Physics have means of 70, 50 and 50 respectively. Just the average of the five numbers.

[In my example, the Means are also equal to the “Medians”. The Median is the “value in the middle”.]

The computation of the Standard Deviation is more complicated. But if you’re interested, you can read about it here. I’ll just give you the answer. Chemistry and Maths both have a Standard Deviation of 15.81. Physics has a Standard Deviation of 31.62. This matches our intuition. Physics has double the “variation” of Chemistry and Maths.

Problems Solved

The Z-Score system solves the problem of different subjects having different means and standard deviations, by “normalising” the raw marks as follows.

Z-Score = (Raw Mark — Mean) / St. Dev

After the normalisation, the raw marks are transformed into z-scores as follows:

Chemistry
-1.26, -0.63, 0, 0.63, 1.26Maths
-1.26, -0.63, 0, 0.63, 1.26Physics
-1.26, -0.63, 0, 0.63, 1.26

Marks below the mean have negative Z-scores. And marks above the mean have positive z-scores. Hence, the effect of different means is “normalised out”.

Similarly, dividing by standard deviation “normalises out” differences in variation.

So, how did Alice do?

Alice’s Z-scores are:

Chemistry
0.63Maths
1.26Physics
0.63

And her average z-score is 0.84 ([0.63 + 1.26 + 0.63] / 3).

Does it solve the problems we observed earlier? Yes!

Alices Maths performance now carries more weight. Compared to Physics and Chemistry.
Also, if Alice got 50 for Maths and 90 for Physics her Z-scores, her respective Z-scores would be 0 and 1.26, giving her an average z-score of 0.63 ([0.63 + 0 + 1.26] / 3). Lower than her current average z-score.

Concluding Disclaimers

Z-Scores are not perfect. They don’t solve all the problems. Just two: that of subjects having different means and different standard deviations.

As promised, I hope to analyse the broader problems in a future article.