Units-free variables

The standardised form of a variable X is found by subtracting its mean then dividing by its standard deviation,

standardised value,   

The resulting values are called z-scores.

Whether a variable consists of temperatures in degrees Celsius and Fahrenheit, its z-scores will be the same. In the same way, standardised distances are the same whether the original values were recorded in miles or kilometers.

Properties of z-scores

A standardised variable always has zero mean and standard deviation one.

Since approximately 95% of values are within 2 standard deviations of the mean in most distributions and almost all are within 3 standard deviations of the mean (see the 70-95-100 rule-of-thumb),

An individual's z-score tells you how many standard deviations it is above the mean. From its value, you can tell whether the value is very high (say over +2) or low (say under -2) in relation to the other values of the variable.

Class marks in exams

The diagram below initially shows the marks obtained by 20 students in a Maths exam.

Click on individual crosses to see how that student's z-score is obtained from the raw exam mark and to see how it is interpreted.

The z-score tells you the position of a student within the class.

Note that original values have units 'correct answers out of 100' whereas the z-scores are unit-less quantities.


Use the pop-up menu to see how the same students performed in an English exam. The mean mark for the English exam is lower, so the z-scores do not correspond to the same raw marks as for the Maths exam.

A z-score of 0 always corresponds to the mean raw mark in the class. The best students are still getting z-scores over +1 and the weakest students are still getting z-scores under −1. (For any variable, we expect around 70% of z-scores to be between −1 and +1.)

Since the z-scores for each exam are relative to that exam's mean and st devn, they correct the raw marks for the different levels of difficulty of the different exams.

The table below shows all z-scores together.

Student Maths English Science
Simeon
Suzanne
Carolyn
Marie
Melanie
Lorna
Leith
Julian
Daniel
Andrew
Craig
Aaron
Benjamin
Gar
Katie
Gavin
Kamini
Tracy
Scott
Samantha
-1.51
-1.44
-1.19
-0.86
-0.80
-0.60
-0.80
-0.73
-0.28
0.37
-0.09
-0.22
0.11
0.69
1.01
1.08
1.53
0.82
1.27
1.66
-1.45
-0.74
-1.23
-1.12
0.06
-0.91
-0.26
-0.69
0.33
-0.53
-0.42
-0.15
-0.37
0.93
0.23
0.50
1.52
1.20
0.60
2.49
-1.54
-1.37
-1.04
-0.87
0.06
-0.36
-0.70
0.49
-0.44
-1.12
-0.02
-0.36
0.66
0.74
0.82
-0.61
1.67
1.25
0.99
1.76

Observe that Simeon obtained the lowest mark in all tests. All three of his z-scores are therefore around −1.5. Similarly, Samantha got the highest mark in all tests so all of her z-scores are between +1.5 and +2.5.