Z-Score Normalisation (Standardisation)

In some decision problems, what matters about an alternative is not its absolute value but how unusual that value is relative to the other alternatives. Z-score normalisation does exactly this: it finds each column's mean and standard deviation, then subtracts the mean from each value and divides by the standard deviation. Values above the mean come out positive, values below it come out negative, and values close to zero mean "around the average." This procedure was defined by Zdzisław Hellwig in 1968, as the standardisation step of the taxonomic method he developed for classifying countries by their level of development; in later years it was adopted as an alternative normalisation option in TOPSIS and many similar methods.

The question behind z-score normalisation is this: "How far above or below the mean does this alternative sit, relative to the others?" Vector or linear max normalisation scales a value against the column's magnitude or largest value; z-score instead scales a value against the column's centre (its mean) and its spread (its standard deviation). This is a statistical viewpoint: each alternative is judged by where it stands within its own group's distribution. One consequence of this approach is that normalised values can be negative, a feature that differs from other forms of normalisation and requires the next step (weighting, summation) to handle negative numbers correctly.

The procedure runs in a single step, but with two sub-calculations.

First, finding the mean and standard deviation. For every criterion column, the mean and standard deviation of its values are calculated. DecisionMind uses the population standard deviation (divisor: the number of alternatives) in this calculation; the sample standard deviation (divisor: the number of alternatives minus one) gives a different number for small alternative sets.

Second, scaling and direction correction. For a "higher is better" criterion, the mean is subtracted from the value and the result divided by the standard deviation; a large value gets a large z-score. For a "lower is better" criterion, this is reversed, the value is subtracted from the mean; a small raw value then gets a large (good) z-score. This is the second respect in which z-score normalisation differs from vector normalisation: direction correction is done within the normalisation step itself, rather than being left to a later step.

The formulas behind the step are given on the DecisionMind method page; this card carries no formulas.

A z-score states how many standard deviations an alternative sits from the mean on that criterion. Zero means "average," positive means "above average," negative means "below average"; but it cannot be converted into a ratio such as "twice as good" or "50 per cent worse." The size of the value depends on how narrow the distribution in that column is: in a narrow distribution, even a small raw difference can turn into a large z-score gap, while in a wide distribution, a large raw difference gives only a small z-score gap. A z-score table cannot be compared with one from a different set of alternatives, because the mean and standard deviation are calculated from that set's own values.

Thus instead of writing:

"The z-score is 1.22, so this alternative is 22 per cent better"

the report should read:

"This alternative sits about 1.22 standard deviations above this group's mean; this is a relative position, not a ratio"

It works with crisp data: a single number per cell, no empty cells. A minimum of two alternatives is required, because a standard deviation cannot be calculated from a single value; in a two-alternative table, the standard deviation always gives the same absolute difference, and the z-scores form only a symmetric pair such as +1/-1. If every value in a column is equal, the standard deviation comes out at zero and the division becomes undefined.

This is a building-block card; DecisionMind holds no separate extension of it (fuzzy, grey, intuitionistic version). Five other normalisation options do the same job with a different logic: vector normalisation (in two separate variants), linear max normalisation, linear sum normalisation and logarithmic normalisation; each is described in its own building-block card. Whether the population or sample standard deviation is used visibly changes the result in small alternative sets (three to five alternatives); DecisionMind fixes the population standard deviation and states this in the report.

Z-score normalisation suits situations where what matters is not an alternative's absolute value but its relative position against the group's mean; it is especially useful where the spread of distributions differs greatly between criteria (one criterion's values clustered together, another's widely spread), because z-score automatically balances out that difference. It also resolves the direction of "lower is better" criteria within the normalisation step itself.

It should not be used where the number of alternatives is very small (two or three) and the standard deviation does not represent a meaningful distribution; in such cases z-score can give excessively sensitive results. It is also confusing where the decision-maker wants to read the normalised value against an intuitive reference such as "the best alternative should get 1.0."

Distribution spread differs greatly between criteria, relative position matters → Z-score normalisation

An intuitive reading such as "the best alternative gets 1.0" is wanted → Linear max normalisation

Preparing data for a method based on Euclidean distance (TOPSIS) → Vector normalisation

Number of alternatives is very small (two or three) → A normalisation not based on standard deviation should be preferred

Z-score normalisation's greatest strength is that it automatically balances out differences in the spread of distributions between criteria; a small difference in a narrow-distribution criterion and a large difference in a wide-distribution criterion are both translated into the same statistical language, units of standard deviation. Because it resolves direction correction within itself, it needs no separate operation for "lower is better" criteria. Resting on a familiar, widely used statistical concept, standardisation, it also makes communication within a team easier.

It has limitations. First, it produces negative values; the summation and weighting steps that follow must handle these correctly, or an interpretation error results. Second, the standard deviation is unstable in small alternative sets; adding or removing a single extreme alternative can shift all the z-scores substantially. Third, if it is not stated which standard deviation (population or sample) was used, the results become impossible to compare with another calculation. Fourth, the choice of normalisation form (z-score or vector) can change the final ranking (Jahan and Edwards, 2015; Milani et al., 2005).

The most common mistake is reading a z-score as a percentage or a ratio; a z-score is only a distance from the mean, measured in standard deviations. A second mistake is confusing the population and sample standard deviation and not stating in the report which was used; in a small set, the two give visibly different numbers. A third mistake is applying the direction correction for a "lower is better" criterion both within the normalisation step and again in the following step; this flips the criterion's direction twice, sending it the wrong way. A fourth mistake is using z-score on a small set of two or three alternatives and assuming the results are stable.

The governing principle is this:

Z-score positions an alternative against the group's mean; it does not give an absolute measure of quality, and which standard deviation was used, and how stable the number of alternatives makes this measure, must be stated clearly in the report.