Methods · Objective weighting
SD-WEIGHT (Standard Deviation Weighting)
The plainest objective weighting method there is: it derives a criterion's weight not by asking an expert but from how much the alternatives differ from one another on that criterion.
Base method's data type: Classical
What Is the Method?
SD-WEIGHT is not a ranking method; it does not rank alternatives, it produces weights for criteria. Given a numerical decision table of alternatives and criteria, the method examines the spread within each column and returns a weight vector that sums to 1. These weights then feed into a ranking method such as TOPSIS, VIKOR or SAW.
The idea of turning standard deviation into criterion weight is a very old and general principle in statistics; it cannot be tied to a single founding paper. Mukhametzyanov (2021) treated this principle alongside Entropy and CRITIC under the name "the SD method" and compared all three within one framework. DecisionMind implements this method on the basis of that framework.
The Philosophy Behind It
The idea behind SD-WEIGHT comes from the same family as Entropy: the more alike the alternatives are on a criterion, the less that criterion affects the decision. It differs in how it measures the spread. Entropy looks at the evenness of a probability distribution, whereas SD-WEIGHT uses statistics' most familiar measure of spread directly, the standard deviation. If a criterion's normalised values are spread across a wide range among the alternatives, its standard deviation comes out large and its weight increases; if the values sit close together, the standard deviation shrinks and the weight falls.
A consequence of this philosophy is, once again, that it draws no distinction between "importance" and "discriminating power." A criterion that is critical in the decision-maker's eyes but comes out similar across all the alternatives receives a low weight under SD-WEIGHT. The method does not count this a flaw; the question it asks is not "which criterion is important" but "which criterion is speaking in this table." If the decision-maker's priority must show up in the result, a subjective weighting method, or a combination of the two, is needed.
How It Works
The method proceeds through two steps.
First, scale equalisation. Criteria are expressed in different units and cannot be compared directly. SD-WEIGHT scales every column to between 0 and 1 according to its direction: for a "more is better" criterion, the largest value becomes 1 and the smallest 0; for a "less is better" criterion, this is reversed. As a result, all columns become unit-free and sit on the same scale.
Second, turning spread into weight. The standard deviation within each column is calculated. A column with a wide-ranging standard deviation is a sign that this criterion discriminates well among the alternatives. The standard deviations of all the columns are summed, and each column's share of that total is taken; the result is a weight vector that sums to 1.
The formulas behind each step and the intermediate tables are given on the DecisionMind method page; this card carries no formulas.
How to Read the Output
The SD-WEIGHT weight shows not a criterion's importance in the decision-maker's eyes but how much it discriminates among the alternatives in the table at hand. A high weight does not mean "this criterion is the most important"; it means "in this alternative set, most of the difference between the alternatives comes from this criterion." The same criterion can take on an entirely different weight in a different alternative set, because the weight is not a fixed property of the criterion but a property of the table.
The scale-equalisation step looks only at the smallest and largest value; how close the intermediate values sit to one another is determined by their position between 0 and 1. As a result, two criteria that look very different in their raw numbers can show the same spread once scaled and so receive equal weight. This is not a computational error; it is a natural consequence of the method's using relative positions rather than raw numbers.
Thus instead of writing:
"The SD-WEIGHT analysis proved that manoeuvrability is the most important criterion"
the report should read:
"In this alternative set, the criterion that discriminates most among the alternatives is manoeuvrability; the SD-WEIGHT weight reflects this discriminating power, not the decision-maker's order of priority"
Data Type and Inputs
SD-WEIGHT works with crisp data: a single number in every cell. DecisionMind holds no fuzzy, grey or intuitionistic extension of this method; if your data type differs, the relevant members of the Entropy or CRITIC family, which carry the same principle, can be considered instead.
You need alternatives in rows, criteria in columns, a number in every cell and no empty cells; direction information ("more is better" or "less is better") is required for every criterion. No weights are entered; the method produces the weights. A minimum of two alternatives and two criteria is required; three to twelve criteria work comfortably. If all the alternatives carry the same value in one column, that column's standard deviation comes out zero and so does its weight; this is not a computational error but a sign that the criterion says nothing in this table.
When to Use It, When Not To
If no expert opinion is available, if experts cannot agree, or if weights are wanted quickly and repeatably from the data itself, SD-WEIGHT is a suitable choice. Its computation is even simpler than Entropy's; it is a practical starting point for large indicator sets and frequently repeated assessments.
The cases where it should not be used follow from its philosophy. Where the decision-maker clearly considers one criterion a priority, SD-WEIGHT cannot see this. If a strong relationship exists among criteria, one rising as another rises, SD-WEIGHT does not account for it, and the weight of the related criteria is implicitly counted twice; CRITIC is more suitable in this case. If the number of alternatives is very small, three or four, a single outlying value can drag a criterion's weight.
No expert opinion available or wanted, weights should be quick and data-derived → SD-WEIGHT
The relationship between criteria should also be accounted for → CRITIC
Not just the spread but the evenness of the distribution should be measured → Entropy
The decision-maker's priority should show up in the result → AHP, BWM, SWARA (subjective)
A criterion is constant across all alternatives → remove that criterion from the table; SD-WEIGHT's zero weight is expected
Strengths
SD-WEIGHT's greatest advantage is its simplicity. Its computation fits into two steps, requires no extra parameter, and can be applied immediately to any crisp data table. The same table gives everyone the same weight; no survey, pairwise comparison or expert panel is needed. Because standard deviation is statistics' most familiar measure of spread, the result is easily explained to a statistician.
Weaknesses
Its limitations arise from its simplicity. First, the method does not see the correlation between criteria; two criteria carrying the same information can both receive a high weight, and that information is counted twice in the weights (Wang and Luo, 2010). Second, the weights depend on the alternative set; when an alternative is added or removed, the scaling changes and all the weights are recalculated. Third, "discriminating power" is not the same as "importance"; the method does not see the decision-maker's values. Fourth, it is sensitive to outliers; in small tables, a single outlying value can widen the scaling range and drag that criterion's weight.
Common Mistakes
The most common mistake is reporting an SD-WEIGHT weight as "importance." The sentence "the data showed that manoeuvrability is the most important criterion" is wrong; the data shows that manoeuvrability is the criterion that discriminates most.
A second mistake is marking criterion direction (more is better, less is better) wrongly; in this case, the scaling is reversed and the weights, while not meaningless, are attached to the wrong criterion. A third mistake is placing related criteria in the table together without noticing the relationship; because SD-WEIGHT does not see this relationship, the same information is weighted twice. A fourth mistake is carrying weights from one study over to a different alternative set; the weight is a property of the table, not of the criterion. A fifth is leaving a constant criterion in the table and mistaking its zero weight for a method error.
The governing principle is this:
An SD-WEIGHT weight is a property of the table, not of the criterion; the weight changes when the alternative set changes, and the report must call this "discriminating power," not "importance."
Cases
Each case opens with a decision table, describes in words what the method does to it, and shows how to read the resulting weights. The first case is DecisionMind's validation example; the figures have been recomputed and verified in Python. The remaining cases are illustrative constructions.
1. Illustrative example: Three alternatives, three criteria (DecisionMind validation example)
This example is not a literature case; since no founding paper for SD-WEIGHT could be verified, it is presented instead through a small, hand-traceable table. Three alternatives are assessed on three criteria; the first two criteria are "more is better," the third, a cost-type criterion, is "less is better."
| Alternative | K1 | K2 | K3 (cost) |
|---|---|---|---|
| A1 | 3 | 5 | 4 |
| A2 | 5 | 3 | 2 |
| A3 | 4 | 4 | 3 |
| Direction | more is better | more is better | less is better |
The method first scales every column to between 0 and 1; because K3 is a cost, its scaling is reversed. In the scaled table, all three columns carry the same pattern: one alternative sits at 0, one at 1, and one exactly in the middle, at 0.5. The standard deviation of each column is then calculated and divided by the total to convert it into a weight.
| Criterion | Standard deviation | Weight |
|---|---|---|
| K1 | 0.408 | 0.333 |
| K2 | 0.408 | 0.333 |
| K3 | 0.408 | 0.333 |
The result reads as follows. Although the three criteria look different from one another in raw numbers (K1 runs 3-5-4, K3 runs 4-2-3), once scaled all three settle into the same pattern of "one at the bottom, one exactly in the middle, one at the top." This pattern's standard deviation always comes out the same, because SD-WEIGHT looks not at raw numbers but at the alternatives' relative position between 0 and 1. The result is that all three criteria count as equally weighted.
The analyst preparing the decision hesitates as follows: what would happen if A3's value on K3 were 3.5 rather than 4? The same calculation raises K1's weight from 0.333 to 0.342 and lowers K2 and K3 to 0.329. It is clear that even a decimal-sized difference shifts the weight; in small, three-alternative tables, weights should therefore be read cautiously.
In the report: "The weights were derived from the standard deviation of the scaled columns; in this table, all three criteria are equally discriminating, which does not mean the three are equally important in the decision-maker's eyes."
Source: DecisionMind SD-WEIGHT manifest, validation example. The J.source field carries the entry "Various, 1980, MCDM literature," which is not a traceable book or article. The verified source treating the standard-deviation principle alongside Entropy and CRITIC is Mukhametzyanov (2021).
2. Agriculture: A cooperative's comparison of wheat varieties
An agricultural cooperative will compare five wheat varieties on four measures: yield per decare, drought-resistance score, disease-resistance score and seed cost. Cost is "less is better," the other three are "more is better." Rather than conduct an expert survey, the cooperative's management has decided to let the weights come from the measurement data.
The method reverses the cost column, scales all four columns to between 0 and 1, and measures how much each measure discriminates among the varieties. Suppose yield showed large differences across the varieties and so received the highest weight, while disease resistance came out similar across all the varieties and so received the lowest weight.
The management's hesitation: disease resistance receiving a low weight does not mean the cooperative considers this measure unimportant. The five varieties are already close to one another on this measure, and it does not determine the ranking. It should also be noted that if a further variety is added, or a year's data changes, the weights must be recalculated; last year's weights are not carried over to this year.
In the report: "Weights were derived with SD-WEIGHT according to discriminating power across varieties; the low weight for disease resistance reflects its small variation across varieties, not its unimportance."
3. Tourism: A municipality's inspection scoring of beach concessions
A coastal municipality will inspect twelve beach concessions on five measures: cleanliness score, lifeguard adequacy, disability access score, waste management score and number of complaints. Number of complaints is "less is better," the other four are "more is better." The municipality has asked that the inspection scores be combined not with a subjective weight but with a weight derived from that year's data.
The method reverses the complaints column, scales the five columns and calculates discriminating power. Suppose the number of complaints varied greatly across the concessions, some at zero, some very high, and so received the highest weight, while disability access came out at a similarly low level in almost all the concessions and so received the lowest weight.
The municipality's hesitation: disability access receiving a low weight might give the impression that this issue is considered unimportant, whereas the real reason is that all the concessions are already inadequate and similar to one another on this measure. The municipality has decided to track this measure separately, as a threshold independent of the weight.
In the report: "Inspection weights were derived from that year's data with SD-WEIGHT; the low weight for disability access stems from the concessions being collectively inadequate on this measure and is being tracked separately with its own threshold."
4. What Not to Do
In the illustrative example, had K3 (cost) been entered without reversal, the most expensive alternative would have received a score of 1 and the weight would have been built up against, rather than in favour of, low cost. A second error is reporting the three criteria's equal weight of 0.333 as "these three criteria are also equally important in the decision-maker's eyes"; the equality applies only to discriminating power for these three alternatives. A third error is taking the weights that come from these three alternatives and using them unchanged for a different study's five alternatives; the weight is a property of the table.
Sources
For the formulas behind each step, the intermediate tables and citation formats, see the DecisionMind method page: decisionmind.app/library/sd-weight
Mukhametzyanov, I. (2021). Specific character of objective methods for determining weights of criteria in MCDM problems: Entropy, CRITIC and SD. Decision Making: Applications in Management and Engineering, 4(2), 76-105. DOI: 10.31181/dmame210402076i
Diakoulaki, D., Mavrotas, G., & Papayannakis, L. (1995). Determining objective weights in multiple criteria problems: The CRITIC method. Computers & Operations Research, 22(7), 763-770. DOI: 10.1016/0305-0548(94)00059-h
Wang, Y.-M., & Luo, Y. (2010). Integration of correlations with standard deviations for determining attribute weights in multiple attribute decision making. Mathematical and Computer Modelling, 51(1-2), 1-12. DOI: 10.1016/j.mcm.2009.07.016
Odu, G. O. (2019). Weighting methods for multi-criteria decision making technique. Journal of Applied Sciences and Environmental Management, 23(8), 1449. DOI: 10.4314/jasem.v23i8.7