This article outlines the methodology used in Quick Risk Matrix for the development of risk matrices, also known as risk assessment matrices.

A risk matrix is a chart in which one axis is subdivided to indicate categories of likelihood while the other axis is subdivided to indicate categories of consequence. The cells of the matrix are colored to indicate how any given pair set {likelihood-category, consequence-category} maps to a risk category. We term the risk categories "risk priority levels."

Likelihood may be expressed as probability or as frequency. We will use the term probability from this point on, but it should be remembered that it could equally well be frequency.

Risk matrices are used in risk assessments. Potential unwanted events are identified. Each event is assigned to a probability category and to a consequence category. The probability category indicates the likelihood that the event will occur. The probability category indicates the severity of the consequences. For a given pair of categories, the risk matrix indicates the risk priority level. The organization's risk management procedures typically make use of the risk priority levels when specifying acceptability, urgency, priority, required level of management attention, etc.

The most popular size for a risk matrix is 5 x 5 but some organizations are using larger risk matrices with up to 10 rows and columns. Quick Risk Matrix imposes no limit on the number of rows or columns.

An example risk matrix is shown below. The categories are labeled and also defined by their numeric ranges. Note that we often use E-notation to keep the chart tick labels compact so that they do not overlap.

When the probability or consequence variable spans a large range, the axis will use a logarithmic scale for legibility. This example uses log scales. The choice of logarithmic or linear scale makes no difference to the coloring pattern of the risk matrix; it is only a legibility issue.

**Figure 1: Example risk matrix**

In the figure above, the probability and consequence categories have been defined using numeric scales (e.g. 0.1, 1, 10, …) and also with labels (e.g. Very Low, Low, ...).

The numeric scales are known as "ratio" scales, because we can compare the sizes of quantities expressed on these scales (i.e. calculate ratios). Most scientific and engineering measurements are made on ratio scales.

The category labels are known as “ordinal” scales. An ordinal scale shows the order in which a variable increases but often tells us nothing about its magnitude.

Some organizations omit ratio scales from their risk matrices and use only ordinal scales. A risk matrix that uses only ordinal scales is almost unverifiable. The only consistency check that can be made is whether a cell above and/or to the right of another cell has an equal or higher risk priority level. So, for example, referring to the above figure, the cell {High, Likely} must have a risk priority level equal to or higher than the cell {High, Unlikely}, because both the consequence and the probability in the former cell are equal to or higher than in the latter cell. But, there is no way to verify the ranking of a pair of cells when one cell is to the right and below another cell. For example, the cell pair {High, Likely} and {Very High, Unlikely} cannot be ranked because the former has lower consequence but higher probability than the latter, and there is no way of knowing whether the consequence or the probability dominates.

A further disadvantage of purely ordinal scales is that linguistic terms like “low”, “moderate”, “likely” and so on will be interpreted very differently by different people.

Since a risk matrix that uses only ordinal scales cannot be fully verified, it follows that its use cannot normally be defended. There is an exception to this rule, which we will describe below.

The exception is that a matrix with a ratio scale for probability and an ordinal scale for consequences is sometimes reasonable when consequences are difficult or controversial to measure. A prime example might be a risk matrix for workforce safety. Let’s say the consequence scale has ordinal categories A–E, the workforce population size is 200, and the categories are defined as “A. One or more injuries, not severe,” “B. One or more severe, possibly permanently disabling, injuries,” “C. 1–4 fatalities,” “D. 5–50 fatalities,” “E. 51–200 fatalities.” The difficulty here is that the measurement units are mixed, being number of injuries at the lower end and number of fatalities at the higher end. One way round this difficulty is to temporarily assign a ratio scale, either based on the monetary value of averting an injury/fatality or based on treating an injury as a fraction of a fatality. Let’s say we treat an injury as a fraction of a fatality. Then the tick marks on the ratio scale might be (for example) 0, 0.5, 1, 5, 50, 200. (If using a log scale, zero should be replaced with a small non-zero number since zero cannot be plotted on a log scale.) The matrix is then designed based on the temporary consequence scale and a suitable ratio scale for probability. Finally, the temporary consequence scale is hidden since the injury to fatality equivalence is uncertain and possibly controversial.

When ratio scales are used, the risk matrix may be developed with mathematical precision. Quick Risk Matrix requires the use of ratio scales, even if only temporary as described in the previous paragraph.

When setting up a ratio scale for consequences, any units of measurement relevant to the type of risk you are assessing may be used, for example, dollars, number of fatalities, volume of chemical discharged to the environment, hectares of land contaminated by a spill, percentage budget overrun, weeks of schedule delay, and so on.

Another requirement for objective risk matrix design is clarity on how risk is to be measured. There is a high degree of consensus that risk is to be computed by combining consequence and likelihood (see for example ISO Guide 73:2009). By far the most common method of forming the combination is to multiply the values of the probability and consequence variables (i.e. risk = probability times consequence). Note that the product of the two variables is the statistical "expected value," i.e. the average value of the consequence measure in a large number of identical situations. We call "expected value" "expected loss" when the consequences are detrimental. In a future article, we will look at an alternative to expected value that is available in the Premium edition of Quick Risk Matrix.

A "risk graph" (our term) is a logical predecessor to a risk matrix. The risk graph uses the same axes as the risk matrix. On it, we plot contours of equal risk (iso-risk contours) to define the boundaries between risk priority levels. The risk graph corresponding to our example risk matrix is presented below. Note that because we have used log-log scales, the risk contours are straight lines.

**Figure 2: Risk graph**

**A note on terminology.** What we term a risk graph should not be confused with the quite different type of chart with the same name in the IEC 61511-3 standard on functional safety of safety instrumented systems. We thought about calling the above type of graph a probability-consequence graph but that's a mouthful and, if we did that, then perhaps we should use the term probability-consequence matrix in place of risk matrix. On balance, it seemed preferable to use the term risk graph as the direct analog of the term risk matrix.

We can construct the risk matrix (Figure 1) from the risk graph (Figure 2) in two steps. We begin by coloring the cells of the risk matrix that are *not* split by the risk contours. These cells can be colored based on the coloring of the risk graph, which, for our example, gives the result in Figure 3 below. There may be a lot of blank cells at this stage, especially if you are trying to use many colors in a small matrix.

**Figure 3: Risk matrix while under construction**

You can see that some of the cells of the risk matrix have a known coloring that follows directly from the risk graph. But the cells that are split by the risk contours cannot be assigned accurately to any of the qualitative risk priority levels. That is because some points in the split cells lie in one level and other points lie in another. With closely spaced risk contours, it is possible for a cell to be split not just between two but between three or more risk levels.

At this stage in the risk matrix design, when we want to color the split cells, we are forced to make an approximation. Quick Risk Matrix Premium allows this approximation to be made in any of five ways, which we briefly refer to as (1) fuzzy interface, (2) round up, (3) predominant color, (4) round down and (5) geometric center. Quick Risk Matrix Standard offers the first three of these algorithms.

The "fuzzy interface" procedure treats the group of cells intersected by any iso-risk contour as if it is a separate risk level in its own right. If we were to do that in this example, we would end up with seven risk priority levels (colors) rather than the original four.

In the "round up" method, the color of a split cell is governed by the color of the highest risk zone in the cell. This means that the level of some risks in the cell (those risks corresponding to points lying under the highest contour intersecting the cell) will be overestimated. The merit of the rounding up method is that no risk will ever be assigned to a level that is too low. But it may be overly conservative for some circumstances, in which case one of the other methods may be used.

"Round down" colors each split cell according to the color of the lowest risk zone in the cell. It is only intended to be used when designing opportunity matrices (an opportunity matrix is similar to a risk matrix but is used to rate potential gains rather than potential losses). The round-down method is conservative for opportunity matrices.

With the "predominant color" method, the color of a split cell is taken to be the color of the risk zone that occupies the largest area within the cell. This is often the most accurate method of designing a risk matrix, but it is not conservative.

The "geometric center" method colors each split cell according to the color of the risk graph at the geometric center of the cell. This often, but not always, gives the same result as the predominant color method.

The Premium edition of the program includes a simulation tool to enable the performance of the different coloring methods to be compared.

If we color our example matrix using the predominant color algorithm, we obtain the following result:

**Figure 4: Risk matrix after coloring the split cells using the predominant color algorithm**

If we use the round-up algorithm, we obtain the following:

**Figure 5: Risk matrix after coloring split cells using the rounding-up algorithm**

At this stage, the risk matrix is finished. You may hide the contours if you wish, you may hide the X- and Y-axis tick marks if they were only a construction aid, and you may export the risk graph and risk matrix to various picture formats. With the Premium edition, you may also export the risk matrix to Excel along with a risk register template.

## Summary

What makes the Quick Risk Matrix methodology so accurate is:

- The use of user-defined ratio scales for the probability and consequence axes (as opposed to ordinal scales).
- The definition of risk priority levels using contours of equal risk ("iso-risk contours"), the numeric values of which are defined by the user.
- The recognition that cells split by the iso-risk contours can only be assigned to a single risk level via an approximation — and the approximation may be made in various ways.
- The provision of several reasonable ways to make the above approximation