In Quantitative Risk Analysis, many of the outputs are talked about in terms of things like “P50” or “P80” or “P95 – P5”, as well as statistical terms like Mean, Median, Mode, Standard Deviation, Variance, Sigma…

It is important to understand the differences between mean, median and mode:

Mean represents the average value of the distribution, i.e. all the results added up and divided by the number of iterations.

Median represents the value that is in the middle of the distribution i.e. half of the iterations will have produced a number higher than the median, and half will have produced a lower number.  

Mode or Most Likely represents the value in the distribution that occurs most frequently.

The graph below summarises these:


P80 (for example) is shorthand for the 80th percentile. In the context of the output from a QRA, it means that 80% of the results coming out of the simulation are lower than that P80 value.

Or to put it another way, based on the simulation, we can be 80% confident that the final outcome will be lower than the P80.


One needs to be very careful when using P80 results, particularly for Cost QRAs. This is because they represent points on a statistical distribution and statistical distributions cannot be added (if they could we wouldn’t need to do Monte Carlo simulations!). The one exception to adding statistics together is that you can add Mean values together (becuase they are averages! and already made up of values being added).

So, when you have a table of risk outputs presented to you by an eager risk analyst, do not be surprised when you do what almost everybody does – add up all the values in a column to check that they add up, and find that they don’t. (P.S. This is how the insurance industry makes a profit – by adding lots of risks together and using the premiums from all those customers who don’t claim to cover the claims of those customers who do)

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