Simulated Result Statistics

In additional to looking at simulated results of an output graphically, there are a number of statistical measures worth investigating. In the @RISK interface, many of these statistics can be found at the right of output graphs and in the Statistics Detail window. Alternatively, @RISK statistics functions can be placed directly in the spreadsheet to report any statistical result.

Measures of Central Tendency

Measures of central tendency are fundamental summary statistics for a distributed variable. @RISK provides the three most common measures: the mean, the median, and the mode. Each of these statistics is appropriate under different circumstances.

  • Mean – the most commonly used measure, calculated by adding up all the samples and dividing by the number of values.
  • Median - The median is the value separating the higher half from the lower half of the samples.
  • Mode – The most likely sample obtained.

  • Please note: For a continuous distribution, the mode is a relatively unstable statistic because it can only be approximately calculated.

The mean, median, and mode are displayed in the statistics tables to the right of all distribution graphs, as well as in the Detailed Statistics window. In addition they can be added directly to a spreadsheet model using the RiskMean, RiskPercentile/RiskPtoX, and RiskMode functions.

Measures of Dispersion

Measures of dispersion indicate the spread of a distribution. The most important measures are the variance and the standard deviation. These two contain the same information, since the standard deviation is simply the square-root of the variance. The standard deviation has the advantage of being in the same units as the variable itself.

The standard deviation appears to the right of all distribution graphs. In the Detailed Statistics window, both the variance and standard deviation are reported for all the inputs and outputs of a model. In addition, they can be added directly to a spreadsheet model using the RiskVariance and RiskStdDev functions.

There are some more esoteric measures of dispersion that are available in @RISK only by using statistic functions:

  • Mean Absolute Deviation - the mean of the data's (unsigned) deviations around the mean. Available using the RiskMeanAbsDev function.
  • Coefficient of Variation – the ratio of the standard deviation to the mean. Available using the RiskCoeffOfVariation function.
  • Semi Standard Deviation – returns the standard deviation of the data values below the mean. This is useful for measuring down-side risk. Available using the RiskSemiStdDev function.
  • Semi Variance– returns the variance of the data values below the mean. This is useful for measuring down-side risk. Available using the RiskSemiVariance function.

Boundary Limits

The boundary limits of a output are the minimum and maximum data values sampled. Note: for many distributions these statistics will not be either stable or useful. Many distributions are unbounded, meaning that they might not have a true minimum or maximum value, but for any finite number of iterations, there will always be some minimum or maximum reported by @RISK. In addition, even for those distributions that are bounded, the reported boundary values will be heavily biased.

The minimum and maximum of a simulated output are displayed to the right of all distribution graphs, as well as in the Detailed Statistics Window. In addition, they can be added directly to a spreadsheet model using the RiskMin and RiskMax functions.

The RiskRange function is a convenient additional function that reports the difference between the maximum and minimum.

Measures of Shape

Measures of shape indicate the nature of the form of a distribution, without reference to its location or width. There are two measures of shape reported by @RISK, the skewness and kurtosis. These are also known as the 3rd and 4th central moments of a distribution. (The first and second moments are the mean and variance.)

  • Skewness – indicates the degree of asymmetry in a distribution. With skewed distributions, one tail is typically longer than the other. A positive skewness indicates a long right tail, while a negative skewness indicates a long left tail. Zero skewness usually indicates a symmetric distribution:
  • Kurtosis - indicates how flat or peaked the distribution is. The higher the kurtosis value, the more peaked the distribution. Confusingly, there are two common definitions of kurtosis in use, one that gives the normal distribution a kurtosis of 3 and one which gives the normal distribution a kurtosis of 0. @RISK uses the former definition. The latter definition is known as the “kurtosis excess”.

The skewness and kurtosis of a simulated output are displayed to the right of all distribution graphs, as well as in the Statistics Detail window. In addition, they can be added directly to a spreadsheet model using the RiskSkewness and RiskKurtosis functions.

Percentiles and Targets

In many cases, it is useful to know how much probability falls to the left (or right) of a certain value. For example, what is the probability that a venture will lose money (have a profit of less than zero.)? Or, what is the probability that it will make more than one million dollars? The reversal of this kind of question is also common. What is the boundary value for the profit that consists of the worst 5% of all cases? These types of questions involve the concepts of percentiles and targets.

Percentile – A percentile is the value of a distribution that have a specified fraction of the probability at or less than that value.

Target – A target is the probability of achieving a particular output value or less.

The easiest way to understand the difference between these two operations is by using the delimiters of an output graph. For example, after typing 5% to the left part of the delimiter bar of the graph below, @RISK reports that that the 5th percentile is -16.4. On the other hand, after typing 1.2 on the number just above the delimiter arrow, @RISK reports that this target value has 54.2% of the probability below it.

As shown above, the easiest way to calculate percentiles and targets is by using the delimiters of an output graph. A set of default percentiles are also shown to the right of all distribution graph and in the Statistics Detail window. The default percentiles shown are configurable in @RISK’s General Preferences dialog.

Some practitioners use an opposite definition of percentiles where the probability is to the right of the target value instead of the left. Using so-called “cumulative descending percentiles” doesn’t change the display of the delimiter bars in a graph, but it will change how percentile values are shown in tables. This setting is called also in @RISK’s General Preferences dialog.

There are a large number of @RISK statistic functions to calculate targets and percentiles. The RiskPercentile and RiskTarget functions accomplish this task. For cumulative descending versions of these functions use RiskPercentileD and RiskTargetD.

Some users gets confused about the difference between percentiles and targets, so each of these functions has an alias which explicitly indicates what will be calculated using the letters X, P, and Q. X refers to a target value, P refers to the probability at or less than that X values, and Q refers to the probability greater than the X value.