Difficulty of interpreting the vertical scale | Vose Software

# Difficulty of interpreting the vertical scale

The most common mistake in interpreting a histogram plot of generated values from a Monte Carlo simulation is to read off the y-scale value as the probability of the x-value occurring. This is correct only if the output variable is discrete and there is only one allowed value in each of the histogram classes. In most situations, the output is continuous, and so the probability of any individual value occurring is zero (technically, it is infinitely small).

ModelRisk offers two options for scaling the vertical axis: density and relative frequency plots.

### Density histogram plots

Figure 1: Histogram 'density' plot. The vertical scale is calculated so that the sum of the histogram bar areas equals unity. This is only appropriate for continuous outputs (left). Simulation software won't recognise if an output is discrete (right), so treats the generated output data in the same way as a continuous output. The result is a plot where the probability values make no intuitive sense - in the right plot the probabilities appear to exceed unity. To be able to tell the probability of the output being equal to 4, for example, we first need to know the width of the histogram bar.

### Relative frequency histogram plots

Figure 2: Histogram 'relative frequency' plot. The vertical scale is calculated as the fraction of the generated values that fall into each histogram bar's range. Thus, the sum of the bar heights equals unity. Relative frequency is only appropriate for discrete variables (right), where the histogram heights now sum to unity. For continuous variables (left), the area under the curve no longer sums to unity.

Read on: Effect of varying number of bars