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HomeTechniques and Tips@RISK Simulation: Graphical ResultsInterpreting or Changing the Y Axis of a Histogram

7.1. Interpreting or Changing the Y Axis of a Histogram

Applies to:
@RISK for Excel 4.5–7.x
@RISK for Project 4.0 and 4.1

How do I interpret the y axis of the histogram that is created from the results of my simulation?

@RISK can show the histogram of your result data in two different formats, probability density or relative frequency. This just a matter of different scaling for the y axis; the shape of the histogram doesn't change. The default histogram is probability density for continuous data, and relative frequency for discrete data.

TIP: Most people find relative frequency easier to understand than probability density. Especially for presentations, you may want to use the relative frequency format, or simply suppress the y axis. (See "How do I select the y axis format," below.)

How do I interpret relative frequency numbers on the y axis?

If a bar is as high as the 2% mark, for example, you know that 2% of all iterations fell within that bar. In other words, the height of each bar represents the proportion of the data (the fraction of all the iterations) in that bar. Since every data point must be in some bar of the histogram, the heights of all the bars add up to 100%.

(Before @RISK 6.2, relative frequencies were shown as decimals, for example 0.02 rather than 2%, but you read them the same way.)

How do I interpret probability density numbers on the y axis?

This is harder. Unlike the case of relative frequency, the height of a histogram bar isn't meaningful on its own. What matters is the area of the bar.

Consider the example at right. (You can click on it to get a larger image.) Look at the bar for $74,000 to $76,000. Its width on the x axis is $2,000, and its height on the y axis is about 4.9×10-5. As with any rectangle, you find its area by multiplying width and height: $2000×4.9×10-5 = 0.098 or 9.8%. The height of that bar by itself doesn't tell you anything, but in conjunction with the width it tells you that 9.8% of the iteration data for this input fell between $74,000 and $76,000. The total area of all the bars is 1 (or 100%).

When you're looking at a theoretical probability curve for an input, or in a fitted distribution, it will be presented as probability density. Again, the height of the curve doesn't tell you anything useful on its own. But the area under part of the probability density curve tells you what percentage of the data should fall within that region, theoretically. For example, the area under the curve to the left of $72,104 is 5.0% according to the bar at the top of the graph. This tells you that theory says 5% of the data for a Normal(80000,4800) should be less than $72,104.

Technically, the area under a part of the curve is the integral of the height of the curve, from the left edge of the region to the right edge. Thus, the 5% was found by integrating the height of the density curve from minus infinity to $72,104. Just as the total area of the bars in a histogram is 1, the total area under a probability density curve is 1.

How does @RISK create the y axis for a probability density histogram?

  1. Divide the data into intervals — see Number of Bins in a Histogram.
  2. Count the number of data points in each interval.
  3. Divide the counts by the total number of data points.
  4. Divide that result by the interval width as shown on the x axis, to obtain the height of the bar along the y axis.

In a probability density histogram or curve, the larger the numbers on the x axis, the smaller the numbers on the y axis must be to keep the total area at 1.

How do I select the y axis format?

In @RISK 5.x–7.x, click the histogram icon at the bottom of the Browse Results window and select Relative Frequency or Probability Density.  If you prefer, you can suppress the numbers on the y-axis entirely: right-click on any of the numbers on the vertical axis and select Axis Options. Then on the Y-Axis tab, under Display, remove the check mark by Axis.

TIP: If you find yourself changing the y axis often, you might want to change the default. In Utilities » Application Settings » Simulation Graph Defaults, change Preferred Distribution Format to Relative Frequency, or whatever you prefer.

In @RISK 4.x, right-click the histogram and select Format Graph...; then select the Type tab. In the Histogram Options section, click the drop-down arrow next to the Format field and choose Density or Relative Frequency.

Last edited: 2018-09-21

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