Monte Carlo Simulation

Overview

The Monte Carlo Simulation calculator repeats an expression many times while randomly sampling one variable from a chosen probability distribution, then summarizes the resulting outcomes as a histogram, a table, and summary statistics.

It is useful when an input is uncertain and you want to see the range of likely results rather than a single answer for one assumed value.

Monte Carlo randomly samples values. It does not sweep a fixed range and is not the same as Redo, which performs a deterministic parameter sweep.

How It Works

For each of the requested trials, Monte Carlo:

1. Draws a random value for the selected variable from the chosen distribution.
2. Substitutes the value for the variable in the expression.
3. Evaluates the resulting expression.
4. Extracts a scalar result value.
5. Repeats until all trials are drawn, then summarizes the collected results.

random samples (per trial)
    -> substitute variable
    -> evaluate expression
    -> collect scalar result
    -> histogram + table + summary stats

A trial that fails to produce a usable numeric result (invalid substitution, evaluation error, or a non-numeric result) is counted separately and excluded from the statistics, rather than stopping the whole run.

Inputs

Expression

The expression or calculator call to repeat. The expression should contain the variable named in Variable.

The default expression is:

sine('x deg')

Vary By

Select what each sampled value changes:

• Parameters (p) replaces a named parameter inside a calculator call, such as inflation='x pct/yr'.
• Variables (v) replaces a standalone expression variable, such as x in x * 120 or sine('x deg').

Use Parameters for calculator or function arguments and Variables for direct expression variables. The default is Parameters.

Variable

The variable name to replace with a randomly sampled value on each trial, for example:

x

Distribution

The probability distribution to sample the variable from:

• normal — bell-shaped, can produce values on either side of the mean.
• uniform — every value between the low and high bound is equally likely.
• triangular — a low, high, and most likely (mode) value; useful for simple estimates.
• lognormal — right-skewed and always positive; useful for prices, durations, or growth rates that cannot go negative.

Param1, Param2, Param3

The meaning of these depends on the selected distribution:

Distribution	Param1	Param2	Param3
normal	mean	stdev	not used
uniform	low	high	not used
triangular	low	high	mode (defaults to midpoint)
lognormal	mu	sigma	not used

mu/sigma for lognormal describe the underlying normal distribution, not the mean/stdev of the sampled values themselves.

Trials

The number of random samples to draw and evaluate. More trials produce a smoother histogram and more stable statistics, at the cost of more computation. Trials are capped by the configured range limit.

Bin Count

The number of histogram bins used to group the results.

Round Off

The number of decimal places used to round each sampled value before substitution.

Result Columns / Result Units / Chart Column

Optional filters, matching the same convention as Redo, for when the expression returns more than one scalar value. Chart Column selects which single result column the histogram is built from; if more than one column would otherwise qualify, specify one explicitly.

Show

Controls whether the result is displayed as a table, chart (histogram), or both. Summary statistics are always computed regardless of this setting.

Chart Title

An optional title for the histogram.

Outputs

Along with the table/chart (depending on Show), Monte Carlo always returns:

• trials_used — number of trials that produced a usable numeric result.
• trials_failed — number of trials excluded (evaluation error or non-numeric result).
• mean, stdev, min, max — summary statistics over the used trials.
• p5, p50, p95 — the 5th, 50th (median), and 95th percentiles, a common way to express a confidence range (e.g. "90% of outcomes fall between p5 and p95").

Example: Angle with Measurement Uncertainty

Use:

sine('x deg')

with:

Variable: x
Distribution: normal
Param1 (mean): 180
Param2 (stdev): 90
Trials: 500

Even though x is sampled from a symmetric bell curve, sine() is nonlinear, so the resulting histogram is not a bell curve — it piles up near +1/-1. This illustrates why Monte Carlo is most useful for nonlinear expressions: a linear/identity expression would just reproduce the input distribution's shape.

Example: Cost Estimate with a Triangular Distribution

Use an expression that computes a total cost from an uncertain unit price:

x * 120

with:

Variable: x
Distribution: triangular
Param1 (low): 8
Param2 (high): 15
Param3 (mode): 10
Trials: 1000

This reflects a typical estimation scenario: a low, high, and most-likely value for an uncertain input, producing a realistic spread of possible total costs rather than one fixed number.

Choosing Distribution Parameters

• Use normal when you have a typical value and know how much it usually varies (mean and standard deviation).
• Use uniform when you only know a plausible low and high bound, with no reason to favor any value in between.
• Use triangular when you have a low, high, and best-guess (most likely) value — common in cost and schedule estimation.
• Use lognormal when the quantity cannot be negative and tends to have a long tail on the high side (e.g. prices, durations).

Expression Rules

The expression must remain valid after substitution. If a quantity requires a unit, include the unit around the variable, for example:

q('x usd') * 1.2

Avoid using a short variable such as x when the expression also contains similarly named identifiers, e.g. x_value.

Errors and Performance

A trial can fail when the substituted value makes the expression invalid. Common causes include:

• A value outside a calculator's valid range.
• A missing or incompatible unit.
• An invalid expression after substitution.
• A result that cannot be converted into a single scalar number.

Failed trials are counted in trials_failed and excluded from the statistics rather than stopping the whole simulation. If every trial fails, an error is raised instead of returning empty statistics.

Each trial re-evaluates the full expression, so a large number of trials on an expensive expression (e.g. one that calls another involved calculator) can take noticeably longer than a single ordinary calculation. Start with a moderate trial count, such as 200 to 500, before increasing it.

Redo and Monte Carlo

Redo is a deterministic parameter sweep:

known values
    -> one value at a time
    -> repeat calculation
    -> chart sensitivity

Monte Carlo is a random simulation:

randomly sampled values
    -> many repeated calculations
    -> probability distribution of outcomes

Use Redo first to understand the controlled, deterministic relationship between a variable and the result. Use Monte Carlo afterward, once you have a plausible distribution in mind for an uncertain input, to see the realistic range and likelihood of outcomes.

Summary

Use Monte Carlo to:

• Repeat a qCalc expression many times with a randomly sampled variable.
• Model an input whose exact value is uncertain but whose typical range is known.
• Generate a histogram, table, and summary statistics (mean, stdev, min, max, percentiles) of the outcomes.
• Understand how uncertainty in one input propagates through a (often nonlinear) calculation.