Six Sigma Toolbox
The premier tool for advanced Six Sigma
and Lean Six Sigma Implementation
Full Feature Brochure
Six Sigma and Lean Six Sigma
Six Sigma is a structured approach to business management that concentrates on
improving quality by reducing process variability and eliminating major failure
mechanisms. Since it relies heavily on the collection and analysis of data, statistical
programs such as Statgraphics are a vital component of all serious Six Sigma
implementations. Lean Six Sigma adds important concepts from lean manufacturing,
which concentrates on the elimination of unnecessary resource utilization.
The Statgraphics Centurion Six Sigma Toolbox provides a large selection of tools for
use in Lean and Six Sigma programs. The areas covered include:
1. Quality Management 3
2. Measurement Systems Analysis (Gage Studies) 6
3. Process Capability Analysis 9
4. Statistical Process Control Charts 14
5. Design of Experiments 24
6. Life Data Analysis and Reliability 32
7. Acceptance Sampling 37
8. Monte Carlo Simulation 39
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1. Quality Management
When beginning a Six Sigma project, it is important to understand the problem to be addressed and
the factors that most effect quality and cost. Statgraphics Centurion XVII contains several
procedures that are helpful in this area.
Pareto Chart
Pareto charts are often used to determine the "vital few" causes that are responsible for the majority
of defects or complaints regarding a product or service.
More: Pareto Analysis.pdf
Quality Function Deployment Matrix
QFD is a customer-driven planning process by which products and services are matched to the
needs of customers. STATGRAPHICS illustrates the relationship between customer needs and
design requirements by constructing a "House of Quality”.
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More: QFD Matrix.pdf
Cost of Quality Trend Analysis
A COQ (Cost-of-Quality) trend analysis illustrates the cost of poor quality by constructing a chart
displaying prevention, appraisal, and failure costs over time. Runs tests are also performed to search
for significant trends.
More: Cost-of-Quality Trend Analysis.pdf
Cause-and-Effect Diagram
Cause-and-effect or fishbone diagrams illustrate the causes of a problem or effect by creating a
diagram resembling the skeleton of a fish.
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Raw card Solder process Inspection
Splatter
Chain speed
Short circuit Measurement
Height
Wave pump
Flow
Shroud Test coverage
Flux
Time
Setup
Moisture content Inspector
Temperature Temperature
Control
Circuit
board defects
Wrong component
Wrong part Missing component
Functional failure Autoposition
Alignment
Operator
Vendor
Missing from reel
Setup
Crimp
Components Component insertion
More: Cause-and-Effect Diagram.pdf
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Process Map
Process maps are commonly used to create flow charts and other step-by-step diagrams. An
example of a simple map is shown below.
More: Process Mapping.pdf
2. Measurement Systems Analysis (Gage Studies)
When implementing any statistical method that relies on data, it is important to be sure that the
systems that collect that data are both accurate and precise. A set of procedures, often referred to
as "Gage Studies", are widely used to assess the quality of measurement systems.
Variable Measurement Systems - Repeatability and Reproducibility
For measurement systems that result in quantitative measurements such as weight, concentration,
or strength, it is important to determine the magnitude of any error in the resulting measurements. If
the error is large, it may be impossible to determine whether or not an individual sample is within
spec. In addition, designed experiments rely on the ability to separate real effects of making changes
from the background noise and could be sabotaged by an inadequate measurement system
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When quantifying measurement error, it is common to separate the error into repeatability (error due
to the instrument or measurement procedure) and reproducibility (error due to the appraiser).
STATGRAPHICS implements the procedures suggested by the AIAG (Automotive Industry Action
Group), including the average and range method, ANOVA method (with and without interaction), and
the range method (for short studies).
More: Gage Study Setup.pdf, Gage R&R SnapStat.pdf, Gage Studies - ANOVA Method.pdf, Gage Studies -
Average and Range Method.pdf, Gage Studies - Range Method.pdf
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Variable Measurement Systems - Linearity and Accuracy
While repeatability and reproducibility studies concentrate on the variability or precision of a
measurement system, Linearity and Accuracy studies quantify the bias. In these studies, multiple
measurements are made on reference samples and an equation is constructed for the bias of the
measurements.
More: Gage Linearity and Accuracy.pdf
Attribute Measurement Systems
When the results of a measurement system are PASS or FAIL rather than a quantitative value, special
procedures are necessary. STATGRAPHICS Centurion provides three procedures outlined by the
AIAG to deal with such systems: the risk analysis method, the signal theory method, and the analytic
method. In the risk analysis method, multiple appraisers measure samples with known
characteristics. Statistics are calculated based on how often the appraisers correctly characterize
each sample and how frequently they agree with themselves and each other.
More: Gage Studies - Analytic Method.pdf,
Gage Studies - Risk Analysis Method.pdf,
Gage Studies - Signal Theory Method.pdf
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3. Process Capability Analysis
An important technique used to determine how well a process meets a set of specification limits is
called a process capability analysis. A capability analysis is based on a sample of data taken from a
process and usually produces:
1. An estimate of the DPMO (defects per million opportunities).
2. One or more capability indices.
3. An estimate of the Sigma Quality Level at which the process operates.
STATGRAPHICS provides capability analyses for the following cases:
Capability Analysis for Measurement Data from a Normal Distribution
This procedure performs a capability analysis for data that are assumed to be a random sample from
a normal distribution. It calculates capability indices such as Cpk, estimates the DPM (defects per
million), and determines the sigma quality level (SQL) at which the process is operating. It can handle
two-sided symmetric specification limits, two-sided asymmetric limits, and one-sided limits.
Confidence limits for the most common capability indices may also be requested.
More: Capability Assessment SnapStat.pdf
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Capability Analysis for Measurement Data from Non-Normal Distributions
This procedure performs a capability analysis for data that are not assumed to come from a normal
distribution. The program will fit up to 25 alternative distributions and list them according to their
goodness-of-fit. For a selected distribution, it then calculates equivalent capability indices, DPM,
and the SQL.
More: Capability Analysis (Variable
Data).pdf
Capability Analysis for Correlated Measurements
When the variables that characterize a process are correlated, separately estimating the capability of
each may give a badly distorted picture of how well the process is performing. In such cases, it is
necessary to estimate the joint probability that one or more variables will be out of spec. This
requires fitting a multivariate probability distribution. This procedure calculates capability indices,
DPM, and the SQL based on a multivariate normal distribution
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More: Multivariate Capability Anaysis.pdf
Capability Analysis for Counts or Proportions
When examination of an item or event results in a PASS or FAIL rather than a measurement, the
capability analysis must be based on a discrete distribution. For very large lots, the relevant
distribution is the binomial. For small lots or cases of limited opportunities for failure, the
hypergeometric distribution must be used
More: Capability Analysis (Percent Defective).pdf
Capability Analysis for Rates
When the relevant measure of performance is a rate, then the capability analysis is based on: a
Poisson distribution if failures occur randomly; a negative binomial distribution if failures tend to
occur in clumps.
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More: Capability Analysis (Defects per Unit).pdf
Six Sigma Calculator
The STATGRAPHICS Six Sigma Calculator converts between various commonly used quality
metrics. After entering the value of any one metric, the equivalent values of the others are calculated.
More: Six Sigma Calculator.pdf
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4. Statistical Process Control Charts
One of the most important actions that can help maintain the quality of any good or service is to
collect relevant data consistently over time, plot it, and examine the plots carefully. All statistical
process control charts plot data (or a statistic calculated from data) versus time, with control limits
designed to alert the analyst to events beyond normal sampling variability.
STATGRAPHICS Centurion provides one of the most extensive collection of control charts available.
All control charts can be used for Phase I studies, in which the data determine the location of the
control limits, and Phase II studies, in which the data are compared against a pre-established
standard. A special procedure is also provided to help design a control chart with acceptable power.
E-mail alerts can be generated when points fall outside the control limits or when a runs rule is
violated.
Basic Variables Charts
The classical type of control chart, originally developed back in the 1930's, is constructed by
collecting data periodically and plotting it versus time. If more than one data value is collected at the
same time, statistics such as the mean, range, median, or standard deviation are plotted. Control
limits are added to the plot to signal unusually large deviations from the centerline, and runs rules are
employed to detect other unusual patterns.
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More: X-Bar and R Charts.pdf, X-Bar and S Charts.pdf, X-Bar and S-Squared Charts.pdf, Median
and Range Charts.pdf, Individuals Control Charts.pdf
Basic Attributes Charts
For attribute data, such as arise from PASS/FAIL testing, the charts used most often plot either rates
or proportions. When the sample sizes vary, the control limits depend on the size of the samples. On
most control charts, colored zones may be used to indicate the distance to 1, 2, and 3-sigma.
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More: P Chart.pdf, NP Chart.pdf, U Chart.pdf, C Chart.pdf, U’ Chart.pdf, P’ Chart.pdf
Moving Average and EWMA Charts
When data are collected one sample at a time and plotted on an individuals chart, the control limits
are usually quite wide, causing the chart to have poor power in detecting out-of-control situations.
This can be remedied by plotting a weighted average of the data instead of just the most recent
observation. The most common "time-weighted" charts are the moving average (MA) chart and the
exponentially weighted moving average (EWMA). The average run length of such charts is usually
much less than that of a simple X chart.
More: Moving Average (MA) Charts.pdf , EWMA Charts.pdf
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Cusum Charts
A useful chart for plotting measurements from a continuous process is the CUSUM chart, which
plots at each time point the sum of all deviations from a target value up to and including the most
recent observation. When the "V-mask" format for a Cusum chat is used, the process is deemed to
be in control if all points on the chart fall within the mask. If any points fall outside the mask, as in the
chart at the left, then an out-of-control alert is generated.
.
More: Cusum Charts (V-Mask).pdf , Cusum Charts (Tabular).pdf
Multivariate Control Charts
When data for more than one variable are collected, separate control charts are frequently plotted for
each variable. If the variables are correlated, this can lead to missed out-of-control signals. For such
situations, STATGRAPHICS provides several types of multivariate control charts: T-Squared charts,
Generalized Variance charts, and Multivariate EWMA charts. In the case of two variables, the points
may be plotted on a control ellipse.
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More: Multivariate T-Squared Control
Chart.pdf, Multivariate EWMA Control
Chart.pdf
ARIMA Control Charts
With today's automated data collection systems, samples are frequently collected at closely spaced
increments of time. Any sort of process dynamics introduces correlation into successive
measurements, which causes havoc with standard control charts that assume independence
between successive samples. In such cases, a control chart that captures the dynamics of the
process must be used to properly detect unusual events when they occur.
The proper chart for such situations is an ARIMA control chart, which is based upon a parametric
time series model for process dynamics. Such charts either plot the residual shocks to the system at
each time period, or they display varying control limits based upon predicted values one period
ahead in time.
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More: ARIMA Charts.pdf
Toolwear Charts
Control charts can also be used to monitor processes in which the mean measurement is expected
to change over time. This commonly occurs when monitoring the wear on a tool, but also arises in
other situations. The control charts for such cases have a centerline and control limits that follow the
expected trend.
More: Toolwear Charts.pdf
Acceptance Control Charts
For processes with a high Cpk, requiring the measurements to remain within 3 sigma of the
centerline may be unnecessarily restrictive. In such cases, the process may be allowed to drift, as
long as it does not come too close to the specification limits. A useful type of control chart for this
case is the Acceptance Control Chart, which positions the control limits based on the specification
limits rather than the process mean.
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More: Acceptance Charts.pdf
Cuscore Charts
When monitoring a real-world process, the types of out-of-control situations that are likely to occur
may be known ahead of time. For example, a pump that begins to fail may introduce an oscillation
into the measurements at a specific frequency. In such cases, specialized CuScore Charts may be
constructed to watch for that specific type of failure.
STATGRAPHICS will construct CuScore charts to detect: spikes, ramps, bumps of known duration,
step changes, exponential increases, sine waves with known frequency and phase, or any custom
type of pattern that the user wishes to specify.
More: Cuscore Charts.pdf
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Control Charts for Rare Events
G charts and T charts are used to monitor the occurrence of rare events, such as infections in a
hospital. By monitoring the length of time between consecutive events, the charts can detect
changes in the underlying rate.
t Chart for Infection time
Shape parameter = 1.94302; Scale parameter = 3049.88
(X 1000)
10
8 8059.90
Time between events
2525.58
2
0 101.75
0 20 40 60 80 100
Observation
More: G Charts.pdf, T Charts.pdf
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Control Chart Design
For a control chart to be effective, it must be able to distinguish between situations in which the
process is operating as expected and situations in which it has deviated seriously from its target
values. STATGRAPHICS provides a procedure for designing control charts that will detect deviations
of a specified magnitude within an acceptable time. In a typical application, the user specifies a
target mean and the desired average run length before a deviation of that magnitude is detected.
The procedure then determines the number of samples and/or smoothing parameter that will
achieve the desired performance.
More: Control Chart Design.pdf
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E-mail Alerts
The control chart procedures can be automated using the dynamic updating capabilities of
Statgraphics Centurion. In such cases, it can be useful to generate e-mail alerts whenever an
unusual event occurs. Alerts may be created when points on the control charts fall outside the
control limits, or when a run rule is violated. If specification limits exist for the variable being plotted,
alerts may also be generated whenever estimated capability indices fall below a threshold value or
when the estimated DPM is too large.
More: StatFolio Alerts.pdf
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5. Design of Experiments
STATGRAPHICS contains extensive capabilities for the creation and analysis of statistically designed
experiments. The designs that can be created are divided into several types.
Screening Designs
Screening designs are intended to determine the most important factors affecting a response. Most
of the designs involve only 2 levels of each factor. The factors may be quantitative or categorical.
Included are 2-level factorial designs, mixed level factorial designs, fractional factorials, irregular
fractions, and Plackett-Burman designs. For designs of less than full resolution, the confounding
pattern is displayed. Blocking and randomization are options.
More: DOE Wizard - Screening Designs.pdf
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Response Surface Designs
Response surface designs are intended to determine the optimal settings of the experimental
factors. The designs involve at least 3 levels of the experimental factors. Included are central
composite designs, Box-Behnken designs, 3-level factorials, and Draper-Lin designs.
More: DOE Wizard - Response
Surface Designs.pdf
Mixture Experiments
Mixture experiments involve components of a mixture, where the levels of the components are
constrained to sum to 100% (or some other fixed value). Upper and lower constraints may be
specified for each component. Included are simplex-lattice, simplex-centroid, and extreme vertices
designs.
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More: DOE Wizard - Mixture Experiments.pdf
Computer-Generated Optimal Designs
Computer-generated designs may be created given:
1. A statistical model to be fit.
2. The desired number of runs.
3. The optimality criterion to be maximized (A-efficiency, D-efficiency, G-efficiency, or I-
efficiency).
4. Constraints that may exist limiting the combinations of factor levels at which experimentation
is possible.
These designs are commonly used when the number of runs must be as small as possible, when the
design region is constrained, or when additional runs need to be added to an undesigned
experiment to improve its statistical properties.
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More: DOE Wizard – Computer-Generated Designs.pdf, DOE Wizard - Multilevel Factorial
Designs.pdf
Robust Parameter Designs
Statgraphics can create experimental designs for use in robust parameter design (RPD). In such
experiments, two types of factors are varied: controllable factors that the experimenter can
manipulate both during the experiment and during production, and noise factors that can be
manipulated during the experiment but are normally uncontrollable. The goal of RPD is to find levels
of the controllable factors where the response variables are relatively insensitive to changes in the
noise factors.
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RPD designs may be created using either the crossed approach of Taguchi (with inner and outer
arrays) or the combined approach of Montgomery (both types of factors varied in a single array).
More: DOE Wizard - Robust Parameter Designs.pdf
Single Factor Categorical Designs
Single Factor Categorical designs are used to compare levels of a single non-quantitative factor.
They include completely randomized designs, randomized block designs, balanced incomplete block
(BIB) designs, Latin Squares, Graeco-Latin Squares, and hyper-Graeco-Latin Squares
More: DOE - Single Factor Categorical
Designs.pdf
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Multi-Factor Categorical Designs
Multi-Factor Categorical designs are used to study multiple non-quantitative factors, with several
levels of each. They are analyzed using a multifactor analysis of variance.
More: DOE - Multi-Factor Categorical Designs.pdf
Variance Component Designs
Variance Component (hierarchical) designs are used to study the effect of two or more nested
factors on the variability of a response. Estimates of the contribution of each factor to the overall
variability are obtained.
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More: DOE - Variance Component Designs.pdf
Design of Experiments Wizard
STATGRAPHICS Centurion contains a wizard that assists users in constructing and analyzing
designed experiments. It guides the user through twelve important steps. The first 7 steps are
executed before the experiment is run. The final 5 steps are executed after the experiment has been
performed.
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More: Design of Experiments Wizard.pdf
Multiple Response Optimization
In order to find a combination of the experimental factors that provides a good result for multiple
response variables, the DOE Wizard uses the concept of desirability functions. Desirability functions
provide a way to balance the competing requirements of multiple responses, which may be
measured in different units. Users specify the target value or acceptable range for each response,
together with its relative importance. The program then finds the best combination of the
experimental factors.
More: Design of Experiments Wizard.pdf
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6. Life Data Analysis and Reliability
Determining the reliability of manufactured items often requires performing a life test and analyzing
observed times to failure. Such data is frequently censored, in that some items being tested may not
have failed when the test is ended. In addition, it may be necessary to accelerate failure times by
changing the value of an influential variable such as temperature. For all of these reasons, special
tools are needed to deal with this type of data.
Life Tables
In analyzing life data, interest commonly centers on estimating the probability that a unit will still be
operating at any given time. A common way of estimating this survival function, without making any
assumption about functional form or error distribution, is to tabulate the data and calculate the
survivor function directly from the observed failures. When censoring is present, the estimates are
calculated using the Kaplan-Meier approach.
More: Life Tables (Intervals).pdf, Life Tables (Times).pdf
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Distribution Fitting with Censored Data
If sufficient data is available, it may be possible to fit a specific distribution to the failure times.
Maximum likelihood methods can be easily adapted to the presence of censored data.
STATGRAPHICS will automatically fit up to 45 probability distributions for any sample of data and
rank them according to goodness-of-fit.
More: Distribution Fitting (Censored data).pdf
Weibull Analysis
Experience has shown that failure data can often be well modeled by a Weibull distribution. A
common method to check the fit of a Weibull distribution is through a Weibull plot. Uncensored
failure times should fall approximately along a straight line.
In the STATGRAPHICS Weibull Plot, you may add a histogram of censored failure times and
confidence limits for failure percentiles.
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More: Weibull Analysis.pdf
Arrhenius Plot
When failures do not occur often enough under normal operating conditions, it is necessary to
accelerate the failures by increasing the stress caused by one or more variables. A very common
accelerant is temperature. By analyzing failure rates at high temperatures and fitting an Arrhenius
model, it is often possible to extrapolate the data back to a normal operating temperature (usually
expressed in Kelvin).
Model: Arrhenius Plot.pdf
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Life Data Regression
To describe the impact of external variables on failure times, regression models may be fit.
Unfortunately, standard least squares techniques do not work well for two reasons: the data are
often censored, and the failure time distribution is rarely Gaussian. For this reason, STATGRAPHICS
provides a special procedure that will fit life data regression models with censoring, assuming either
an exponential, extreme value, logistic, loglogistic, lognormal, normal or Weibull distribution.
More: Life Data Regression.pdf
Cox Proportional Hazards
The Cox Proportional Hazards procedure is an alternative method for fitting a life data regression
without assuming any specific distributional form. Instead, it is assumed that the predictor variables
affect the hazard function in a multiplicative manner. Like the parametric life data regression
procedure, the predictor variables can be either quantitative or categorical.
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More: Cox Proportional Hazards.pdf
Analysis of Repairable Systems
The Repairable Systems procedures are designed to analyze data consisting of failure times from
systems that can be repaired. It is assumed that when the system fails, it is immediately repaired and
placed in service again. Further, it is assumed that the repair time is negligible compared to the time
between failures. The goal of the analysis is to develop a model that can be used to estimate failure
rates or quantities such as the MTBF (mean time between failures).
This procedure differs from the Distribution Fitting and Weibull Analysis procedures in that it allows
for a failure rate that changes as the system ages.
More: Repairable Systems (Times).pdf, Repairable Systems (Intervals).pdf,
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7. Acceptance Sampling
When lots containing a relatively large number of items require inspection, acceptance sampling
plans can provide reasonable protection against shipping or receiving an unacceptable fraction of
non-conforming items without inspecting 100% of the lot. The Acceptance Sampling procedures
generate acceptance sampling plans by on either attributes or variables.
STATGRAPHICS generates several types of acceptance sampling plans:
• OC Plans - plans that control the alpha and beta risks, i.e., the probability of accepting a bad
lot and the probability of rejecting a good lot. For such a plan, “good” and “bad” must be
well-defined.
• AOQL Plans - plans that minimize the average outgoing quality limit, i.e., the maximum
fraction of non-conforming items accepted on average. Such a plan requires 100%
inspection and rectification of all rejected lots.
• LTPD Plans - plans that minimize total inspection while controlling the risk of rejecting a bad
lot, where “bad” must again be well-defined. Such a plan also requires 100% inspection and
rectification of all rejected lots.
• MIL-STD 105E, 414, and 1916 – plans developed by the U. S. Department of Defense and
widely adopted by many companies.
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8. Monte Carlo Simulation
Monte Carlo simulation is used to estimate the distribution of variables when it is impossible or
impractical to determine that distribution theoretically. It is used in many areas, including
engineering, finance, and DFSS (Design for Six Sigma). A typical Monte Carlo simulation includes:
1. One or more input variables X, some of which usually follow a probability distribution.
2. One or more output variables Y, whose distribution is desired.
3. A mathematical model coupling the X’s and the Y’s.
More: Monte Carlo Simulation.pdf
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