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Latitude Plots Tutorial

A worked example, from a model to a plot you can read. For what every control does, see Latitude Plot and its Options page; this page is about reading the result.

QXL Monte Carlo New > Monte Carlo > Contribution Tools > Latitude Plot

The question a Latitude Plot answers

A simulation tells you how often the output falls outside its specification. It does not tell you which combinations of inputs put it there.

A Latitude Plot draws that region. It takes a pair of inputs, spreads them across their ranges, and marks the boundary where an output reaches a specification limit. You get a map of the pair's space with the acceptable part fenced off, and your process's own variation drawn on the same map.

The model

Two parts stacked end to end. Part A and Part B are made separately, and the stack has to fit a slot between 25 and 35 cm.

Press Copy for Excel, then paste the table into a blank worksheet.

Stack Value
Part A 23
Part B 7
Stack height =B2+B3

Click cell A1 and press Ctrl+V. B2 is Part A, B3 is Part B and B5 is the stack height.

Mark the two inputs with Mark Input, both Normal:

Cell Distribution Mean Standard deviation
B2, Part A Normal 23 2
B3, Part B Normal 7 1

Mark B5 as an output with Mark Output, lower limit 25 and upper limit 35.

What the model says before you plot anything

This model is a straight sum of two independent normals, so its answer is exact arithmetic and you can check the simulation against it:

Value Where it comes from
Mean of the stack 30 23 + 7
Standard deviation 2.2361 sqrt(2² + 1²) = sqrt(5)
Distance to each limit 2.2361 standard deviations the band is 25 to 35, so 5 either side of 30
Outside the band 2.53%, or 25,347 DPM 1.27% below 25 and the same above 35
Cpk 0.75 5 / (3 x 2.2361)

Run the model first. Run Model, and the histogram should put its mean near 30 with a visible tail past each limit. If it does not, fix that before plotting: the Latitude Plot reads the same model.

Drawing the plot

  1. QXL Monte Carlo New > Monte Carlo > Contribution Tools > Latitude Plot.
  2. Select both inputs. Every pair of the inputs you select becomes one chart, so two inputs give one chart.
  3. Give each a low, a high and a nominal. The dialog suggests these from the engine, so leaving them alone is a reasonable first run.
  4. Leave Display Dots and Display Expected Variation Region ticked, and leave Confidence Interval at 0.99. Both boxes start ticked.
  5. Finish.

Reading the chart

The axes are the two inputs, Part A on one and Part B on the other, in centimetres. Every point on the chart is one combination of the two.

The specification contours: the latitude window

Two lines cross the chart, one where the stack reaches 25 and one where it reaches 35. The band between them is the region where the stack is in specification, and it is what gives the plot its name: the latitude you have.

On this model those lines are straight and run corner to corner, because the stack is A + B. Every combination adding to 25 sits on one line and every combination adding to 35 sits on the other, so the band runs diagonally across the chart at a slope of minus one. A model with a product or a power in it gives curved contours, and that is worth seeing once: the shape of the band is the shape of your equation.

A contour is missing when the whole plotted region is on one side of that limit. If nothing in the ranges you gave could reach the lower limit, there is no lower line to draw, and the plot records that rather than drawing an edge that is not there.

The dots

One dot per evaluated combination across the pair's space, marked pass or fail. They fill in what the contours outline, and they are the quickest way to see which side of a boundary is which.

Turn them off with Display Dots when the contours alone are clearer.

The Expected Variation Region: what your process actually does

An ellipse centred on the pair's nominal values, covering the proportion of the joint variation you asked for. At the default Confidence Interval of 0.99 it covers 99 per cent of the pairs the two inputs would produce together.

It tilts when the pair is correlated, because it is fitted to the joint variation rather than drawn as a box around each input's range. On this model the two parts are independent, so the ellipse sits square to the axes: wider along Part A, which has a standard deviation of 2, than along Part B, which has 1.

It is not drawn at all when it cannot be built: too few points, an input with no variation, or a confidence interval that is not strictly between 0 and 1.

The reading that matters: the ellipse against the window

Put the two together and the chart answers one question: does your process fit in the room you have?

What you see What it means
The ellipse sits well inside the band the pair's variation is comfortably within specification
The ellipse crosses a contour some combinations the process really produces are out of specification, and the part sticking out is where they are
The ellipse is inside but hard against one contour it fits, with nothing to spare on that side
The band is far wider than the ellipse in every direction there is more latitude than the process is using

On this model the ellipse crosses both contours, which is the same finding the simulation reported as 2.53 per cent outside the band. The plot adds where. The histogram says how often; the Latitude Plot says which combinations of Part A and Part B.

Now change one thing at a time

Halve Part A's standard deviation to 1 and run the model and the plot again.

The stack's standard deviation falls to sqrt(1² + 1²) = 1.4142, the distance to each limit rises to 3.54 standard deviations, and the fraction outside the band drops to 0.041%, or 407 DPM, from 25,347. On the chart the ellipse shrinks along the Part A axis and pulls clear of both contours.

Then put Part A back to 2 and move its mean to 25 instead. The stack now averages 32, the spread is unchanged, and the distance to the upper limit falls to 3 / 2.2361 = 1.34 standard deviations while the lower limit moves to 3.13. Nine per cent of stacks are now out of specification, against 2.53 per cent before, and every one of them is over rather than under. The ellipse does not change size; it slides toward the upper contour.

Those two changes are the two things you can do to a process, and the plot tells them apart at a glance: a spread problem changes the size of the ellipse, and a centring problem moves it. A histogram of the output shows both as "more defects".

What this page does not describe

How the plot is computed is not documented, deliberately. This tutorial and the Latitude Plot pages say what the tool draws, what every control does and how to read the result. They do not describe the method behind it.

See Also