One-sigma and two-sigma solution ellipses with principal directionsLive current inputs
Math & Statistics
Linear System Distribution Calculator
Propagate uncertainty in two correlated right-hand constants through an invertible two-equation system and visualize the resulting joint distribution of x and y.
Mean solution x-
Mean solution y-
Standard deviation of x-
Standard deviation of y-
Solution correlation-
Major-axis standard scale-
Minor-axis standard scale-
Coefficient determinant-
JOINT SOLUTION DISTRIBUTION
One-sigma and two-sigma solution ellipses with principal directions
The ellipse rotates and stretches as coefficient geometry, input uncertainty, and c-f correlation change. Crosshairs mark the mean solution.
COVARIANCE PROPAGATION
From uncertain constants to uncertain unknowns
The live matrix ledger exposes every variance and covariance term.
| Quantity | Matrix position | Entered or derived value | Units | Interpretation |
|---|
UNCERTAINTY SETUP
Describe the joint input, not only separate spreads
- Enter the fixed coefficient matrix first.
- Use standard deviations in the same units as c and f.
- Estimate correlation from paired observations when possible.
- Keep |ρ| at or below one.
- Read ellipse direction together with the determinant.
ELLIPSE GEOMETRY
The matrix can rotate uncertainty even when the inputs are uncorrelated
The inverse coefficient matrix mixes c and f into both unknowns. Its geometry therefore determines the orientation of the solution cloud.
A small determinant makes the inverse large and can inflate uncertainty dramatically even when the mean solution appears ordinary.
LINEAR UNCERTAINTY TRANSFORM
Transform the full covariance matrix, not two error bars in isolation
Because x = A⁻¹b is linear, the mean and covariance propagate exactly when A is fixed. Correlation in the constants can either amplify or cancel uncertainty in a solution direction.
Detailed calculation process and general formulas
μz = A⁻¹ μbΣb = [[σc², ρσcσf], [ρσcσf, σf²]]Σz = A⁻¹ Σb (A⁻¹)ᵀσx = √Σz,11corr(x,y) = Σz,12 / (σxσy)Symbols, meanings, and units
- A
- fixed 2 by 2 coefficient matrixcoefficient units
- μb
- mean right-hand vector [c,f]ᵀequation units
- Σb
- input covariance matrixequation units²
- Σz
- solution covariance for [x,y]ᵀsolution units²
- ρ
- correlation between c and fdimensionless
DISTRIBUTION READING
Center, spread, and dependence answer different questions
The live ellipse carries all three.
Center
-The crosshair is the solution obtained from the mean constants.
Marginal spread
-Horizontal and vertical projections give σx and σy.
Joint direction
-Rotation reveals how the solved unknowns move together.
Decision takeaway: Do not combine x and y error bars independently when their covariance is material.
ASSUMPTIONS TO DOCUMENT
What must remain fixed for this transform
- Coefficient matrix treated as exact
- Linear equations remain valid
- Input covariance estimated on the same population
- No truncation or inequality constraints
- Normal-looking ellipse is a display convention, not a distribution test
Applied decisions
How correlation changes the same marginal inputs
Positively correlated constants
c and f rise together across observations.
What the result clarifies: One solution direction can widen while the orthogonal direction narrows.
Nearly singular coefficients
The two equation rows become almost parallel.
What the result clarifies: The ellipse elongates sharply because the inverse problem is poorly conditioned.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
The covariance propagation is exact for a fixed invertible linear matrix, but the displayed ellipses do not prove normality or define guaranteed coverage. Coefficient uncertainty and nonlinear constraints are excluded.
Linear System Distribution Calculator | Uncertain Inputs and Solution Ellipse FAQ
Is the two-sigma ellipse a 95% confidence region?
Not exactly. Joint two-dimensional coverage depends on the chosen probability model and chi-square threshold.
Why is x-y correlation different from c-f correlation?
The inverse coefficient matrix rotates and mixes the two input directions.
What happens near determinant zero?
The inverse becomes unstable and solution uncertainty can grow very large.
Can standard deviations be zero?
Yes. The corresponding input is treated as fixed.