A derivative density paired with its coefficient-variance anatomyLive current inputs
Math & Statistics
Derivative Distribution Calculator
Propagate independent uncertainty in the cubic, quadratic, and linear coefficients of a cubic model into a normal approximation for its derivative at a selected x-value.
Mean derivative -
Derivative SD -
Central derivative interval -
Probability derivative > 0 -
Cubic variance share -
Quadratic variance share -
Linear variance share -
Dominant uncertainty source -
PROPAGATED SLOPE DISTRIBUTION
A derivative density paired with its coefficient-variance anatomy
The upper panel shows the normal slope approximation and zero threshold; the lower panel decomposes variance rather than confusing coefficient magnitude with uncertainty influence.
UNCERTAINTY CONTRIBUTION TABLE
Each coefficient's sensitivity, standard deviation, and variance contribution
The derivative is linear in a, b, and c at fixed x0, so independent coefficient variances add exactly under the stated model.
| Coefficient | Derivative sensitivity | Coefficient SD | Slope SD contribution | Variance share |
|---|
UNCERTAINTY SETUP
Enter coefficient uncertainty, not raw-data spread
- Use coefficient means from the fitted cubic model.
- Use standard errors or justified coefficient SDs from the same fit.
- Enter x0 in the model's original scale.
- Confirm the independence assumption before interpreting variance shares.
- Choose coverage to describe uncertainty, not to force a desired sign conclusion.
SENSITIVITY CHANGES WITH X
The same coefficient uncertainties create different slope uncertainty at different locations
The sensitivity to a grows with x0 squared, while sensitivity to b grows linearly with x0. Far from the origin, the cubic coefficient can dominate even when its SD is numerically small.
The probability of a positive slope is conditional on the normal and independence assumptions. It is not the probability that the underlying scientific theory is true.
LINEAR UNCERTAINTY PROPAGATION
Differentiate first, then propagate only the coefficients that remain
For f(x)=ax^3+bx^2+cx+d, the derivative at fixed x0 is a linear combination of a, b, and c. The intercept d contributes no derivative variance.
Detailed calculation process and general formulas
M = 3a x0^2 + 2b x0 + cV = (3x0^2 sa)^2 + (2x0 sb)^2 + sc^2S = sqrt(V)CI = M +/- z SPr(f'>0) = Phi(M/S)Symbols, meanings, and units
- M
- mean derivative at x0f-unit/x-unit
- V
- derivative variance(f-unit/x-unit)^2
- S
- derivative standard deviationf-unit/x-unit
- sa,sb,sc
- coefficient standard deviationscoefficient-specific
- Phi
- standard normal cumulative distributionprobability
- z
- normal critical value for coveragedimensionless
MODEL DIAGNOSTICS
Use the decomposition to decide what to estimate better
A wide slope interval can have a specific coefficient source.
Direction evidence
-The zero threshold converts the slope density into positive and negative mass.
Dominant source
-The largest variance share identifies the most influential coefficient uncertainty.
Location sensitivity
-Changing x0 immediately changes the a and b multipliers.
Decision takeaway: Improve the coefficient responsible for variance at the decision-relevant x, not simply the coefficient with the largest raw SD.
ASSUMPTION CHECKS
Before treating the curve as a probability model
- Coefficient estimates came from the same model and scale.
- Covariances are negligible or intentionally omitted.
- The evaluation point is within a defensible model region.
- Normal coefficient approximations are reasonable.
- Structural model uncertainty is documented separately.
Applied decisions
How derivative uncertainty changes a practical conclusion
Growth-rate sign
A cubic trend is evaluated near a suspected turning point.
What the result clarifies: The density shows whether a positive mean is distinguishable from coefficient uncertainty.
Calibration priority
A far-from-origin prediction has a wide slope range.
What the result clarifies: The variance bars reveal why the cubic coefficient dominates at that x-value.
Worked default scenario
Current-input substitution and reconciliation
Method references
References for this calculator's specific method
Scope and limitations
This calculator assumes independent normally distributed coefficient estimates and a fixed x0. Real regression coefficients are often correlated. Omitting covariance can overstate or understate the derivative uncertainty.
Derivative Distribution Calculator | Coefficient-Uncertainty Propagation FAQ
Why does d not appear?
The derivative of the constant intercept is zero.
Are the variance shares based on coefficient size?
No. They use derivative sensitivity multiplied by coefficient uncertainty.
What if coefficient covariances are important?
Use the full covariance matrix and the gradient-vector formula instead of this independent model.
Can the interval cross zero?
Yes. That indicates the entered uncertainty does not isolate one derivative sign at the selected coverage.