What do skewness and kurtosis tell me?

Skewness measures the asymmetry of the output distribution and excess kurtosis measures the heaviness of its tails. When both are near zero the distribution is close to normal and interpretations like k=2 covering about 95% hold. Once |skewness| exceeds roughly 0.5 or |excess kurtosis| exceeds 1, normal-based coverage intervals become unreliable and Monte Carlo evaluation is recommended.

What formulas can I enter?

Any expression in x and y (up to two independent, normally distributed inputs): arithmetic operators, powers with ^, parentheses, functions sin/cos/tan/asin/acos/atan/exp/log/sqrt/abs, and constants pi and e. Examples: pi*x^2*y (cylinder volume), x/y (density or ratio), atan(x/y) (angle).

How are the derivatives computed?

Partial derivatives up to 4th order (including cross terms) are computed numerically with 5-point central finite-difference stencils around the mean, building a 4th-order Taylor polynomial. Its mean, variance, skewness, and kurtosis are then evaluated with 12-point Gauss–Hermite quadrature, which is exact for polynomials up to degree 23.

Higher-Order Taylor Error Propagation Simulator — Mean Shift, Skewness & Kurtosis to 4th Order

Q: How does this differ from first-order (GUM) propagation?

Standard GUM uncertainty propagation linearizes the measurement equation (first-order Taylor expansion). This tool extends the expansion to 4th order, quantifying effects that linearization cannot show: the shift of the output mean (2nd order), variance corrections, distribution skewness (3rd order), and kurtosis (4th order). You can see numerically where the first-order approximation breaks down for strongly nonlinear formulas or large relative uncertainties.

Q: How does it relate to the Monte Carlo method (JCGM 101)?

JCGM 101 (GUM Supplement 1) specifies propagating full distributions with the Monte Carlo method. This tool runs a 200,000-sample Monte Carlo calculation in parallel under identical conditions and shows how closely each Taylor order approaches the Monte Carlo reference. A large gap means the Taylor approximation itself is at its limit.

Higher-Order Taylor Error Propagation Simulator

Enter a measurement formula and go beyond first-order GUM propagation: a 4th-order Taylor expansion yields the mean shift, variance correction, skewness, and kurtosis of the output, compared against a Monte Carlo reference (in the spirit of JCGM 101) to judge whether the normal approximation is trustworthy.

Formula & Input Quantities

Presets (common in practice)

Measurement formula f(x, y)

Available: x, y, + − * / ^ ( ), sin cos tan atan exp log sqrt abs, pi, e. Without y the formula is treated as single-variable.

Mean of x, μx

Best estimate of input quantity x

Standard uncertainty u(x)

Standard deviation of N(μx, u²)

Mean of y, μy

Best estimate of input quantity y

Standard uncertainty u(y)

Standard deviation of N(μy, u²)

Monte Carlo samples (reference)

Inputs are assumed independent and normal, N(μ, u²) — the representative case of GUM / JCGM 101.

Results

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f(μ) (nominal)

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Mean shift Δ (2nd+)

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u_c 1st order (GUM)

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u_c 4th-order Taylor

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Skewness γ₁ (3rd+)

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Excess kurtosis γ₂ (4th+)

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u_c Monte Carlo ref.

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Normal approx. validity

Convergence by Order — Mean Shift, u_c, Skewness, Kurtosis

Statistics of the Taylor polynomial at each order; the rightmost MC column is the Monte Carlo reference (JCGM 101). At 1st order Δ, γ₁, and γ₂ are always zero — that IS the normal approximation.

Output Distribution — Monte Carlo vs 1st-Order Normal

Histogram = actual distribution from Monte Carlo. Curve = GUM first-order normal approximation N(f(μ), u₁²). The larger the gap, the more dangerous the linearization.

Theory & Key Formulas

\( y = f(\mu) + \sum_i \frac{\partial f}{\partial x_i}\delta_i + \frac{1}{2}\sum_{i,j} \frac{\partial^2 f}{\partial x_i \partial x_j}\delta_i\delta_j + \cdots \)

Taylor expansion of the measurement equation about the mean (δᵢ = xᵢ − μᵢ). This tool builds all terms up to 4th order, including cross terms, by numerical differentiation.

\( u_c^2 \approx \sum_i \left(\frac{\partial f}{\partial x_i}\right)^2 u_i^2 \qquad \Delta = E[y] - f(\mu) \approx \frac{1}{2}\sum_i \frac{\partial^2 f}{\partial x_i^2} u_i^2 \)

Left: the first-order (linear) GUM propagation law. Right: the mean shift produced by second-order terms — it appears as a systematic offset in nonlinear measurements.

\( \gamma_1 = \frac{E[(y-\bar y)^3]}{u_c^3} \qquad \gamma_2 = \frac{E[(y-\bar y)^4]}{u_c^4} - 3 \)

Skewness γ₁ (driven mainly by 3rd-order terms) and excess kurtosis γ₂ (4th-order terms); both are zero for a normal distribution. Moments of the Taylor polynomial are evaluated exactly with 12-point Gauss–Hermite quadrature.

A Conversation: Why Expand to 4th Order?

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I always use the GUM formula \(u_c^2=\sum (\partial f/\partial x_i)^2 u_i^2\) for my uncertainty budgets. When is that not enough?

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It is fine while the formula is nearly linear and relative uncertainties are small. But that formula is a first-order Taylor expansion — it replaces your measurement equation with its tangent plane. For strongly curved equations it misses three things: the output mean shifts, the spread estimate changes, and the distribution becomes non-normal.

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The mean shifting feels counterintuitive. The inputs scatter symmetrically around μ, yet the output mean moves away from f(μ)?

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The second-order term \(\tfrac12 f''u^2\) is responsible. Take f=x² with x=1±0.1: the output mean is 1.01, not 1. Curvature maps a symmetric input scatter onto an asymmetric output. In the lab this shows up as a systematic offset that does not average away, e.g. with instruments that have curved calibration functions.

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And how should I read the 3rd and 4th orders? The request mentioned skewness and kurtosis.

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Third-order terms generate the asymmetry of the distribution — skewness γ₁. A skewed distribution means a k=2 interval no longer covers "95%". Fourth-order terms set the tail weight — excess kurtosis γ₂. Positive γ₂ means extreme outliers occur more often than a normal distribution predicts. So orders 3 and 4 tell you how far you can trust the interpretation of the expanded uncertainty U = k·u_c.

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If I expand to 4th order, will I always match Monte Carlo?

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No — and that is the most important practical lesson. For f=eˣ with u=0.3, the 4th-order u_c is actually farther from the reference than the 1st-order value, because moments of a truncated Taylor series do not converge monotonically. That is why this tool always displays the JCGM 101 Monte Carlo reference alongside. When Taylor and MC disagree, trust MC.

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What does the normality badge actually check?

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|γ₁| and |γ₂| of the Monte Carlo distribution. As a rule of thumb: |γ₁|≤0.3 and |γ₂|≤0.5 means the normal approximation is practically safe; |γ₁|>0.5 or |γ₂|>1 means interval evaluation should be redone with Monte Carlo (JCGM 101). The thresholds are conventional — tighten them for demanding applications.

What Is Higher-Order Taylor Error Propagation?

For a measurand y = f(x₁, x₂, …), the usual uncertainty evaluation linearizes the measurement equation around the mean (first-order Taylor expansion) and combines sensitivity coefficients with input uncertainties (GUM: JCGM 100). When the equation is strongly nonlinear or relative uncertainties are large, linearization fails to reproduce the mean, variance, and shape of the output distribution. Extending the expansion to 2nd, 3rd, and 4th order successively introduces corrections for the mean shift and variance, the skewness (asymmetry), and the kurtosis (tail weight). This tool quantifies that hierarchy on one screen, decomposing how much each order contributes.

Numerical Method and Assumptions

The entered formula is interpreted by a safe expression parser. Partial derivatives up to 4th order around the mean — including the cross terms f_xy, f_xxy, f_xyy, f_xxyy, f_xxxy, f_xyyy for two variables — are computed with 5-point central finite-difference stencils. For the resulting 4th-order Taylor polynomial, the mean, variance, skewness, and kurtosis under independent normal inputs N(μᵢ, uᵢ²) are evaluated with 12-point Gauss–Hermite quadrature, which is exact for polynomials up to degree 23 — so the only approximation is the Taylor truncation itself. In parallel, a Monte Carlo calculation under identical conditions (Box–Muller, 200k samples by default) provides the reference. Assumptions: (1) inputs are mutually independent, (2) each input is normally distributed, (3) the formula is smooth (4-times differentiable) near the mean.

Where It Helps in Practice

The classic use case is building uncertainty budgets for calibration and inspection. Product-and-power formulas such as the cylinder volume V=πr²h develop a non-negligible 2nd-order mean shift once relative uncertainties exceed a few percent. Ratios R=V/I and reciprocals 1/x skew to the right (γ₁>0) when the denominator uncertainty is large, making k=2 intervals asymmetric. Exponential and logarithmic responses (thermocouple EMF, decay rates, pH) develop both skewness and kurtosis, so normal-based intervals overstate their confidence level. Check the gap against MC here, and switch to a JCGM 101 Monte Carlo evaluation for the risky cases.

Common Misconceptions and Caveats

"Higher order is always more accurate" is a misconception: moments built from a truncated Taylor series do not converge monotonically, and the eˣ preset with u=0.3 reproduces a case where 4th order is temporarily worse than 1st. The higher-order corrections here also assume independent, normally distributed inputs; uniform or correlated inputs require different moment algebra — use the covariance extension of GUM 5.2 or the Monte Carlo method. Finally, formulas with non-smooth points (absolute values, thresholds) break numerical differentiation; make sure the neighborhood of the mean is smooth and watch the error display.

Frequently Asked Questions

How does this differ from first-order (GUM) propagation?

Standard GUM propagation is based on linearizing the measurement equation (first-order Taylor expansion). This tool expands to 4th order and quantifies what linearization cannot show in principle: the mean shift (2nd order), variance corrections, skewness (3rd), and kurtosis (4th) — decomposed by order so you can see which effect matters.

How does it relate to the Monte Carlo method (JCGM 101)?

JCGM 101 propagates full input distributions by Monte Carlo. This tool always runs a parallel Monte Carlo calculation under the same conditions and shows whether the Taylor statistics approach the reference (or fail to). When they disagree, adopt the Monte Carlo result.

How should I use skewness and kurtosis?

As a validity check for interpreting the expanded uncertainty U = k·u_c. If γ₁ and γ₂ are both near zero, "k=2 covers about 95%" holds. Once |γ₁|>0.5 or |γ₂|>1, coverage intervals become asymmetric and heavy-tailed, and Monte Carlo interval evaluation (e.g. the shortest coverage interval of JCGM 101) is recommended.

Can I change the input distributions or add correlation?

The current version supports independent normal inputs. Uniform or triangular distributions and correlated inputs change the higher-order moment algebra and are not supported; evaluate such cases by Monte Carlo (your own computation or JCGM 101-compliant software).

Is the numerical differentiation accurate enough?

Step sizes scale automatically (1% of |μ|), and 5-point stencils are used up to the 4th derivative. The implementation is validated against known solutions: higher-order terms vanish exactly for linear formulas, and for f=x² the mean shift and variance match the analytic values. Near non-smooth points or singularities (zero denominators) the results become unstable — watch the error message.