Taguchi Loss Function Calculator

Understanding the Taguchi loss function calculator — concept and principle

The Taguchi loss function puts a money value on deviation from a target. Instead of treating every part inside the tolerance as equally good and every part outside as scrap, it says that loss grows smoothly as a characteristic drifts from its ideal value. Enter a measured value, the target, the tolerance half-width Δ and the cost incurred at the tolerance limit, and the calculator gives the loss per unit and the loss coefficient k.

Loss rises with the square of the distance from target, even inside the tolerance.

How it is calculated

L = k × (y − T)², with k = A ÷ Δ²

y: measured value of the characteristic.
T: target (nominal) value.
Δ: distance from target to the tolerance limit, in the same unit as y.
A: cost per unit when the deviation equals Δ (repair, scrap, complaint or warranty cost).

L and A are in the same currency; k is in currency per unit of deviation squared.

The principle

Taguchi argued that a product exactly on target gives the least loss to society, and that any deviation, even inside the specification, costs something through wear, poorer fit or customer dissatisfaction. Near the target, any smooth loss curve looks like a parabola (the first term of a Taylor expansion about a minimum is quadratic), so a squared deviation is the simplest reasonable model. Fixing the curve so that the loss equals A at y = T ± Δ gives k = A/Δ². Over a batch, the average loss becomes k[σ² + (μ − T)²], which shows that reducing variation is as valuable as centring the mean.

Worked example

Inputs: y = 10.3, T = 10, Δ = 0.5, A = 20.
k = 20 ÷ 0.5² = 20 ÷ 0.25 = 80.
Deviation = 10.3 − 10 = 0.3; squared = 0.09.
L = 80 × 0.09 = 7.2 per unit.

Each point is the calculator re-run with only “Measured value” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Measured value	Loss per unit
2.58	4,410.45
5.15	1,881.8
7.72	414.05
10.3	7.2
12.9	661.25
15.4	2,376.2
20.6	8,988.8

How to read the result

This part is within tolerance, yet it already carries 36% of the limit cost (0.3 is 60% of Δ, and 0.6² = 0.36). Halving the deviation to 0.15 cuts the loss to a quarter, 1.8. Multiply the loss per unit by annual volume to judge whether a better process or tighter adjustment is worth its cost.

Common mistakes

Using the full tolerance width instead of the half-width Δ, which makes k four times too small.
Using a nominal-the-best formula for a characteristic where smaller is better (L = k y²) or larger is better (L = k / y²).

Keep in mind: The quadratic loss is an approximation that is most reasonable near the target. It assumes symmetric loss on both sides; if going over costs more than going under, use separate coefficients for each side. Results are estimates for decision support, not accounting figures.

Frequently asked questions

Why is the loss not zero inside the specification?

Because performance and customer satisfaction usually degrade gradually as a part moves away from target, long before it fails inspection.

How do I find the cost at the tolerance limit?

Use the cost of what happens when a part at the limit reaches the customer or is caught: rework, scrap, replacement or warranty handling.

How does this relate to process capability?

Average loss depends on both the spread σ and the offset μ − T, the same quantities behind Cpk and Cpm; Cpm in particular is built on the deviation from target.

Sources: G. Taguchi, S. Chowdhury and Y. Wu, Taguchi's Quality Engineering Handbook, Wiley (2005) · D. C. Montgomery, Introduction to Statistical Quality Control, Wiley

Checked 8 Nov 2026. Standard textbook formulas in SI units, computed in your browser. For learning and estimates — real designs must follow your local codes and be checked by a qualified engineer.

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