Error Propagation Calculator

Enter the values X and Y and their relative changes in the calculator & it will calculate the error of propagation.

Operator:

\( Z = X + Y\)

\( ΔZ = \sqrt{(ΔX)^2 + (ΔY)^2}\)

ΔX:

ΔY:

Related

The Error Propagation Calculator estimates the uncertainty in the final outcome of physical quantities. It combines measurement errors from variables to provide a more reliable estimate.

Standard Error Propagation

Propagation of error (or uncertainty) shows how uncertainties in individual variables affect the result of a function. This calculator computes the final uncertainty based on errors in measured quantities like X, Y, and Z.

Error Propagation Formulas

Addition and Subtraction:

Z = X + Y or Z = X - Y
\(\Delta Z = \sqrt{(\Delta X)^2 + (\Delta Y)^2}\)

Multiplication and Division:

Z = X * Y or Z = X / Y
\(\Delta Z = Z \cdot \sqrt{\left(\dfrac{\Delta X}{X}\right)^2 + \left(\dfrac{\Delta Y}{Y}\right)^2}\)

Practical Example

Suppose we have two rods:

First rod: X = 850 cm, ΔX = 50 cm
Second rod: Y = 850 cm, ΔY = 30 cm

We want to calculate the propagated error for their combined length.

Step 1: Calculate sum

Z = X + Y = 850 + 850 = 1700 cm

Step 2: Calculate propagated uncertainty

\(\Delta Z = \sqrt{(50)^2 + (30)^2} = \sqrt{3400} \approx 58.31 \text{ cm}\)

Result: Z ± ΔZ = 1700 ± 58.31 cm

Using the Error Propagation Calculator

Input:

Select the operation (addition, subtraction, multiplication, division)
Enter values and uncertainties: X, ΔX, Y, ΔY
Click "Calculate"

Output:

Calculated value Z and propagated uncertainty ΔZ
Step-by-step explanation of how ΔZ was computed

FAQs

What is Measurement Error?

Measurement error is the difference between a measured quantity and its true value. The propagated standard deviation reflects this uncertainty accurately.