In various scientific and engineering fields, when we make measurements or do calculations, there is usually some uncertainty or margin of error involved. That's how this error propagation calculator can be used to evaluate how these uncertainties or errors in the mathematical calculations can affect the accuracy of the final result.

## How to use our error propagation calculator?

Calculating error propagation becomes effortless when using the error propagation calculator provided by Calculatored.

- First,
**select the mathematical operation**in which you want to calculate

- Enter the values of
**X, ΔX, Y,**and**ΔY**into the designated field

- Hit the
**Calculate**button

- Using the error propagation equation, this calculator will present you with the values of
**Z**and**ΔZ**and the step-by-step solution

## What is error propagation?

Error propagation, also known as **propagation of uncertainty**, is the process where uncertainties in measured values affect the precision of calculated results during mathematical operations.

When mathematical operations such as **addition, subtraction, multiplication, or division** are performed using measured values with associated** uncertainties,** the uncertainties can **propagate** to the final result.

## Error propagation formulas:

When dealing with uncertainties in measured quantities, our error propagation calculator uses the following formulas to estimate errors in different mathematical operations.

### Addition or Subtraction:

\(ΔZ = \sqrt{(ΔX)^2 + (ΔY)^2}\)

### Multiplication or Division:

\(ΔZ = Z \cdot \sqrt{(\dfrac{ΔX}{X})^2 + (\dfrac{ΔY}{Y})^2}\)

### Examples:

**Addition (Z = X + Y):**

\(ΔZ = \sqrt{(ΔX)^2 + (ΔY)^2}\)

**Subtraction (Z = X - Y):**

\(ΔZ = \sqrt{(ΔX)^2 + (ΔY)^2}\)

**Multiplication (Z = X * Y):**

\(ΔZ = Z \cdot \sqrt{(\dfrac{ΔX}{X})^2 + (\dfrac{ΔY}{Y})^2}\)

**Division (Z = X/Y):**

\(ΔZ = Z \cdot \sqrt{(\dfrac{ΔX}{X})^2 + (\dfrac{ΔY}{Y})^2}\)

## How do I calculate error propagation? - Example

Suppose you have two measurements:

**Given values:**

**X**= 10 ± 0.5**Y**= 5 ± 0.2

Now, calculate the error propagation under different situations by using their corresponding formulas:

**Addition:**

**Z**= X + Y

**Z**= 10 + 5 = 15**ΔZ**= √(0.5^2 + 0.2^2) ≈ 0.7**Result:**Z = 15 ± 0.7

**Subtraction:**

**Z**= X - Y**Z**= 10 - 5 = 5**ΔZ**= √(0.5^2 + 0.2^2) ≈ 0.7**Result:**Z = 5 ± 0.7

**Multiplication:**

**Z**= X * Y**Z**= 10 * 5 = 50**ΔZ**= 50 * √((0.5/10)^2 + (0.2/5)^2) ≈ 2.3**Result:**Z = 50 ± 2.3

**Division:**

**Z**= X/Y**Z**= 10 / 5 = 2**ΔZ**= 2 * √((0.5/10)^2 + (0.2/5)^2) ≈ 0.9

**Result:** Z = 2 ± 0.9