Error Propagation in Scientific Calculations

Errors in direct measurements carry over to indirect measurements, which are calculated by combining direct measurements using mathematical operations.

Error propagation is a critical concept in any scientific or technical field that involves measurements, as every experimental observation is inherently subject to some degree of uncertainty.

When we use mathematical operations to combine measurements and derive new quantities—referred to as indirect measurements—the errors in the original measurements propagate as well, influencing the final results.

For instance, imagine calculating the area of a rectangle by measuring its length ($x$) and width ($y$) with a ruler. Each measurement has an associated uncertainty, say $\Delta x$ and $\Delta y$. When we compute the area $A = x \cdot y$, how do these uncertainties affect the resulting error $\Delta A$? The answer lies in the rules of error propagation, which explain how initial uncertainties combine during mathematical operations.

Key Principles of Error Propagation
Practical Example
Why Is Error Propagation Important?

Key Principles of Error Propagation

The way errors propagate depends on the type of operation being performed (e.g., addition, multiplication, division) and follows specific mathematical rules.

Addition or Subtraction
When adding or subtracting measured quantities, the absolute errors are summed. For example, if $z = x \pm y$, the resulting error is: $$ \Delta z = \Delta x + \Delta y $$ This reflects the fact that uncertainties accumulate as measurements are combined.
For instance, when calculating the total length of two sticks measured independently, the combined uncertainty accounts for the inaccuracies in both measurements.
Multiplication or Division
For multiplication or division, the propagation of error involves summing relative errors. If $z = x \cdot y$ or $z = \frac{x}{y}$, then: $$ \frac{\Delta z}{z} = \frac{\Delta x}{x} + \frac{\Delta y}{y} $$ Here, the relative error is the ratio of the absolute uncertainty to the measured value.
For example, when calculating speed as $v = \frac{s}{t}$ (distance divided by time), the uncertainties in both $s$ and $t$ contribute to the final uncertainty in $v$.
Multiplication or Division by a Constant
When multiplying a measurement $x$ by a constant $k$, the absolute error scales proportionally: $$ \Delta z = |k| \cdot \Delta x $$ Similarly, dividing a measurement by a constant reduces the absolute error proportionally: $$ \Delta z = \frac{\Delta x}{|k|} $$
Exponentiation
When a measured quantity is raised to a power, such as $z = x^n$, the relative error is scaled by the absolute value of the exponent: $$ \frac{\Delta z}{z} = |n| \cdot \frac{\Delta x}{x} $$
For example, if calculating the volume of a sphere as $V = \frac{4}{3} \pi r^3$, any uncertainty in the radius $r$ will propagate to the volume, amplified by a factor of 3 (the power of $r$).

Practical Example

Let’s consider the following measurements for the surface of a table:

Length: $x = 10.0 \pm 0.1 \, \text{cm}$,
Width: $y = 5.0 \pm 0.2 \, \text{cm}$.

We calculate the area of the rectangle as:

$$ A = x \cdot y = 10.0 \cdot 5.0 = 50.0 \, \text{cm}^2 $$

The relative errors in the measurements are:

$$ \frac{\Delta x}{x} = \frac{0.1}{10.0} = 0.01, \quad \frac{\Delta y}{y} = \frac{0.2}{5.0} = 0.04 $$

Since the area is a product of two measurements, the relative errors are added:

$$ \frac{\Delta A}{A} = 0.01 + 0.04 = 0.05 $$

Converting the relative error into an absolute error:

$$ \Delta A = 0.05 \cdot 50.0 = 2.5 \, \text{cm}^2 $$

Thus, the area with its uncertainty is:

$$ A = 50.0 \pm 2.5 \, \text{cm}^2 $$

Example 2

Now, let’s calculate the sum of two measurements, applying the error propagation rule for addition:

Suppose we have two ropes with the following measurements:

$x = 10.0 \pm 0.1 \, \text{cm}$,
$y = 5.0 \pm 0.2 \, \text{cm}$.

The combined length is given by $z = x + y$, with its uncertainty $\Delta z$ calculated as:

$$ \Delta z = \Delta x + \Delta y $$

Add the measurements:

$$ z = x + y = 10.0 + 5.0 = 15.0 \, \text{cm} $$

And sum the absolute errors:

$$ \Delta z = \Delta x + \Delta y = 0.1 + 0.2 = 0.3 \, \text{cm} $$

The combined length with its uncertainty is:

$$ z = 15.0 \pm 0.3 \, \text{cm} $$

This means the actual length could range from $14.7 \, \text{cm}$ to $15.3 \, \text{cm}$, accounting for the uncertainties.

Example 3

Now let’s consider error propagation in exponentiation. Suppose the radius $r$ of a sphere is measured as:

$$ r = 5.0 \pm 0.1 \, \text{cm} $$

We calculate the volume $V$ and its uncertainty $\Delta V$:

$$ V = \frac{4}{3} \pi r^3 $$

Substituting the radius:

$$ V = \frac{4}{3} \pi (5.0)^3 = \frac{4}{3} \pi \cdot 125.0 \approx 523.6 \, \text{cm}^3 $$

The relative error in $r$ is:

$$ \frac{\Delta r}{r} = \frac{0.1}{5.0} = 0.02 $$

Multiply by the exponent $n = 3$:

$$ \frac{\Delta V}{V} = 3 \cdot 0.02 = 0.06 $$

Convert this to an absolute error:

$$ \Delta V = 0.06 \cdot 523.6 \approx 31.4 \, \text{cm}^3 $$

The volume with its uncertainty is:

$$ V = 523.6 \pm 31.4 \, \text{cm}^3 $$

This demonstrates how a small error in one measurement (radius) can significantly impact a derived value like volume.

Example 4

Let’s calculate the total length of a rope measured as $L = 10.0 \pm 0.2 \, \text{m}$ when multiplied by 3:

$$ L_{\text{total}} = 3 \cdot L = 3 \cdot 10.0 = 30.0 \, \text{m} $$

When multiplying by a constant, the absolute error is also scaled by the same constant:

$$ \Delta L_{\text{total}} = 3 \cdot 0.2 = 0.6 \, \text{m} $$

The total length with its error is:

$$ L_{\text{total}} = 30.0 \pm 0.6 \, \text{m} $$

This means the total length can vary between $30.0 - 0.6 = 29.4 \, \text{m}$ and $30.0 + 0.6 = 30.6 \, \text{m}$.

Why Is Error Propagation Important?

Error propagation helps quantify the reliability of results derived from experimental measurements.

It provides insight into how much confidence we can place in our conclusions and ensures transparency when communicating scientific results.

By understanding and accounting for uncertainties, we can make informed decisions and improve the quality of experimental work.