Understanding measurement uncertainty

Every measurement taken in an experiment has some degree of uncertainty. This uncertainty arises from various factors, including the limitations of measuring instruments and human error.

Key points about measurement uncertainty:

• Uncertainty is expressed using the ± symbol, indicating a range within which the true value likely lies.
• The maximum difference between your measured value and the true value is called the margin of error.

Types of uncertainty

There are two primary measures of uncertainty:

• Absolute uncertainty - The total uncertainty for a measurement, expressed in the same units as the measurement.
• Percentage uncertainty - The uncertainty expressed as a percentage of the measurement.

Note: Measuring larger values typically reduces the percentage uncertainty.

Combining uncertainties in calculations

When performing calculations with uncertain values, you need to combine their uncertainties. The method depends on the mathematical operation.

Adding or subtracting:

When adding or subtracting values, the uncertainty can be found by adding the absolute uncertainties.

Example:

When investigating the extension of springs during a Hooke's law practical, the following data was recorded:

• original length of spring = 2.0 $\pm$ 0.1 cm
• final length of spring = 6.3 $\pm$ 0.1 cm

Extension = final length - original length = 6.3 - 2.0 = 4.3 cm

Total uncertainty = 0.1 + 0.1 = 0.2 cm

The extension is expressed as 4.3 $\pm$ 0.2 cm

Multiplying or dividing values:

When multiplying or dividing values, the uncertainty can be found by adding the percentage uncertainties.

Example:

When investigating the resistance of a resistor, the following data was recorded:

• V = 5.2 $\pm$ 1% V
• I = 2.5 $\pm$ 2% A

$\text{R = }\frac{V}{I}\text{ = }\frac{5.2}{2.5}\text{ = 2.08 }\mathrm{\Omega }$

Total uncertainty = 1% + 2% = 3%

The resistance is expressed as 2.08 $\pm$ 3% $\mathrm{\Omega }$.

Power rules:

When a variable has a power, the percentage uncertainty must be multiplied by the power.

Example:

Calculate the volume of a ball with a radius of 15 $\pm$ 2% cm.

V = $\frac{4}{3}\pi {\text{ r}}^3\text{ = }\frac{4}{3}\pi {\text{ x 15}}^3{\text{ = 14,137 cm}}^3$

Total uncertainty = percentage uncertainty x power = 2 x 3 = 6%

The volume would be expressed as 14,137 $\pm$ 6% cm^3^.

Worked example - combining uncertainties in addition

Calculate the extension of a wire stretched from 5.1 ± 0.2 cm to 6.5 ± 0.1 cm.

Step 1: Calculate the difference in length

6.5 - 5.1 = 1.4 cm

Step 2: Add the absolute uncertainties

0.2 + 0.1 = 0.3 cm

Step 3: Express the result with its uncertainty

Extension = 1.4 ± 0.3 cm

Worked example - combining uncertainties in multiplication

A current of 3A ± 1% flows for 8 ± 0.25 s.

Calculate the charge and its percentage uncertainty.

Step 1: Calculate charge

Q = I x t = 3 x 8 = 24 C

Step 2: Calculate percentage uncertainty in time

$\%\text{ uncertainty in t = }\frac{\text{absolute uncertainty}}{\text{value}}\times 100\text{ = }\frac{0.25}{8}\times 100\text{ = 3.125 }\%$

Step 3: Add percentage uncertainties

Total uncertainty = 1% + 3.125% = 4.125 %

Step 4: Express the result with its uncertainty

Charge = 24 C ± 4.125 %

Worked example - combining uncertainties with powers

A cylinder of length 15 m ± 5% has a radius of 4 m ± 3%.

Calculate the volume of the cylinder and its percentage uncertainty.

Step 1: Calculate the volume

$\text{V = 2 }\pi {\text{ r}}^2\text{ l}\text{ = 2 }\pi {\text{ x 4}}^2{\text{ x 15 = 1,508 m}}^3$

Step 2: Combine percentage uncertainties

Because the radius is squared we multiply the percentage uncertainty in the radius by 2.

Total uncertainty = (2 x 3) + 5 = 11 %

Step 3: Express the result with its uncertainty

V = 1,508 m${}^3$ $\pm$ 11 %