Mathematics, Statistics & Geometry

Percentage Error Calculator

Calculate the percentage error between an observed and true value. Understand experimental accuracy with a clear step-by-step breakdown.

Percentage Error
2%
Absolute Error0.2
Relative Error0.02
Calculation StepsPercentage Error = (|Observed - True| / |True|) x 100 = (|10.2 - 10| / |10|) x 100 = (0.1999999999999993 / 10) x 100 = 0.02 x 100 = 2%

Calculated locally in your browser. Fast, secure, and private.

The Complete Guide to Percentage Error

A percentage error calculator measures how far an experimental or observed measurement deviates from a known, accepted, or true value. It is one of the most essential tools in science, engineering, and quality control, enabling you to quantify the accuracy of any measurement.

Percentage Error=∣Vobserved−Vtrue∣∣Vtrue∣×100\begin{aligned} \text{Percentage Error} = \frac{|V_{observed} - V_{true}|}{|V_{true}|} \times 100 \end{aligned}

Where:
VobservedV_{observed}=
The measured or experimental value
VtrueV_{true}=
The accepted, theoretical, or reference value

How to Calculate Percentage Error Step by Step

  1. Find the absolute error: Subtract the true value from the observed value, then take the absolute value: |Observed - True|.
  2. Divide by the absolute true value: This normalises the error relative to the expected measurement.
  3. Multiply by 100: Convert the decimal to a percentage.

The result is always positive because the formula uses absolute value. It does not matter whether you over-measured or under-measured; the percentage error reflects the magnitude of the deviation.

Worked Example: Chemistry Lab

A student measures the boiling point of water and records 101.3 degrees Celsius. The accepted true value is 100 degrees Celsius.

  • Step 1: |101.3 - 100| = 1.3
  • Step 2: 1.3 / |100| = 0.013
  • Step 3: 0.013 x 100 = 1.3%

A 1.3% error is excellent for a school laboratory experiment.

Worked Example: Engineering Measurement

A machined part should measure exactly 50.000 mm. The inspector measures 49.82 mm.

  • Step 1: |49.82 - 50.000| = 0.18
  • Step 2: 0.18 / |50.000| = 0.0036
  • Step 3: 0.0036 x 100 = 0.36%

Understanding the Three Types of Error

  • Absolute Error: The raw numerical difference between observed and true values (e.g. 0.18 mm). Useful for understanding the physical magnitude of the deviation.
  • Relative Error: The absolute error divided by the true value, expressed as a decimal (e.g. 0.0036). Useful in mathematical contexts.
  • Percentage Error: The relative error multiplied by 100 (e.g. 0.36%). The most intuitive and widely used format for reporting experimental accuracy.

Our calculator displays all three so you can choose the most appropriate measure for your context.

Common Sources of Error

  • Systematic errors: Consistent biases in equipment calibration or methodology that push all measurements in the same direction.
  • Random errors: Unpredictable fluctuations caused by environmental conditions, human judgement, or instrument precision limits.
  • Gross errors: Outright mistakes such as misreading a scale, recording data incorrectly, or using damaged equipment.

Understanding the source of your error is just as important as quantifying it.

Real-World Applications

  • Science: Validating experimental results against theoretical predictions in physics, chemistry, and biology.
  • Manufacturing: Ensuring machined parts fall within specified tolerance bands.
  • Finance: Comparing forecasted revenue against actual figures.
  • Medicine: Verifying the accuracy of diagnostic equipment against calibrated standards.

Frequently Asked Questions

The vertical bars represent absolute value, a standard mathematical notation. |x| means 'the positive magnitude of x, ignoring its sign'. For example, |5| = 5 and |-5| = 5. In the percentage error formula, the absolute value ensures the error is always expressed as a positive percentage, regardless of whether the observed value is above or below the true value.

Percentage error measures how far an observed value deviates from a known true value. The true value always serves as the reference. Percentage change, by contrast, measures the shift between any two values over time, where the earlier value is the reference. Percentage error is always positive because it uses absolute value, while percentage change can be positive or negative.

It depends on the field. In chemistry and physics, an error below 5% is generally considered acceptable for school-level experiments. Professional laboratories typically aim for errors below 1%. In engineering and manufacturing, tolerances are often specified in fractions of a percent. Context matters: a 2% error in measuring a room's length is minor, but a 2% error in pharmaceutical dosing could be critical.

Yes, but only when the observed value exactly equals the true value. In practice, this is rare because all measurements involve some degree of uncertainty. An error of exactly 0% would indicate a perfect measurement.

The formula requires dividing by the true value. Since division by zero is undefined in mathematics, the percentage error cannot be calculated when the true value is exactly zero. In such cases, scientists typically report only the absolute error.

Absolute error is the raw difference between the observed and true values (e.g. an error of 0.3 grams). Percentage error expresses that same difference as a proportion of the true value (e.g. 2% error). Percentage error is more useful for comparing accuracy across different scales and units.

Yes. If you measure 25 grams when the true value is 10 grams, the percentage error is |25 - 10| / |10| x 100 = 150%. This simply means your measurement was off by more than the true value itself.

Related Calculators in Mathematics, Statistics & Geometry

Angle Between Vectors Calculator

Try It Now

ANOVA Calculator

Try It Now

Antilog Calculator

Try It Now

Arc Length of Curve Calculator

Try It Now

Area of Annulus Calculator

Try It Now

Area of Circle Calculator

Try It Now