Audited 06 Jun 2026·Last updated 27 Jul 2026·5 citations·Tier 1·0 uses

Percent Error Calculator

Calculate percent error between experimental and theoretical values. Used in laboratory science, engineering, and measurement uncertainty analysis.

Percent Error Calculator

Your measured or observed value
Accepted or reference value, must be non-zero
Percent error
100.00
Absolute percent error between experimental and theoretical values

Background.

Percent error quantifies the discrepancy between an experimentally measured value and an accepted theoretical or reference value, expressing that discrepancy as a percentage of the reference. It is one of the most commonly reported metrics in introductory physics and chemistry laboratories, analytical chemistry, materials testing, and engineering validation. The calculation is straightforward, but its proper interpretation requires understanding the difference between percent error and percent difference, the role of measurement uncertainty, and the conditions under which the theoretical value is genuinely known.

The concept of comparing a measured value to a standard value has roots in the development of precision measurement during the Scientific Revolution. When Antoine Lavoisier conducted quantitative chemical experiments in the late eighteenth century, he compared his measured masses of reactants and products to the theoretical predictions of the conservation of mass. The modern formulation of percent error emerged from twentieth-century metrology, where national standards laboratories such as NIST in the United States and NPL in the United Kingdom maintain primary physical standards and characterize the accuracy of transfer standards by comparing them to reference values. The percent error formula is now taught in virtually every physical science curriculum and is a standard output of calibration certificates and inter-laboratory comparison reports.

In experimental practice, percent error serves multiple purposes. It provides a normalized measure of accuracy that is independent of the magnitude of the measured quantity, enabling comparison across different experiments and different units. A percent error of 2 percent in a length measurement and a percent error of 2 percent in a voltage measurement are directly comparable, whereas absolute errors of 2 millimeters and 2 millivolts are not.

Percent error also helps identify systematic bias: consistently positive percent errors across repeated measurements suggest the instrument reads high, while consistently negative errors suggest it reads low. Random errors, by contrast, produce percent errors that fluctuate in sign and magnitude around zero.

What is percent error calculator?

Percent error is the absolute difference between an experimental value and a theoretical or accepted value, divided by the theoretical value and multiplied by 100 percent. The formula is percent error = |experimental − theoretical| / |theoretical| × 100%. The result is a dimensionless percentage that indicates how large the measurement discrepancy is relative to the reference value. A percent error of zero indicates perfect agreement, while larger values indicate greater relative inaccuracy. Percent error is distinct from percent difference, which compares two experimental values without designating either as the reference. Percent error is also distinct from relative standard uncertainty, which propagates error components according to the GUM framework rather than comparing a single measurement to a single reference. The percent error formula assumes the theoretical value is exact; when it has its own uncertainty, the percent error should be accompanied by an uncertainty budget. The formula is undefined when the theoretical value is zero because the denominator vanishes, and it can produce values exceeding 100 percent when the experimental value has the opposite sign from the theoretical value or when the experimental value is more than twice the theoretical value in magnitude.

How to use this calculator.

  1. Enter your experimentally measured value in the Experimental value field.
  2. Enter the accepted theoretical or reference value in the Theoretical value field.
  3. Ensure the theoretical value is not zero, as division by zero is undefined.
  4. Click Calculate to compute the absolute percent error.
  5. Review the result, expressed as a percentage with two decimal places.
  6. Compare the percent error to the acceptable tolerance for your application or laboratory protocol.

The formula.

PE = |E − T| ⁄ |T| × 100
PE · percent error (%)E · experimental valueT · theoretical value

The percent error formula derives from the definition of relative error. Let E denote the experimental value and T denote the theoretical value. The absolute error is |E − T|, which measures the magnitude of the discrepancy in the original units. To make this error comparable across quantities with different scales, it is divided by the magnitude of the theoretical value, yielding the relative error |E − T| / |T|. Multiplying by 100 converts the relative error to a percentage. The use of the absolute value in the numerator ensures that the percent error is always non-negative, reflecting magnitude of discrepancy without regard to direction. The theoretical value appears in the denominator because it represents the accepted standard against which the measurement is judged. In metrology, this standard is typically a certified reference value with a stated uncertainty traceable to the International System of Units. When the experimental value is closer to the theoretical value than the theoretical value's own expanded uncertainty, the percent error is not statistically significant, and the measurement is considered consistent with the reference. The formula does not account for the uncertainty of either value; it is a descriptive statistic, not an inferential test. Dimensional analysis confirms the consistency of the formula. If E and T both have units of meters, the numerator has units of meters, the denominator has units of meters, and the ratio is dimensionless. Multiplying by 100 percent, which is itself dimensionless, yields a pure percentage. The absolute value in the denominator prevents cancellation of signs when T is negative.

A worked example.

Example

A student measures the acceleration due to gravity using a simple pendulum and obtains a value of 9.8 meters per second squared. The accepted standard value at sea level is g = 9.80665 m/s², though the local theoretical value used in the classroom is 9.81 m/s². The calculator computes the absolute difference: |9.8 − 9.81| = 0.01 m/s². The theoretical value has magnitude |9.81| = 9.81. The relative error is 0.01 / 9.81 = 0.001019. Multiplying by 100 gives a percent error of 0.1019 percent, which the calculator rounds to 0.10 percent. This small error indicates that the student's measurement is in good agreement with the theoretical value. The error could arise from several sources: the length of the pendulum may have been measured with a ruler accurate only to the nearest millimeter, the timing of 20 oscillations with a stopwatch may have a reaction-time error of 0.2 seconds, and air resistance was neglected in the theoretical formula T = 2π√(L/g). To reduce the percent error, the student could use a photogate timer, measure the pendulum length more precisely, and perform multiple trials to average out random timing errors.

theoretical Value9.81
experimental Value9.8

Frequently asked questions.

What is the difference between percent error and percent difference?
Percent error compares an experimental value to a known theoretical or accepted value, with the theoretical value in the denominator. Percent difference compares two experimental values of equal standing, using their average in the denominator. Percent error is used when a reference standard exists, such as a physical constant or a certified material property. Percent difference is used when comparing two independent measurements of the same quantity, neither of which is designated as the truth. For example, if two students measure the density of water as 0.998 g/mL and 1.002 g/mL, their percent difference is |0.998 − 1.002| / ((0.998 + 1.002)/2) × 100 = 0.4 percent. If one student compares their 0.998 g/mL to the theoretical 1.000 g/mL at 4 degrees Celsius, the percent error is 0.2 percent.
Can percent error be greater than 100 percent?
Yes. Percent error exceeds 100 percent whenever the absolute difference between the experimental and theoretical values is larger than the magnitude of the theoretical value. For example, if the theoretical value is 10 and the experimental value is 25, the percent error is |25 − 10| / 10 × 100 = 150 percent. If the experimental value has the opposite sign from the theoretical value, the percent error can also exceed 100 percent: with E = −5 and T = 2, the error is |−5 − 2| / 2 × 100 = 350 percent. Large percent errors indicate gross measurement errors, instrument malfunction, or a fundamental misunderstanding of the experimental setup.
Why must the theoretical value be non-zero?
Division by zero is undefined in real arithmetic, and the percent error formula has the theoretical value in the denominator. If the theoretical value were zero, the formula would require dividing the absolute error by zero, which has no finite or meaningful result. Even in the limit, as the theoretical value approaches zero, the percent error grows without bound for any fixed non-zero absolute error, making the percentage scale useless as a measure of accuracy. In physics and chemistry, a theoretical value of zero sometimes arises, such as the ideal voltage across a short circuit or the expected displacement of an object at rest. In these cases, percent error is not the appropriate metric; absolute error or a normalized error relative to a different reference, such as the measurement range, should be used instead.
Does percent error tell me if my measurement is good or bad?
Percent error provides a normalized measure of accuracy, but whether a particular value is acceptable depends on the context and the experimental constraints. In an introductory physics laboratory using meter sticks and stopwatches, a percent error of 5 percent is often considered acceptable. In pharmaceutical manufacturing, a percent error of 0.1 percent in active ingredient dosage may be unacceptable because it could affect patient safety. In fundamental physics experiments testing conservation laws, percent errors below 0.001 percent may be required. The appropriate threshold is determined by the field, the specific application, regulatory standards, and the cost of improving precision.
Should I use the absolute value in the numerator?
Yes, for standard laboratory reports and most engineering applications, the absolute value in the numerator ensures that percent error reports the magnitude of discrepancy without regard to whether the measurement overestimates or underestimates the theoretical value. This convention allows straightforward comparison of accuracy across multiple trials and different experimenters. However, some disciplines prefer signed percent error, defined as (E − T) / |T| × 100, which preserves information about the direction of bias. Signed error is useful for detecting systematic trends, such as an instrument that consistently reads high. This calculator reports absolute percent error; users requiring signed error can compute the signed difference manually and interpret the sign separately.
What if both the experimental and theoretical values have uncertainty?
The standard percent error formula ignores uncertainty in both values, treating the theoretical value as exact. When both values are subject to comparable uncertainty, the percent error alone can be misleading. A more rigorous approach follows the Guide to the Expression of Uncertainty in Measurement (GUM), which propagates standard uncertainties through the percent error formula using partial derivatives. The combined standard uncertainty of the percent error depends on the relative uncertainties of both the experimental and theoretical values. If the experimental value is within one expanded uncertainty of the theoretical value, the measurement is consistent with the reference regardless of the percent error magnitude.
Can I use percent error for categorical or qualitative data?
No. Percent error requires numerical experimental and theoretical values that can be subtracted and divided. It is not defined for categorical variables, ordinal scales, or qualitative descriptions. For binary classifications, accuracy, precision, recall, and F1 score are the appropriate metrics. For ordinal data, rank correlation coefficients such as Kendall's tau or Spearman's rho measure agreement. For count data compared to expected frequencies, the chi-square statistic or standardized residuals are used. Applying percent error to non-numeric data would be mathematically meaningless.
How do I reduce percent error in an experiment?
Reducing percent error requires identifying and controlling the sources of measurement variation. Systematic errors, which bias results in one direction, can be reduced by calibrating instruments against traceable standards, applying correction factors, and improving experimental technique. Random errors, which cause scatter around the true value, can be reduced by increasing the number of repeated measurements and computing the mean, because the standard error of the mean decreases as 1/√n. Environmental factors such as temperature, humidity, and vibration should be controlled. Parallax errors in reading analog scales can be eliminated by using digital instruments or viewing scales perpendicularly.
Is percent error the same as relative standard deviation?
No. Percent error compares a single measurement or mean to an external theoretical value. Relative standard deviation (RSD) is the standard deviation of a set of repeated measurements divided by their mean, expressed as a percentage. RSD quantifies precision, which is the reproducibility of measurements under the same conditions. Percent error quantifies accuracy, which is the closeness of a measurement to the true value. A measurement can be precise (low RSD) but inaccurate (high percent error) if the instrument is miscalibrated. Conversely, a measurement can be accurate (low percent error) but imprecise (high RSD) if the mean of scattered measurements happens to fall near the true value.
Can percent error be negative?
Not in the standard definition used by this calculator, because the absolute value in the numerator ensures a non-negative result. However, if you compute signed percent error as (E − T) / |T| × 100, the result can be negative when the experimental value is less than the theoretical value. For example, if E = 8 and T = 10, the signed percent error is −20 percent. The sign indicates the direction of deviation but does not change the magnitude. Some textbooks and laboratory manuals use signed percent error to help students identify whether their measurements are systematically high or low. This calculator adheres to the absolute value convention because it is the dominant standard in physics, chemistry, and engineering metrology.

References& sources.

  1. [1]Taylor, B.N. and Kuyatt, C.E. (1994). Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results. NIST Technical Note 1297.
  2. [2]ISO/IEC Guide 98-3:2008. Uncertainty of Measurement — Part 3: Guide to the Expression of Uncertainty in Measurement (GUM).
  3. [3]Bevington, P.R. and Robinson, D.K. (2003). Data Reduction and Error Analysis for the Physical Sciences, 3rd ed. McGraw-Hill.
  4. [4]Ku, H.H. (1969). "Precision Measurement and Calibration." NIST Special Publication 300, Vol. 1.
  5. [5]Box, G.E.P., Hunter, W.G., and Hunter, J.S. (1978). Statistics for Experimenters. Wiley.
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