Percent Error Calculator

A percent error calculator computes the relative difference between an observed (experimental) value and a true (theoretical) value, then expresses that difference as a percentage. Enter an experimental value and a theoretical value, and the calculator instantly returns the percent error, absolute error, and relative error.

Percent error calculations are used across chemistry, physics, engineering, and science to measure how accurate an experiment or measurement is. A small percent error signals that the observed value is close to the true value. A large percent error signals that the measurement needs to be reviewed or repeated.

What Is Percent Error?

Percent error is the relative size of the difference between an experimental value and a theoretical value, expressed as a percentage. It tells you how far off a measured result is from the expected result.

In science, measured values rarely match true values exactly. When measuring the density of a material, the standard acceleration due to gravity, or the boiling point of a liquid, the observed value will differ from the true value due to human error, instrument limitations, or environmental factors. Percent error gives a consistent way to quantify that variation.

A percent error of 1% means the observed and true values are very close. A percent error of 90% means something went significantly wrong with the measurement or experiment.

The Percent Error Formula

The percent error formula calculates the absolute difference between the experimental value and theoretical value, divides it by the theoretical value, and multiplies by 100 to produce a percentage.

Percent Error Formula

Percent Error = |E − T| / T × 100%

Where E = Experimental (observed) value  |  T = Theoretical (true) value

The absolute value signs ensure the result is always positive, because in most scientific contexts only the size of the error matters, not its direction. If directional error matters for your work, see the section on signed (negative) percent error below.

Absolute Error Formula

The absolute error is the raw numerical difference between the experimental value and the theoretical value, expressed in the same units as the measurement.

Absolute Error = |E − T|

For example, if a scale reads 1.2 pounds but the true weight is 1.5 pounds, the absolute error is 0.3 pounds. This tells you the raw size of the discrepancy in original units.

Relative Error Formula

The relative error is the absolute error divided by the theoretical value. It expresses the error as a proportion of the true value rather than in original units, making it more useful than absolute error when comparing measurements of different scales.

Relative Error = |E − T| / T

Using the scale example above: Relative Error = 0.3 / 1.5 = 0.2. Multiplying by 100 gives a percent error of 20%.

How to Calculate Percent Error

To calculate percent error, follow these 4 steps:

1. Subtract the theoretical value from the experimental value.
2. Find the absolute value of the result to ensure it is positive.
3. Divide the absolute error by the theoretical value.
4. Multiply the final result by 100 to convert it into a percentage.

Percent Error Calculation Example 1 — Boiling Point of Water

Percent Error Calculation Example 2 — Speed of Sound

Percent Error Calculation Example 3 — Product Weight Check

Can Percent Error Be Negative? (Signed vs. Absolute Error)

By default, the standard percent error formula uses absolute values, so the result is always a positive number. This is intentional — in most scientific contexts, only the magnitude of the error matters, not its direction.

However, a signed (negative) percent error is possible and meaningful in some contexts. To get a signed percent error, simply remove the absolute value from the formula:

Signed Percent Error = (E − T) / T × 100%

With an experimental value of 7 and a theoretical value of 9:

(7 − 9) / 9 × 100% = −22.22%

A negative percent error means the experimental value is lower than the theoretical value. A positive percent error means the experimental value is higher than the theoretical value. The ideal result in any case is 0%, meaning perfect agreement between the observed and true value.

Precision vs. Accuracy

Percent error measures accuracy — how close a single measured value is to the true value. It is often confused with precision, which is a separate concept.

• Accuracy describes how close a measurement is to the true or accepted value. A low percent error means high accuracy.
• Precision describes how consistently repeated measurements cluster together, regardless of whether they are close to the true value. A precise instrument gives similar readings every time, even if those readings are systematically wrong.

For example, a thermometer that consistently reads 102°C when the true boiling point is 100°C is precise (consistent) but not accurate (off by 2°C each time). Percent error quantifies the accuracy gap. Standard deviation quantifies the precision of a set of repeated measurements.

Systematic Error vs. Random Error

Understanding why percent errors occur helps you reduce them in future experiments. The two primary sources of experimental error are systematic error and random error.

Systematic Error

Systematic errors push the experimental value consistently in one direction — always too high or always too low. Common causes include an uncalibrated instrument, a flawed experimental setup, or a persistent environmental factor such as temperature. Because systematic errors are repeatable and directional, they can often be identified and corrected through instrument calibration and careful experimental design. Systematic errors are reflected in the sign of a signed percent error.

Random Error

Random errors are unpredictable fluctuations that cause measurements to scatter around the true value. They arise from reading a graduated scale slightly differently each time, environmental vibrations, or natural variability in the phenomenon being measured. Random errors cannot be eliminated entirely but are reduced by taking multiple measurements and averaging the results. Standard deviation is the standard tool for quantifying random error in a dataset.

What Is an Acceptable Percent Error?

There is no universal “good” or “bad” percent error — what is acceptable depends entirely on the field and the precision required. The table below provides benchmark ranges across common disciplines.

When assessing your result, always compare against the standards of your specific field. A 5% error that would fail a pharmaceutical audit is perfectly acceptable in a high school titration experiment.

Percent Error with Significant Figures

One of the most common student pain points is correctly rounding a percent error result using significant figures (sig figs). The general rule is that a percent error should be reported to the same number of significant figures as the least precise measurement used in the calculation.

For example, if your experimental value is 14.3 m/s (3 sig figs) and the theoretical value is 343 m/s (3 sig figs), your percent error should also be reported to 3 significant figures. A raw result of 4.08163% would be rounded and reported as 4.08%.

When values are given to only 2 significant figures, report the percent error to 2 significant figures as well — even if your calculator displays more decimal places.

Percent Error vs. Percentage Difference vs. Percentage Change

These three formulas are often confused because they all involve dividing a difference by some reference value. They answer fundamentally different questions.

Use percent error when a theoretical (accepted) value exists. Use percentage difference when you are comparing two experimental values with no established reference. Use percentage change when tracking how a quantity evolved from one point in time to another.

How to Calculate Percent Error in Excel or Google Sheets

The percent error formula translates directly into a spreadsheet. If your experimental value is in cell A2 and your theoretical value is in cell B2, enter the following formula in cell C2:

=ABS(A2-B2)/B2*100

This returns the percent error as a number (e.g., 4.08). To display it with a percent sign, format the cell as a percentage — but note you would then change the formula to =ABS(A2-B2)/B2 and apply the percentage format, which multiplies by 100 automatically.

To calculate the mean of multiple experimental trials before computing the percent error, use =AVERAGE(A2:A10) as your experimental value.

Solving for Experimental and Theoretical Values

Experimental Value Formula

When the percent error (P) and theoretical value (T) are known, you can solve for the experimental value. Because of the absolute value, there are two possible solutions:

E = T × (1 ± P/100)

Both solutions are mathematically valid. Context determines which one applies — for instance, whether the experimental value is expected to be above or below the theoretical value.

Theoretical Value Formula

When the experimental value (E) and percent error (P) are known but the true value is unknown, you can reverse-engineer it:

T = E / (1 ± P/100)

Percent Error, Margin of Error, and Standard Error

Percent error, margin of error, and standard error are three related but distinct concepts in error analysis and statistics. Each measures a different type of deviation.

Percent Error (PE) measures how far a single observed value deviates from a known theoretical value. It requires both an experimental value and a theoretical value. It is the primary tool in scientific experiments, lab accuracy assessments, and quality control comparisons. Formula: PE = |E − T| / T × 100%.

Standard Error (SE) is a statistical term describing how accurately a sample mean represents the true population mean. It is computed from the standard deviation of a dataset divided by the square root of the sample size. Standard error does not require a known theoretical value — it is computed entirely from the observed data.

Margin of Error (MOE) is used in statistical sampling and survey analysis. It is calculated by multiplying the standard error by a z-score corresponding to a chosen confidence level (such as 95%). Margin of error quantifies the uncertainty in a survey or poll result, not the accuracy of a single measurement against a known value.

In summary: percent error compares a measurement to a known value; standard error describes variability within a sample dataset; and margin of error quantifies uncertainty in a population estimate from sample data. In laboratory settings, percent error is the most relevant measure when a theoretical value is known.

Percentage Error (Same Concept, Different Name)

Percentage error is identical to percent error — the two terms are used interchangeably in science, math, and error analysis. Both use the same formula and produce the same result. The choice of term is typically regional or disciplinary preference.

Percentage error appears in three primary contexts: lab experiments in chemistry and physics (where measured results are compared to known theoretical values), quality control (where manufactured products are compared to target specifications), and data validation (where estimated or predicted values are compared to actual recorded values).

For example, if the observed value is 56.891 and the true value is 62.327:

When the true value is unknown, general percentage calculations or standard deviation can help represent measurement spread as an alternative to percent error.

Notes on Percent Error Calculations

The theoretical value in chemistry, physics, and science is the established, ideal value expected as the result of an experiment. Other terms used for it include: accepted value, actual value, expected value, exact value, and true value. The theoretical value always appears in the denominator of the percent error formula.

The experimental value is the observed result of an experiment. Other terms include: measured value, observed value, estimated value, and approximate value. It appears in the numerator of the formula.

The order of values in the numerator does not affect the final result because the absolute value is taken: |E − T| = |T − E|. If a percent error formula found elsewhere uses a different arrangement, this property explains why both produce the same answer.

Summary

A percent error calculator finds the relative difference between an experimental value and a theoretical value and expresses it as a percentage. The percent error formula is:

Percent Error = |Experimental Value − Theoretical Value| / Theoretical Value × 100%

The three key outputs from a percent error calculation are the absolute error, the relative error, and the percent error itself. A small percent error indicates a precise measurement close to the true value. A large percent error signals a need to review the measurement method, check for systematic errors, or repeat the experiment.

Percent error is used throughout chemistry, physics, and engineering to assess the accuracy of experimental results. It is distinct from margin of error and standard error, which are statistical measures applied to sample datasets rather than individual experimental measurements against a known value.