Calculate percent error between experimental and accepted values instantly, with step-by-step solutions, formulas, worked examples and practical science applications.
Subtract the accepted value from your experimental value, take the absolute value of that difference, divide by the absolute accepted value, then multiply by 100. In symbols: percent error = |experimental − accepted| ÷ |accepted| × 100%. If you measured gravity as 9.6 m/s² against the accepted 9.81 m/s², the absolute error is 0.21, the relative error is 0.021407, and the percent error is 2.14%. Lower is better, and 0% means the measurement matched the accepted value exactly.
Enter your measured result and the accepted (true) value. Plain decimals and scientific notation both work — 1.5e-3 means 0.0015 and 2.3e4 means 23000. Any units are fine as long as both values use the same one.
One pair per line as experimental, accepted — handy for a full set of lab readings. Commas, spaces, tabs and semicolons all work as separators.
| Metric | Value |
|---|
Percent error expresses how far a measured result sits from the value it should have been, as a percentage of that accepted value. It answers the question every science student and lab technician eventually asks: was that measurement close enough?
Expressing the gap as a percentage rather than a raw difference is what makes it useful. Being out by 0.21 means nothing until you know what you were measuring. If you were measuring gravitational acceleration and got 9.6 m/s² against the accepted 9.81 m/s², that 0.21 is a percent error of 2.14% — a respectable result for a school pendulum experiment. If you were out by 0.21 while measuring something with an accepted value of 0.5, the same absolute gap becomes a 42% error and something has gone seriously wrong. Percent error scales the error to the size of the thing being measured, which is exactly what lets you compare results across different experiments, units and magnitudes.
By convention it is reported as a positive number, because it describes the size of the discrepancy rather than its direction. Some courses and some fields do prefer the signed form, which keeps a minus sign when the measurement came in low, so this calculator offers both.
It is the standard way of reporting accuracy, and accuracy is what makes a measurement trustworthy.
Three details matter. The vertical bars mean absolute value, so the answer comes out positive whichever way the measurement missed. Dividing by the accepted value, not the experimental one, is what makes the result comparable between experiments — the accepted value is the fixed reference point. And dividing by the absolute accepted value keeps the sign honest when the accepted value itself is negative, such as a temperature of −18 °C.
These three describe the same discrepancy at increasing levels of usefulness, and they are frequently confused.
| Measure | Formula | Units | What it tells you |
|---|---|---|---|
| Absolute error | |experimental − accepted| | Same as the data (g, m/s², °C) | The raw size of the miss, with no context |
| Relative error | |experimental − accepted| ÷ |accepted| | None — it is a ratio | The miss as a fraction of what was being measured |
| Percent error | Relative error × 100 | Percent | The same fraction in the form people actually read |
Percent error and relative error carry identical information — one is simply a hundred times the other. Absolute error is different in kind, and it still has its uses: instrument specifications are often quoted as an absolute tolerance, such as a balance accurate to ±0.01 g.
Use percent error when one of your two numbers is a known, accepted value. Use percent difference when neither is — two independent measurements of the same unknown quantity, for instance. Percent difference divides by the average of the two values, because there is no reason to privilege either one:
The experimental value — also called the measured or observed value — is what your instrument, calculation or observation produced. The accepted value — also called the true, theoretical, actual or expected value — is the figure you are measuring against.
Accepted values come from a few different places: published physical constants (the speed of light, the gas constant, standard gravity), reference data such as the density or melting point of a pure substance, a theoretical prediction derived from an equation, or a certified calibration standard traceable to a national metrology laboratory. In a school lab the accepted value is usually supplied in the prac sheet.
Percent error measures accuracy only, and the distinction is worth being clear about because a result can be excellent on one count and terrible on the other.
| Aspect | Accuracy | Precision |
|---|---|---|
| Question it answers | How close is the result to the true value? | How close are repeated results to each other? |
| Measured by | Percent error | Standard deviation, range, repeatability |
| Needs a true value? | Yes | No |
| Typical cause of failure | Systematic error — miscalibration, wrong method | Random error — noise, inconsistent technique |
| Fix | Recalibrate, correct the method | Repeat and average, improve technique |
A balance reading 2 g heavy on every measurement is precise but inaccurate: the readings agree beautifully with each other and are all wrong by the same amount. Sloppy stopwatch timing produces the opposite — readings scattered widely whose average may still land close to the true value. The first problem shows up in percent error; the second shows up in the standard deviation of your repeats. Good experimental work needs both checks.
Four steps, always in this order.
| Scenario | Experimental | Accepted | Working | Percent error |
|---|---|---|---|---|
| Physics — gravity | 9.6 m/s² | 9.81 m/s² | 0.21 ÷ 9.81 × 100 | 2.14% |
| Chemistry — titration | 0.1023 mol/L | 0.1000 mol/L | 0.0023 ÷ 0.1 × 100 | 2.30% |
| Chemistry — boiling point | 98.6 °C | 100 °C | 1.4 ÷ 100 × 100 | 1.40% |
| Materials — density | 2.65 g/cm³ | 2.70 g/cm³ | 0.05 ÷ 2.70 × 100 | 1.85% |
| Calibration — balance | 99.2 g | 100 g | 0.8 ÷ 100 × 100 | 0.80% |
| Maths — approximating √2 | 1.41 | 1.41421356 | 0.00421 ÷ 1.41421 × 100 | 0.30% |
| Engineering — pressure gauge | 752 kPa | 760 kPa | 8 ÷ 760 × 100 | 1.05% |
| Construction — steel beam length | 6.032 m | 6.000 m | 0.032 ÷ 6.000 × 100 | 0.53% |
| Hydraulics — system pressure | 17.4 MPa | 18.0 MPa | 0.6 ÷ 18.0 × 100 | 3.33% |
In a teaching lab, percent error is the number that closes the loop between theory and practice. A student who measures the molar volume of a gas, the heat of a reaction or the resistance of a wire compares their result with the accepted figure and reports the percent error alongside an explanation of where it came from. Marks generally hinge more on that explanation than on the size of the number — a 6% error with an identified cause is a better piece of work than a suspiciously perfect 0.1% with no discussion.
Beyond the classroom, laboratories use percent error to verify equipment before it is trusted with real samples. A pipette is checked by dispensing water onto a balance and comparing mass with the nominal volume; a spectrophotometer is checked against a standard of known absorbance. When percent error drifts outside the documented tolerance, the instrument goes out of service until it is recalibrated.
Physics is full of quantities with well-established accepted values, which makes it the natural home of percent error. Pendulum and free-fall experiments are compared against standard gravity of 9.80665 m/s²; optics experiments against known refractive indices; electrical experiments against component tolerances printed on the part. Percent error also gives a fast reality check on a derived result — if calculating the speed of sound from your timing data yields a value 40% off, the problem is almost certainly a method or unit error rather than experimental noise.
Analytical chemistry runs on accuracy. Titrations, gravimetric analysis, and instrumental methods are all judged by how close the determined concentration is to a certified reference value. Chemistry also has a close relative of percent error worth distinguishing: percent yield, which is (actual yield ÷ theoretical yield) × 100. Percent yield is normally below 100% and is not an accuracy measure — it reflects how much product a reaction actually delivered. Percent error, by contrast, measures how far a measurement missed a known value. Students frequently mix the two up in write-ups.
Engineering uses percent error to validate models and monitor equipment. A simulated stress, flow rate or heat-transfer coefficient is compared with a physical test result, and the percent error tells the engineer whether the model can be trusted for design decisions. Instrumentation is specified with an accuracy class stated as a percentage of full-scale reading, so a pressure transmitter rated ±0.5% of a 1,000 kPa span may be out by up to 5 kPa anywhere in its range. Quality control applies the same logic to production tolerances, where parts outside a percentage band are rejected.
Percent error tells you that a measurement missed. Identifying why requires knowing the usual suspects.
| Type | Cause | Effect on results | Example |
|---|---|---|---|
| Systematic error | A consistent flaw in equipment or method | Shifts every reading the same way | A balance not zeroed; a thermometer reading 1 °C high |
| Random error | Unpredictable variation between trials | Scatters readings around a centre | Reaction time on a stopwatch; fluctuating readings |
| Instrument resolution | The finest division the device can show | Caps the achievable accuracy | A ruler marked in millimetres cannot resolve 0.1 mm |
| Parallax error | Reading a scale from an angle | Consistent over- or under-reading | Reading a meniscus above eye level |
| Environmental error | Temperature, humidity, air currents, vibration | Drifts readings over time | A hot draught disturbing a sensitive balance |
| Human error | Misreading, transcription slips, arithmetic mistakes | Anything from small to catastrophic | Recording 12.5 as 1.25 |
| Sample or reagent problems | Contamination, impurity, degradation | Biases results in one direction | An old standard solution that has evaporated slightly |
| Mistake | Why it goes wrong | How to avoid it |
|---|---|---|
| Dividing by the experimental value | The accepted value is the reference and belongs on the bottom | Write the formula out before substituting numbers |
| Forgetting the absolute value | Produces a negative percent error where a positive one is expected | Take the absolute value at step 2, or choose the signed mode deliberately |
| Forgetting to multiply by 100 | You have calculated relative error, not percent error | Check the scale: 0.021 is a relative error, 2.1% is a percent error |
| Using percent error with no true value | Percent error needs a genuine reference figure | Use percent difference when both numbers are measurements |
| Confusing percent error with percent yield | They answer different questions | Yield is actual ÷ theoretical; error is |difference| ÷ accepted |
| Mixing units | Comparing grams with kilograms inflates the error enormously | Convert both values to the same unit first |
| Dividing by zero | Percent error is undefined when the accepted value is 0 | Report absolute error instead, or use percent difference |
| Rounding the difference too early | Small differences lose most of their significant figures | Carry full precision until the final percentage |
| Reporting the error without explaining it | The number alone says nothing about the cause | Identify whether the error is systematic or random, and why |
There is no single threshold for a "good" percent error — it depends entirely on the field, the equipment and the quantity. As a rough guide for school and undergraduate work, under 5% is usually considered acceptable and under 1% is very good, while errors above 10% normally point to a method problem rather than ordinary imprecision. Analytical chemistry and metrology work to far tighter limits, often well under 1%. Biological and field measurements tolerate much more, because the systems themselves vary.
| Percent error | Interpretation | What it usually means |
|---|---|---|
| 0% | Perfect match | Verify it is genuine rather than rounding or back-working the answer |
| Under 1% | Excellent | Careful technique with well-calibrated equipment |
| 1–5% | Very good | Normal, well-executed school or undergraduate work |
| 5–10% | Acceptable in many school labs | Identify and discuss the likely source of the error |
| Above 10% | Review method, equipment or calculations | Often a unit slip, systematic fault or arithmetic mistake rather than noise |
Treat those bands as a guide for teaching laboratories only. Analytical chemistry, metrology and precision engineering work to far tighter limits, while biological and field measurements routinely tolerate much more because the systems themselves vary.
|experimental − accepted||experimental − accepted| / |accepted||experimental − accepted| / |accepted| × 100%(experimental − accepted) / |accepted| × 100%|A − B| / |(A + B)/2| × 100%actual / theoretical × 100%× 100percent / 100 × |accepted|undefined<1% very good <5% acceptable >10% check the method
1) |48 − 50| ÷ 50 × 100 = 4% 2) 5 ÷ 100 × 100 = 5% 3) 0% — the measurement matched exactly 4) |9.7 − 9.81| = 0.11 m/s² 5) 0.5 ÷ 2.0 × 100 = 25%
1) 0.09 ÷ 9.81 × 100 = 0.9174% 2) Percent error = 3.25%; absolute error = 0.0325 × 400 = 13 3) 0.0017 ÷ 0.05 × 100 = 3.4%, an underestimate 4) Percent difference, because neither value is accepted: 0.5 ÷ 12.65 × 100 ≈ 3.9526% 5) |−20 − (−18)| ÷ |−18| × 100 = 2 ÷ 18 × 100 ≈ 11.1111%
What is percent error?
Percent error measures how far a measured or experimental value falls from the accepted true value, expressed as a percentage of that accepted value. It is calculated as |experimental − accepted| ÷ |accepted| × 100%. A percent error of 0% means the measurement matched the accepted value exactly, and larger values mean less accurate measurements. Because it is a percentage rather than a raw difference, it can be compared across experiments of completely different scales.
How do you calculate percent error?
Four steps. Subtract the accepted value from the experimental value. Take the absolute value of that difference, giving the absolute error. Divide by the absolute accepted value, giving the relative error. Multiply by 100 for the percentage. For 21.5 measured against an accepted 20: the difference is 1.5, divided by 20 is 0.075, and multiplied by 100 gives a percent error of 7.5%.
What is the percent error formula?
Percent error = |experimental value − accepted value| ÷ |accepted value| × 100%. The vertical bars mean absolute value, so the result is positive regardless of which direction the measurement missed. The signed version, (experimental − accepted) ÷ |accepted| × 100%, keeps a minus sign when the measurement came in below the accepted value.
What is the difference between percent error and absolute error?
Absolute error is the raw size of the discrepancy in the original units — measuring 9.6 m/s² against 9.81 m/s² gives an absolute error of 0.21 m/s². Percent error scales that difference against the accepted value and expresses it as a percentage, in this case 2.14%. Absolute error tells you how much you missed by; percent error tells you whether that miss was large or small relative to what you were measuring.
What is relative error?
Relative error is the absolute error divided by the absolute accepted value, giving a plain ratio with no units. It contains exactly the same information as percent error, which is simply relative error multiplied by 100. A relative error of 0.0325 is a percent error of 3.25%.
Can percent error be negative?
By standard convention, no — the absolute value in the formula keeps it positive, because percent error describes the size of a discrepancy rather than its direction. Some courses and some fields do use a signed version, where a negative result means the measurement came in below the accepted value and a positive result means it came in above. This calculator offers both forms; check which one your subject expects before reporting.
Is a lower percent error better?
Yes. A lower percent error means the measurement sat closer to the accepted value, so it is more accurate. A percent error of exactly zero deserves a second look though — it can indicate an excellent measurement, but it more often signals that the accepted value was worked backwards into the data or that the result was rounded until it happened to match.
How do you calculate experimental error?
"Experimental error" usually means percent error applied to a laboratory measurement: take your experimental result, subtract the accepted value, take the absolute value, divide by the absolute accepted value and multiply by 100. Report the figure alongside an explanation of its likely cause — systematic problems such as an uncalibrated instrument, or random variation such as inconsistent timing.
What is an accepted value?
The accepted value is the figure your measurement is being compared against — also called the true, theoretical, actual or expected value. It comes from published physical constants, reference data for pure substances, a theoretical prediction from an equation, or a certified calibration standard. In school experiments it is normally provided in the practical instructions.
What is an experimental value?
The experimental value is the result you actually obtained — the measured or observed figure produced by your instrument, calculation or observation. It is the number that goes first in the percent error formula, and it is the one that carries all of your experiment's error.
Why is percent error important?
It is the standard way of reporting accuracy, which makes it the basis for grading experiments, calibrating instruments, comparing measurement methods and setting acceptance tolerances in manufacturing and analysis. Because it is a percentage, it allows a 0.21 m/s² miss on gravity and a 0.0023 mol/L miss on a titration to be compared meaningfully.
How do scientists use percent error?
They use it to verify equipment before trusting it with real samples, to validate computer models against physical test results, and to check whether an unexpected result reflects a genuine finding or a measurement fault. Instruments are specified with an accuracy class stated as a percentage, and when routine checks show percent error drifting outside that band, the instrument is recalibrated or withdrawn from service.
What causes measurement errors?
Systematic errors come from a consistent flaw — an unzeroed balance, a thermometer reading high, a method with a built-in bias — and shift every reading the same way. Random errors come from unpredictable variation such as reaction time or fluctuating readings, and scatter results around a centre. Other common causes are limited instrument resolution, parallax from reading a scale at an angle, environmental factors like draughts and temperature drift, contaminated samples, and simple human slips in reading or recording.
What is a good percent error?
It depends on the field and the equipment. For school and undergraduate science, under 5% is usually acceptable and under 1% is very good, while anything above 10% normally points to a method or unit problem rather than ordinary imprecision. Analytical chemistry and metrology routinely work well under 1%, whereas biological and field measurements tolerate far more because the systems themselves vary.
What is the difference between accuracy and precision?
Accuracy is how close a result is to the true value, and it is what percent error measures. Precision is how close repeated results are to each other, and it is measured by standard deviation or range. A balance reading 2 g heavy every time is precise but inaccurate; sloppy stopwatch timing gives scattered readings that may average out accurately but are imprecise. Good experimental work needs both.
What is the difference between percent error and percent difference?
Percent error compares a measurement with a known accepted value and divides by that accepted value. Percent difference compares two measurements when neither is a known true value, and divides by their average: |A − B| ÷ |(A + B)/2| × 100%. If you cannot say which of your two numbers is the reference, percent difference is the correct choice.
What happens if the accepted value is zero?
Percent error is undefined, because the formula would divide by zero. There is no meaningful way to express an error as a percentage of nothing. Report the absolute error instead, or use percent difference if neither value is a known reference. This calculator detects the situation and explains it rather than returning a nonsense figure.
Does this percent error calculator show the steps?
Yes. Enter your experimental and accepted values and the calculator shows the substituted formula plus each step — the difference, the absolute error, the relative error and the final percentage — along with the direction of the error and the equivalent percent difference. It accepts scientific notation such as 1.5e-3, offers absolute or signed results, lets you set decimal precision, and includes a bulk mode that analyses a whole column of readings at once and reports the mean absolute percent error. Results can be copied with one tap.