# Percent error: how far off is your measurement, and how it differs from percent change

> You measured gravity as 9.6 m/s² and the textbook says 9.81. How wrong are you, in percent? This guide shows the one-line formula, three worked examples, how it differs from percent difference and percent change, and what size of error is acceptable.

Published 2026-10-09 by Toolcloak. 3 minute read. Original: https://toolcloak.pages.dev/articles/percent-error-how-far-off-is-your-measurement-and-how-it-differs-from-percent-change
Related free tool: [Percent Error Calculator](https://toolcloak.pages.dev/tools/percent-error-calculator)

**Percent error** tells you how far a measured or estimated value is from the true (accepted) value, as a percentage of that true value. It turns a gap like '0.21 off' into something you can compare across experiments: 2% off is easy to judge, whatever the units.

> **In short**
> - **Percent error = |measured − true| ÷ |true| × 100.** The bars mean you drop any minus sign.
> - Divide by the **true** value, not the measured one. Swapping them gives a different answer.
> - It needs a known true value. If you are comparing two measurements with no reference, use **percent difference** instead.
> - Low percent error means accurate. Repeating the same wrong number is precise but not accurate.

## The formula in three steps

**Measured g = 9.6 m/s², accepted g = 9.81 m/s²**

1. **Find the gap.** 9.6 − 9.81 = −0.21. Drop the sign: 0.21.
2. **Divide by the true value.** 0.21 ÷ 9.81 = 0.0214.
3. **Multiply by 100.** 0.0214 × 100 = **2.14%**.

*The result says your measurement was about 2.14% below the accepted value. Some teachers keep the sign to show whether you were over or under; the calculator can show both.*

Try it: [Percent Error Calculator](https://toolcloak.pages.dev/tools/percent-error-calculator). Enter the measured and true values to get percent error, with the steps shown.

## Eight examples

| Measured | True value | Gap | Percent error |
| --- | --- | --- | --- |
| 9.6 | 9.81 | 0.21 | 2.14% |
| 101.2 | 100 | 1.2 | 1.20% |
| 48 | 50 | 2 | 4.00% |
| 0.52 | 0.5 | 0.02 | 4.00% |
| 298 | 299.792 | 1.792 | 0.60% |
| 3.14 | 3.14159 | 0.00159 | 0.05% |
| 72 | 70 | 2 | 2.86% |
| 1.9 | 2.0 | 0.1 | 5.00% |

*Look at rows 3 and 4. A gap of 2 and a gap of 0.02 both give 4%, because each is 4% of its own true value. That is the point of using a percentage.*

**Percent error in each example (%)**

|  | Value |
| --- | --- |
| g | 2.14% |
| 100 | 1.2% |
| 50 | 4% |
| 0.5 | 4% |
| c | 0.6% |
| π | 0.05% |
| 70 | 2.86% |
| 2.0 | 5% |

*Even 3.14 for pi, which looks rough, is only 0.05% off the real value.*

## Percent error, percent difference and percent change

Three formulas look alike and are used for different jobs. The difference is what you divide by.

| Name | When to use it | Divide by | Example |
| --- | --- | --- | --- |
| Percent error | Compare a measurement with a known true value | The true value | 9.6 vs 9.81 → 2.14% |
| Percent difference | Compare two measurements, neither is 'right' | Their average | 9.6 vs 9.81 → 2.16% |
| Percent change | How much something grew or shrank over time | The old value | 50 to 60 → +20% |

*Percent error and percent difference are close when the numbers are close, but they are not the same thing.*

**Same two numbers, 50 and 60, three different answers**

|  | Percent |
| --- | --- |
| Percent change, 50 to 60 | 20% |
| Percent error, measured 50, true 60 | 16.7% |
| Percent difference, 50 and 60 | 18.2% |

*Percent change divides by 50, error by 60 and difference by the average, 55. Always say which one you mean.*

## Why the direction matters

**Dividing by the wrong number**

| Divide by the true value | Divide by the measured value |
| --- | --- |
| 4.00% | 4.17% |
| Measured 48, true 50 | Same gap 2 ÷ 48 = 0.0417 |
| Gap 2 ÷ 50 = 0.04 | Gives a slightly different, wrong answer |
| This is the correct percent error | Small here, bigger when the error is large |

*The true value is the fixed reference, so it always goes at the bottom of the fraction.*

## How big an error is acceptable?

There is no universal limit. It depends on the instrument and the purpose. A school pendulum experiment with a stopwatch often lands in the 1% to 5% range. A high-precision lab measurement aims for a tiny fraction of 1%.

**A rough guide for a school lab**

| Zone | From | To |
| --- | --- | --- |
| Excellent | 0 | 2 |
| Good | 2 | 5 |
| Check method | 5 | 10 |
| Likely mistake | 10 | 15 |

*Bands vary by teacher and by experiment. Treat them as a conversation starter, not a grade.*

## Accuracy against precision

**Percent error measures accuracy, not precision**

| Box | What it means |
| --- | --- |
| Accurate and precise | Readings cluster together right on the true value. Percent error is small. |
| Precise, not accurate | Readings agree with each other but sit away from the truth, for example a mis-zeroed scale. Percent error is large. |
| Accurate, not precise | Readings scatter around the true value. The average is close; each reading is not. |
| Neither | Scattered and off-target. Check method, tools and recording. |

*Taking several readings and averaging them reduces random scatter. It does not remove a built-in bias.*

## Common sources of error

1. **Reading the instrument,** such as looking at a ruler from an angle.
2. **Zero error,** where a scale or timer does not start at zero.
3. **Rounding too early,** which adds error to every later step.
4. **Reaction time** with hand-held stopwatches.
5. **Wrong true value,** for example using 9.8 for gravity in a place where 9.78 is more accurate.

> **When the true value is zero:** Percent error cannot be calculated when the accepted value is 0, because you would divide by zero. Report the absolute error (the plain gap) instead.

## Frequently asked questions

### What is the percent error formula?

Percent error = |measured value − true value| ÷ |true value| × 100. It expresses the gap as a percentage of the accepted value.

### Can percent error be negative?

With the absolute value bars it is always positive. Some courses omit the bars to show direction: negative means the measurement was below the true value.

### What is the difference between percent error and percent difference?

Percent error compares a measurement with a known true value and divides by that value. Percent difference compares two measurements and divides by their average.

### What is a good percent error?

It depends on the experiment. For many school labs under 5 percent is good and under 2 percent is excellent, while precision science aims for far smaller values.

### Why can't I calculate percent error when the true value is zero?

Because the formula divides by the true value. Dividing by zero is undefined, so report the absolute error instead.

### Is percent error the same as percent change?

No. Percent change compares an old and a new value over time and divides by the old value. Percent error compares a measurement with a true reference value.