Enter the measured value (the value you actually got in an experiment or measurement) and the true value (the value accepted as correct), and press "Calculate". The percent error, error, absolute error and relative error are calculated together.
The percent error is shown with a sign (plus if the measured value is larger than the true value, minus if it is smaller). The true value cannot be 0.
Result and graph
Enter the measured value and the true value in the fields on the left and press "Calculate". The result and a graph will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter just two values, the measured value and the true value, and get all four results at once - percent error, error, absolute error and relative error
- Use it as is to check the accuracy of a science experiment or measurement, such as "I measured the boiling point of water as 98.6 °C. How many percent off is that from the true value of 100 °C?"
- The percent error is shown with a sign. A plus shows at a glance that you measured higher than the true value, and a minus that you measured lower
- Along with the result, a graph shows where the gap (the error) between the measured value and the true value lies
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
When the true value is 0, the percent error and relative error cannot be calculated (it would mean dividing by 0). In that case, compare using the absolute error (the size of the gap itself).
What is this calculation used for?
Judging the accuracy of a measurement in a lab report (school)
You measure the boiling point of water as 98.6 °C. Compared with the true value of 100 °C, the percent error is \((98.6 - 100) \div 100 \times 100 = -1.4\%\) (1.4% lower).
In a middle or high school lab report, going beyond the raw result to discuss how many percent you were off and why makes a big difference to your grade. Percent error is the standard lab report calculation that starts that discussion.
Dimension checks and quality control (manufacturing)
A part is designed to be 2.000 in long and measures 1.996 in. With the design value as the reference, the percent error is \((1.996 - 2.000) \div 2.000 \times 100 = -0.2\%\).
Factories inspect parts against a tolerance, such as "pass if within some percent (or some thousandths of an inch) of the design value". Tracking errors as percents lets you compare the precision of parts of different sizes on the same scale (in this example the design value takes the place of the true value).
Checking the accuracy of a sales or demand forecast (work and business)
If you forecast "we will sell 110 units next month" and actually sell 100, the forecast's percent error against the actual result is \((110 - 100) \div 100 \times 100 = +10\%\) (10% too high).
In retail and manufacturing, teams check the percent error between forecast and actual every month to improve how they forecast. A run of pluses tends to leave extra stock, and a run of minuses tends to lead to running out. Because the percent error has a sign, it even shows the habit behind your estimates.
Checking how accurate a scale or measuring tool is (home and lab prep)
If you put a 100 g calibration weight on a kitchen scale and it reads 99 g, the percent error is \((99 - 100) \div 100 \times 100 = -1\%\).
Measuring something whose correct value you know tells you how far you can trust the tool. Comparing your result with the tool's stated accuracy (for example, ±1%) is a useful check for lab prep, cooking and health tracking.
Testing your mental math and estimates (estimation practice)
You estimate the items in your cart at "about $15" in your head, and the actual total at checkout is $14.80. The percent error is \((15 - 14.80) \div 14.80 \times 100 \approx +1.35\%\).
In estimation and Fermi problem practice, you measure your skill by how many percent you were off. Once your percent error stays within a few percent, that sense for numbers pays off in many places, from work estimates to managing a household budget.
Formulas and figures
Formula for the percent error (%)
Figure
Standard notation
(the usual math form)
\(E\)
\(=\)
\((\)
\(A\)
\(-\)
\(T\)
\()\)
\(\div\)
\(T\)
\(\times\)
\(100\)
In words
(symbols replaced with words)
⑤
\(E\): percent error (%)
\(=\)
\((\)
①
\(A\): measured value
\(-\)
②
\(T\): true value
\()\)
\(\div\)
③
\(T\): true value
\(\times\)
④
\(100\): per hundred
The formula in words
①
From the
\(A\): measured value
②
subtract the
\(T\): true value
to get the error
③
divide that by the
\(T\): true value
to get it as a rate
④
multiply it by
\(100\): per hundred
to turn it into a percent
⑤
and you get the
\(E\): percent error (%)
Quick example
A thermometer shows the boiling point of water as 98.6 °C. The percent error compared with the true value of 100 °C is
\(E\): percent error (%)
\(=\)
\((\)
measured value (98.6 °C)
\(-\)
true value (100 °C)
\()\)
\(\div\)
true value (100 °C)
\(\times\)
per hundred (100)
\(98.6 - 100 = -1.4\)
\(-1.4 \div 100 = -0.014\)
\(-0.014 \times 100 = -1.4\ \ (-1.4\%)\)
Key idea
The sign of the answer shows the direction of the gap. A minus says you measured lower than the true value (\(-1.4\)% is 1.4% lower), and a plus says you measured higher. The closer the percent error is to 0, the more accurate the measurement.
Always divide by the true value. The same gap of 1.4 is 1.4% of a true value of 100 but 14% of a true value of 10, because the percent error shows how far off you are compared with the true value.
Many US science textbooks define percent error with absolute value bars, \(|A - T| \div T \times 100\), so that it is always positive and shows only the size of the gap (1.4% in this example). The calculator on this page keeps the sign so you can also see the direction. Drop the sign if your class uses the positive version.
Formula for the absolute error
Figure
Standard notation
(the usual math form)
\(D\)
\(=\)
\(|\)
\(A\)
\(-\)
\(T\)
\(|\)
In words
(symbols replaced with words)
④
\(D\): absolute error
\(=\)
③
\(|\)
①
\(A\): measured value
\(-\)
②
\(T\): true value
\(|\)
The formula in words
①
From the
\(A\): measured value
②
subtract the
\(T\): true value
to get the error
③
remove any minus sign with the absolute value bars | | so only the size of the gap is left
④
and you get the
\(D\): absolute error
Quick example
The absolute error between the measured boiling point of 98.6 °C and the true value of 100 °C is
\(D\): absolute error
\(=\)
\(|\)
measured value (98.6 °C)
\(-\)
true value (100 °C)
\(|\)
\(98.6 - 100 = -1.4\)
\(| -1.4 | = 1.4\)
Key idea
The absolute error is how far off you actually were, in the real units such as °C or grams. The direction does not matter, so the absolute value removes the minus sign.
The absolute error alone cannot tell you whether a measurement is good. The same absolute error of 1.4 is a small gap for a true value of 100 but a large gap for a true value of 2. That is why you judge it with the relative error (next) or the percent error, which compare the gap with the true value.
Formula for the relative error
Figure
Standard notation
(the usual math form)
\(R\)
\(=\)
\(D\)
\(\div\)
\(|T|\)
In words
(symbols replaced with words)
③
\(R\): relative error
\(=\)
①
\(D\): absolute error
\(\div\)
②
\(|T|\): size of the true value
The formula in words
①
Take the
\(D\): absolute error
②
divide it by the
\(|T|\): size of the true value
③
and you get the
\(R\): relative error
Quick example
From the absolute error of 1.4 °C and the true boiling point of 100 °C, the relative error is
\(R\): relative error
\(=\)
absolute error (1.4 °C)
\(\div\)
size of the true value (100 °C)
\(1.4 \div 100 = 0.014\)
Key idea
The relative error shows how far off you were compared with the true value, as a decimal rate. Multiply it by 100 and you get a percent that matches the size of the percent error (the percent error without its sign): \(0.014 \times 100 = 1.4\)%.
We divide by \(|T|\) (the absolute value of the true value) so that we always divide by a size, even when the true value is negative. If the true value is positive, it is the same as dividing by \(T\).
Look at the units: the absolute error has units such as °C or grams, but the relative error is a rate with no units. Only the unitless relative error and percent error let you compare the accuracy of measurements in different units (for example, a temperature measurement and a weight measurement).
Error calculations start from "error = measured value − true value". Divide the error by the true value and multiply by 100 to get the signed percent error (%), where the sign shows the direction of the gap (plus = measured higher, minus = measured lower). To look at only the size of the gap, use the absolute error (|measured value − true value|) and the relative error (absolute error ÷ |true value|).
Symbols and terms
Symbols
| \(A\) | A | The measured value, the value you actually got in an experiment or measurement. It is the 98.6 °C in "I measured the boiling point of water as 98.6 °C". |
| \(T\) | T | The true value, the value accepted as correct, such as a theoretical value or a value from a reference book. It is the 100 °C in "water boils at 100 °C at 1 atm". |
| \(A - T\) | A minus T | The error. It shows how far the measured value is from the true value, with a sign and in the real units. A minus shows that you measured lower than the true value. |
| \(E\) | E | The percent error: the error divided by the true value, multiplied by 100 and written as a percent. On this page it has a sign, and \(-1.4\) says "1.4% lower than the true value". |
| \(D\) | D | The absolute error, the absolute value of the error (|measured value − true value|). It shows only the size of the gap, in the real units, whatever its direction. |
| \(R\) | R | The relative error, the absolute error divided by the size of the true value. It is a rate with no units. Multiply it by 100 to get the size of the percent error (0.014 → 1.4%). |
| \(|\ |\) | absolute value bars | The absolute value sign. It gives the size of the number inside with its sign removed. Example - \(|-1.4| = 1.4\) and \(|1.4| = 1.4\). |
| \(\%\) | percent | The percent sign. It shows how many out of 100. The word comes from the Latin per centum, "for each hundred". |
Terms
| error | The gap between the measured value and the true value. In science it is normally defined as "error = measured value − true value". No instrument, however precise, can bring the error all the way to 0, so estimating the size of the error is an important part of measuring. |
| true value (accepted value) | The value a quantity really has. In real measurements the true value itself is often unknown, so a theoretical value, a value from an official reference or a value from a more precise measurement is used in its place to judge the error. US textbooks often call it the accepted value or the theoretical value. |
| measured value (observed value) | The value you actually got in an experiment, observation or measurement. It is also called the observed value or the experimental value. |
| percent error | How many percent of the true value the error is. It is also called the percentage error. Written with a sign it also shows the direction of the gap; written as an absolute value it shows only the size. |
| absolute error | The absolute value of the error. It shows the size of the gap in the real units such as °C, grams or inches. Keep in mind that you cannot judge a measurement without comparing this with the size of the true value. |
| relative error | The absolute error divided by the size of the true value. It has no units, so it lets you compare the accuracy of different kinds of measurements (such as temperature and weight). Multiply it by 100 to show it as a percent (the size of the percent error). |
| absolute value | The distance from 0 on the number line. For a negative number, it is the number with the minus sign removed (example - \(|-1.4| = 1.4\)). It is taught in Grade 6 in US schools. |
| percent | A rate written with the base counted as 100. The sign is %. Multiply a decimal rate by 100 to get the percent (0.014 → 1.4%). |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
If you get stuck, going back over these topics is the quickest way forward.
| Rates and percents (Grade 6) |
|
| Positive and negative numbers (Grades 6–7) |
|
| Absolute value (Grade 6) |
|
| Multiplying and dividing decimals (Grades 5–6) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the percent error (measured boiling point 98.6 °C, true value 100 °C)
| Measured value | 98.6 |
| True value | 100 |
| Percent error (%) | =(B1-B2)/B2*100 |
Table to find the absolute error (same example)
| Measured value | 98.6 |
| True value | 100 |
| Absolute error | =ABS(B1-B2) |
Table to find the relative error (same example)
| Measured value | 98.6 |
| True value | 100 |
| Relative error | =ABS(B1-B2)/ABS(B2) |
After pasting, B1 and B2 are your inputs and B3 is calculated automatically.
"B1" and "B2" in a formula tell Excel to use the number in that cell. "*" is multiplication, "/" is division and "ABS( )" is the absolute value function (it removes a minus sign).
For example, B3 shows -1.4 in the first table (1.4% lower than the true value), 1.4 in the second and 0.014 in the third. Just replace B1 and B2 with your own numbers.
"B1" and "B2" in a formula tell Excel to use the number in that cell. "*" is multiplication, "/" is division and "ABS( )" is the absolute value function (it removes a minus sign).
For example, B3 shows -1.4 in the first table (1.4% lower than the true value), 1.4 in the second and 0.014 in the third. Just replace B1 and B2 with your own numbers.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the percent error (measured boiling point 98.6 °C, true value 100 °C)
| Measured value | 98.6 |
| True value | 100 |
| Percent error (%) | =(B1-B2)/B2*100 |
Table to find the absolute error (same example)
| Measured value | 98.6 |
| True value | 100 |
| Absolute error | =ABS(B1-B2) |
Table to find the relative error (same example)
| Measured value | 98.6 |
| True value | 100 |
| Relative error | =ABS(B1-B2)/ABS(B2) |
The same formulas as in Excel (including the ABS function) work as is. Copy the whole table, paste it into cell A1, and replace B1 and B2 with your own numbers.
How to calculate it in Python
# Find the percent error, absolute error and relative error (length of a part: measured 48 mm, true 50 mm)
observed = 48 # measured value
true_value = 50 # true value
error = observed - true_value # error (with sign)
percent_error = error / true_value * 100 # percent error (%, with sign)
absolute_error = abs(error) # absolute error
relative_error = abs(error) / abs(true_value) # relative error
print(f"Error = {error}")
print(f"Percent error = {percent_error}%")
print(f"Absolute error = {absolute_error}")
print(f"Relative error = {relative_error}")
Runs with the standard library only. abs( ) is the absolute value function (it removes a minus sign). Change the two numbers, observed (the measured value) and true_value (the true value), and run it. This example prints an error of -2, a percent error of -4.0%, an absolute error of 2 and a relative error of 0.04.
How to write it in LaTeX and other math languages (copy and paste)
Formula for the percent error (%)
E = (A − T) ÷ T × 100
E = \frac{A - T}{T} \times 100
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>E</mi>
<mo>=</mo>
<mfrac>
<mrow><mi>A</mi><mo>−</mo><mi>T</mi></mrow>
<mi>T</mi>
</mfrac>
<mo>×</mo>
<mn>100</mn>
</mrow>
</math>
E = (A - T)/T xx 100
(observed - truevalue)/truevalue*100
percenterror := (observed - truevalue)/truevalue*100;
percent_error = (observed - truevalue)/truevalue*100;
E = (A - T)/T × 100
Formula for the absolute error
D = |A − T|
D = \left| A - T \right|
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>D</mi>
<mo>=</mo>
<mrow>
<mo>|</mo>
<mi>A</mi>
<mo>−</mo>
<mi>T</mi>
<mo>|</mo>
</mrow>
</mrow>
</math>
D = |A - T|
Abs[observed - truevalue]
abserror := abs(observed - truevalue);
abs_error = abs(observed - truevalue);
D = |A - T|
Formula for the relative error
R = |A − T| ÷ |T|
R = \frac{\left| A - T \right|}{\left| T \right|}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>R</mi>
<mo>=</mo>
<mfrac>
<mrow>
<mo>|</mo>
<mi>A</mi>
<mo>−</mo>
<mi>T</mi>
<mo>|</mo>
</mrow>
<mrow>
<mo>|</mo>
<mi>T</mi>
<mo>|</mo>
</mrow>
</mfrac>
</mrow>
</math>
R = |A - T|/|T|
Abs[observed - truevalue]/Abs[truevalue]
relerror := abs(observed - truevalue)/abs(truevalue);
rel_error = abs(observed - truevalue)/abs(truevalue);
R = |A - T|/|T|
How to have ChatGPT do the calculation
You are a calculation assistant for percent error. Do the following calculations by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). Find the percent error with its sign ((measured value − true value) ÷ true value × 100). 1. The percent error when the measured value is 98.6 and the true value is 100 2. The percent error, absolute error and relative error when the measured value is 10 and the true value is 11 3. The percent error when the measured value is 123.456 and the true value is 100 Show the formulas you used and the numbers from the execution result.
How to Use
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1Enter your numbersType the numbers you want to calculate with into the input fields
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2CalculatePress the "Calculate" button
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3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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