Percent Error Calculator
Percentage Error
Percentage error is a measurement of the discrepancy between an observed (measured) and a true (expected, accepted, known etc.) value. It is typically used to compare measured vs. known values as well as to assess whether the measurements taken are valid.
When measuring data, whether it be the density of some material, standard acceleration due to gravity of a falling object, or something else entirely, the measured value often varies from the true value. Error can arise due to many different reasons that are often related to human error, but can also be due to estimations and limitations of devices used in measurement. Calculating the percentage error provides a means to quantify the degree by which a measured value varies relative to the true value. A small percentage error means that the observed and true value are close while a large percentage error indicates that the observed and true value vary greatly. In most cases, a small percentage error is desirable, while a large percentage error may indicate an error or that an experiment or measurement technique may need to be re-evaluated. If, for example, the measured value varies from the expected value by 90%, there is likely an error, or the method of measurement may not be accurate.
Computing percentage error
The computation of percentage error involves the use of the absolute error, which is simply the difference between the observed and the true value. The absolute error is then divided by the true value, resulting in the relative error, which is multiplied by 100 to obtain the percentage error. Refer to the equations below for clarification.
| Absolute error = |Vobserved – Vtrue| |
| Relative error = |
|
| Percentage error = |
| × 100% |
For example, if the observed value is 56.891 and the true value is 62.327, the percentage error is:
| × 100% = 8.722% |
The equations above are based on the assumption that true values are known. True values are often unknown, and under these situations, standard deviation is one way to represent the error. Please refer to the standard deviation calculator for further details.
Negative percentage error
Based on the formula above, when the true value is positive, percentage error is always positive due to the absolute value. In most cases, only the error is important, and not the direction of the error. However, it is possible to have a negative percentage error. This occurs if we do not take the absolute value of the error, the observed value is smaller than the true value, and the true value is positive. For example, given an observed value of 7, a true value of 9, and allowing for a negative percentage, the percentage error is:
| × 100% = |
| × 100% | ||||
| = | -22.222% | ||||||
A negative percentage error simply means that the observed value is smaller than the true value. If the observed value is larger than the true value, the percentage error will be positive. Thus, in the context of an experiment, a negative percentage error just means that the measured value is smaller than expected. It does not indicate that the observed value is somehow better than expected, since the best possible outcome for percentage error is that the observed and true values are equal, resulting in a percentage error of 0.
How to Calculate Percent Error
Sample Percent Error Calculation
:max_bytes(150000):strip_icc()/how-to-calculate-percent-error-609584_final-97d164b04ae647bc887f285cd95a3a71.png)
ThoughtCo / Nusha Ashjaee
Percent error or percentage error expresses as a percentage the difference between an approximate or measured value and an exact or known value. It is used in science to report the difference between a measured or experimental value and a true or exact value. Here is how to calculate percent error, with an example calculation.
Key Points: Percent Error
- The purpose of a percent error calculation is to gauge how close a measured value is to a true value.
- Percent error (percentage error) is the difference between an experimental and theoretical value, divided by the theoretical value, multiplied by 100 to give a percent.
- In some fields, percent error is always expressed as a positive number. In others, it is correct to have either a positive or negative value. The sign may be kept to determine whether recorded values consistently fall above or below expected values.
- Percent error is one type of error calculation. Absolute and relative error are two other common calculations. Percent error is part of a comprehensive error analysis.
- The keys to reporting percent error correctly are to know whether or not to drop the sign (positive or negative) on the calculation and to report the value using the correct number of significant figures.
Percent Error Formula
Percent error is the difference between a measured or experiment value and an accepted or known value, divided by the known value, multiplied by 100%.
For many applications, percent error is always expressed as a positive value. The absolute value of the error is divided by an accepted value and given as a percent.
|accepted value — experimental value| \ accepted value x 100%
For chemistry and other sciences, it is customary to keep a negative value, should one occur. Whether error is positive or negative is important. For example, you would not expect to have positive percent error comparing actual to theoretical yield in a chemical reaction. If a positive value was calculated, this would give clues as to potential problems with the procedure or unaccounted reactions.
When keeping the sign for error, the calculation is the experimental or measured value minus the known or theoretical value, divided by the theoretical value and multiplied by 100%.
percent error = [experimental value — theoretical value] / theoretical value x 100%
Percent Error Calculation Steps
- Subtract one value from another. The order does not matter if you are dropping the sign (taking the absolute value. Subtract the theoretical value from the experimental value if you are keeping negative signs. This value is your «error.»
- Divide the error by the exact or ideal value (not your experimental or measured value). This will yield a decimal number.
- Convert the decimal number into a percentage by multiplying it by 100.
- Add a percent or % symbol to report your percent error value.
Percent Error Example Calculation
In a lab, you are given a block of aluminum. You measure the dimensions of the block and its displacement in a container of a known volume of water. You calculate the density of the block of aluminum to be 2.68 g/cm 3 . You look up the density of a block of aluminum at room temperature and find it to be 2.70 g/cm 3 . Calculate the percent error of your measurement.
- Subtract one value from the other:
2.68 — 2.70 = -0.02 - Depending on what you need, you may discard any negative sign (take the absolute value): 0.02
This is the error. - Divide the error by the true value:0.02/2.70 = 0.0074074
- Multiply this value by 100% to obtain the percent error:
0.0074074 x 100% = 0.74% (expressed using 2 significant figures).
Significant figures are important in science. If you report an answer using too many or too few, it may be considered incorrect, even if you set up the problem properly.
Percent Error Versus Absolute and Relative Error
Percent error is related to absolute error and relative error. The difference between an experimental and known value is the absolute error. When you divide that number by the known value you get relative error. Percent error is relative error multiplied by 100%. In all cases, report values using the appropriate number of significant digits.
Uncertainty and Errors
Uncertainty and Errors
TABLE OF CONTENTS
When we measure a property such as length, weight, or time, we can introduce errors in our results. Errors, which produce a difference between the real value and the one we measured, are the outcome of something going wrong in the measuring process.
The reasons behind errors can be the instruments used, the people reading the values, or the system used to measure them.
If, for instance, a thermometer with an incorrect scale registers one additional degree every time we use it to measure the temperature, we will always get a measurement that is out by that one degree.
Because of the difference between the real value and the measured one, a degree of uncertainty will pertain to our measurements. Thus, when we measure an object whose actual value we don ’ t know while working with an instrument that produces errors, the actual value exists in an ‘ uncertainty range ’ .
The difference between uncertainty and error
The main difference between errors and uncertainties is that an error is the difference between the actual value and the measured value, while an uncertainty is an estimate of the range between them, representing the reliability of the measurement. In this case, the absolute uncertainty will be the difference between the larger value and the smaller one.
A simple example is the value of a constant. Let ’ s say we measure the resistance of a material. The measured values will never be the same because the resistance measurements vary. We know there is an accepted value of 3.4 ohms, and by measuring the resistance twice, we obtain the results 3.35 and 3.41 ohms.
Errors produced the values of 3.35 and 3.41, while the range between 3.35 to 3.41 is the uncertainty range.
Let ’ s take another example, in this case, measuring the gravitational constant in a laboratory.
The standard gravity acceleration is 9.81 m/s^2. In the laboratory, conducting some experiments using a pendulum, we obtain four values for g: 9.76 m/s^2, 9.6 m/s^2, 9.89m/s^2, and 9.9m/s^2. The variation in values is the product of errors. T he mean value is 9.78m/s^2.
The uncertainty range for the measurements reaches from 9.6 m/s^2, to 9.9 m/s^2 while the absolute uncertainty is approximately equal to half of our range, which is equal to the difference between the maximum and minimum values divided by two.
The absolute uncertainty is reported as:
In this case, it will be:
What is the standard error in the mean?
The standard error in the mean is the value that tells us how much error we have in our measurements against the mean value. To do this, we need to take the following steps:
- Calculate the mean of all measurements.
- Subtract the mean from each measured value and square the results.
- Add up all subtracted values.
- Divide the result by the square root of the total number of measurements taken.
Let ’ s look at an example.
You have measured the weight of an object four times. The object is known to weigh exactly 3.0kg with a precision of below one gram. Your four measurements give you 3.001 kg, 2.997 kg, 3.003 kg, and 3.002 kg. Obtain the error in the mean value.
First, we calculate the mean:
As the measurements have only three significant figures after the decimal point, we take the value as 3.000 kg. Now we need to subtract the mean from each value and square the result:
Again, the value is so small, and we are only taking three significant figures after the decimal point, so we consider the first value to be 0. Now we proceed with the other differences:
All our results are 0 as we only take three significant figures after the decimal point. When we divide this between the root square of the samples, which is √4, we get:
In this case, the standard error of the mean (σx) is almost nothing.
What are calibration and tolerance?
Tolerance is the range between the maximum and minimum allowed values for a measurement. Calibration is the process of tuning a measuring instrument so that all measurements fall within the tolerance range.
To calibrate an instrument, its results are compared against other instruments with higher precision and accuracy or against an object whose value has very high precision.
One example is the calibration of a scale.
To calibrate a scale, you must measure a weight that is known to have an approximate value. Let ’ s say you use a mass of one kilogram with a possible error of 1 gram. The tolerance is the range 1.002kg to 0.998kg. The scale consistently gives a measure of 1.01kg. The measured weight is above the known value by 8 grams and also above the tolerance range. The scale does not pass the calibration test if you want to measure weights with high precision.
How is uncertainty reported?
When doing measurements, uncertainty needs to be reported. It helps those reading the results to know the potential variation. To do this, the uncertainty range is added after the symbol ±.
Let ’ s say we measure a resistance value of 4.5ohms with an uncertainty of 0.1ohms. The reported value with its uncertainty is 4.5 ± 0.1 ohms.
We find uncertainty values in many processes, from fabrication to design and architecture to mechanics and medicine.
What are absolute and relative errors?
Errors in measurements are either absolute or relative. Absolute errors describe the difference from the expected value. Relative errors measure how much difference there is between the absolute error and the true value.
Absolute error
Absolute error is the difference between the expected value and the measured one. If we take several measurements of a value, we will obtain several errors. A simple example is measuring the velocity of an object.
Let ’ s say we know that a ball moving across the floor has a velocity of 1.4m/s. We measure the velocity by calculating the time it takes for the ball to move from one point to another using a stopwatch, which gives us a result of 1.42m/s.
The absolute error of your measurement is 1.42 minus 1.4.
Relative error
Relative error compares the measurement magnitudes. It shows us that the difference between the values can be large, but it is small compared to the magnitude of the values. Let ’ s take an example of absolute error and see its value compared to the relative error.
You use a stopwatch to measure a ball moving across the floor with a velocity of 1.4m/s. You calculate how long it takes for the ball to cover a certain distance and divide the length by the time, obtaining a value of 1.42m/s.
As you can see, the relative error is smaller than the absolute error because the difference is small compared to the velocity.
Another example of the difference in scale is an error in a satellite image. If the image error has a value of 10 metres, this is large on a human scale. However, if the image measures 10 kilometres height by 10 kilometres width, an error of 10 metres is small.
The relative error can also be reported as a percentage after multiplying by 100 and adding the percentage symbol %.
Plotting uncertainties and errors
Uncertainties are plotted as bars in graphs and charts. The bars extend from the measured value to the maximum and minimum possible value. The range between the maximum and the minimum value is the uncertainty range. See the following example of uncertainty bars:
Figure 1. Plot showing the mean value points of each measurement. The bars extending from each point indicate how much the data can vary. Source: Manuel R. Camacho, StudySmarter.
See the following example using several measurements:
You carry out four measurements of the velocity of a ball moving 10 metres whose speed is decreasing as it advances. You mark 1-metre divisions, using a stopwatch to measure the time it takes for the ball to move between them.
You know that your reaction to the stopwatch is around 0.2m/s. Measuring the time with the stopwatch and dividing by the distance, you obtain values equal to 1.4m/s, 1.22m/s, 1.15m/s, and 1.01m/s.
Because the reaction to the stopwatch is delayed, producing an uncertainty of 0.2m/s, your results are 1.4 ± 0.2 m/s, 1.22 ± 0.2 m/s, 1.15 ± 0.2 m/s, and 1.01 ± 0.2m/s.
The plot of the results can be reported as follows:
Figure 2. The plot shows an approximate representation. The dots represent the actual values of 1.4m/s, 1.22m/s, 1.15m/s, and 1.01m/s. The bars represent the uncertainty of ±0.2m/s. Source: Manuel R. Camacho, StudySmarter.
How are uncertainties and errors propagated?
Each measurement has errors and uncertainties. When we carry out operations with values taken from measurements, we add these uncertainties to every calculation. The processes by which uncertainties and errors change our calculations are called uncertainty propagation and error propagation, and they produce a deviation from the actual data or data deviation.
There are two approaches here:
- If we are using percentage error, we need to calculate the percentage error of each value used in our calculations and then add them together.
- If we want to know how uncertainties propagate through the calculations, we need to make our calculations using our values with and without the uncertainties.
The difference is the uncertainty propagation in our results.
See the following examples:
Let ’ s say you measure gravity acceleration as 9.91 m/s^2, and you know that your value has an uncertainty of ± 0.1 m/s^2.
You want to calculate the force produced by a falling object. The object has a mass of 2kg with an uncertainty of 1 gram or 2 ± 0.001 kg.
To calculate the propagation using percentage error, we need to calculate the error of the measurements. We calculate the relative error for 9.91 m/s^2 with a deviation of (0.1 + 9.81) m/s^2.
Multiplying by 100 and adding the percentage symbol, we get 1%. If we then learn that the mass of 2kg has an uncertainty of 1 gram, we calculate the percentage error for this, too, getting a value of 0.05%.
To determine the percentage error propagation, we add together both errors.
To calculate the uncertainty propagation, we need to calculate the force as F = m * g. If we calculate the force without the uncertainty, we obtain the expected value.
Now we calculate the value with the uncertainties. Here, both uncertainties have the same upper and lower limits ± 1g and ± 0.1 m/s2.
We can round this number to two significant digits as 19.83 Newtons. Now We subtract both results.
The result is expressed as ‘ expected value ± uncertainty value ’ .
If we use values with uncertainties and errors, we need to report this in our results.
Reporting uncertainties
To report a result with uncertainties, we use the calculated value followed by the uncertainty. We can choose to put the quantity inside a parenthesis. Here is a n example of how to report uncertainties.
We measure a force, and according to our results, the force has an uncertainty of 0.21 Newtons.
Our result is 19.62 Newtons, which has a possible variation of plus or minus 0.21 Newtons.
Propagation of uncertainties
See the following general rules on how uncertainties propagate and how to calculate uncertainties. For any propagation of uncertainty, values must have the same units.
Addition and subtraction: if values are being added or subtracted, the total value of the uncertainty is the result of the addition or subtraction of the uncertainty values. If we have measurements (A ± a) and (B ± b), t he result of adding them is A + B with a total uncertainty (± a) + (± b).
Let ’ s say we are adding two pieces of metal with lengths of 1.3m and 1.2m. The uncertainties are ± 0.05m and ± 0.01m. The total value after adding them is 1.5m with an uncertainty of ± (0.05m + 0.01m) = ± 0.06m.
Multiplication by an exact number: the total uncertainty value is calculated by multiplying the uncertainty by the exact number.
Let ’ s say we are calculating the area of a circle, knowing the area is equal to A = 2 * 3.1415 • r. We calculate the radius as r = 1 ± 0.1m. The uncertainty is 2 • 3.1415• 1 ± 0.1m, giving us an uncertainty value of 0.6283m.
Division by an exact number: the procedure is the same as in multiplication. In this case, we divide the uncertainty by the exact value to obtain the total uncertainty.
If we have a length of 1.2m with an uncertainty of ± 0.03m and divide this by 5, the uncertainty is ± 0.03 / 5 or ±0.006.
Data deviation
We can also calculate the deviation of data produced by the uncertainty after we make calculations using the data. The data deviation changes if we add, subtract, multiply, or divide the values. Data deviation uses the symbol ‘ δ ’ .
- Data deviation after subtraction or addition: to calculate the deviation of the results, we need to calculate the square root of the squared uncertainties:
- Data deviation after multiplication or division: to calculate the data deviation of several measurements, we need the uncertainty – real value ratio and then calculate the square root of the squared terms. See this example using measurements A ± a and B ± b:
If we have more than two values, we need to add more terms.
- Data deviation if exponents are involved: we need to multiply the exponent by the uncertainty and then apply the multiplication and division formula. If we have y = (A ± a) 2 * (B ± b) 3, the deviation will be:
If we have more than two values, we need to add more terms.
Rounding numbers
When errors and uncertainties are either very small or very large, it is convenient to remove terms if they do not alter our results. When we round numbers, we can round up or down.
Measuring the value of the gravity constant on earth, our value is 9.81 m/s^2, and we have an uncertainty of ± 0.10003m/s^2. The value after the decimal point varies our measurement by 0.1m/s^2; However, the last value of 0.0003 has a magnitude so small that its effect would be barely noticeable. We can, therefore, round up by removing everything after 0.1.
Rounding integers and decimals
To round numbers, we need to decide what values are important depending on the magnitude of the data.
There are two options when rounding numbers, rounding up or down. The option we choose depends on the number after the digit we think is the lowest value that is important for our measurements.
- Rounding up: we eliminate the numbers that we think are not necessary. A simple example is rounding up 3.25 to 3.3.
- Rounding down: again, we eliminate the numbers that we think are not necessary. An example is rounding down 76.24 to 76.2.
- The rule when rounding up and down: as a general rule, when a number ends in any digit between 1 and 5, it will be rounded down. If the digit ends between 5 and 9, it will be rounded up, while 5 is also always rounded up. For instance, 3.16 and 3.15 become 3.2, while 3.14 becomes 3.1.
By looking at the question, you can often deduce how many decimal places (or significant figures) are needed. Let ’ s say you are given a plot with numbers that have only two decimal places. You would then also be expected to include two decimal places in your answers.
Round quantities with uncertainties and errors
When we have measurements with errors and uncertainties, the values with higher errors and uncertainties set the total uncertainty and error values. Another approach is required when the question asks for a certain number of decimals.
Let ’ s say we have two values (9.3 ± 0.4) and (10.2 ± 0.14). If we add both values, we also need to add their uncertainties. The addition of both values gives us the total uncertainty as | 0.4 | + | 0.14 | or ± 0.54. Rounding 0.54 to the nearest integer gives us 0.5 as 0.54 is closer to 0.5 than to 0.6.
Therefore, the result of adding both numbers and their uncertainties and rounding the results is 19.5 ± 0.5m.
Let ’ s say you are given two values to multiply, and both have uncertainties. You are asked to calculate the total error propagated. The quantities are A = 3.4 ± 0.01 and B = 5.6 ± 0.1. The question asks you to calculate the error propagated up to one decimal place.
First, you calculate the percentage error of both:
The total error is 0.29% + 1.78% or 2.07%.
You have been asked to approximate only to one decimal place. The result can vary depending on whether you only take the first decimal or whether you round up this number.