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/s2. In the laboratory, conducting some experiments using a pendulum, we obtain four values for g: 9.76 m/s2, 9.6 m/s2, 9.89m/s2, and 9.9m/s2. The variation in values is the product of errors. The mean value is 9.78m/s2.
The uncertainty range for the measurements reaches from 9.6 m/s2, to 9.9 m/s2 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.
\[\frac{9.9 m/s^2 - 9.6 m/s^2}{2} = 0.15 m/s^2\]
The absolute uncertainty is reported as:
\[\text{Mean value ± Absolute uncertainty}\]
In this case, it will be:
\[9.78 \pm 0.15 m/s^2\]
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:
\[\frac{3.001 kg + 2.997 kg + 3.003 kg + 3.002 kg}{4} = 3.00075 kg\]
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:
\((3.001 kg - 3.000 kg)^2 = 0.000001 kg\)
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:
\((3.002 kg - 3.000 kg)^2 = 0.000004 kg(2.997 kg - 3.000 kg)^2 = 0.00009 kg(3.003 kg - 3.000 kg)^2 = 0.000009 kg\)
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 \(\sqrt4\), we get:
\(\text{Standard error of the mean} = \frac{0}{2} = 0\)
In this case, the standard error of the mean \((\sigma x\)) 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 1.002 kg 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.
\(\text{Absolute error} = 1.42 m/s - 1.4 m/s = 0.02 m/s\)
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.
\(\text{Relatove error} = \frac{|1.4 m/s - 1.42 m/s|}{1.4 m/s} = 0.014\)
\(\text{Absolute error} = 0.02 m/s\)
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.
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/s2, and you know that your value has an uncertainty of ± 0.1 m/s2.
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/s2 with a deviation of (0.1 + 9.81) m/s2.
\(\text{Relative error} = \frac{|9.81 m/s^2 - 9.91 m/s^2|}{9.81 m/s^2} = 0.01\)
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.
\(\text{Error} = 0.05\% + 1\% = 1.05\%\)
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.
\[\text{Force} = 2kg \cdot 9.81 m/s^2 = 19.62 \text{Newtons}\]
Now we calculate the value with the uncertainties. Here, both uncertainties have the same upper and lower limits ± 1g and ± 0.1 m/s2.
\[\text{Force with uncertainties} = (2kg + 1 g) \cdot (9.81 m/s^2 + 0.1 m/s^2)\]
We can round this number to two significant digits as 19.83 Newtons. Now we subtract both results.
\[\text{Uncertainty = |Force - Force with uncertainties|} = 0.21\]
The result is expressed as ‘expected value ± uncertainty value’.
\[\text{Force} = 19.62 \pm 0.21 Newtons\]
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 an example of how to report uncertainties.
We measure a force, and according to our results, the force has an uncertainty of 0.21 Newtons.
\[\text{Force} = (19.62 \pm 0.21) Newtons\]
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), the 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 \(A = 2 \cdot 3.1415 \cdot r\). We calculate the radius as r = 1 ± 0.1m. The uncertainty is \(2 \cdot 3.1415 \cdot 1 \pm 0.1m\), giving us an uncertainty value of 0.6283 m.
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 \(\pm \frac{0.03}{5}\) 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:
\[\delta = \sqrt{a^2+b^2}\]
- 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:
\[\delta = \sqrt{\frac{|a|^2}{A} + \frac{|b|^2}{B}}\]
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 \(y = (A ± a) 2 \cdot (B ± b) 3\), the deviation will be:
\[\delta = \sqrt{\frac{2|a|^2}{A} + \frac{2|b|^2}{B}}\]
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/s2, and we have an uncertainty of ± 0.10003 m/s2. The value after the decimal point varies our measurement by 0.1m/s2; 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:
\(\text{B percentage error} = \frac{|5.6-5.7|}{5.6} \cdot 100 = 1.78 \%\)
\(text{A percentage error} = \frac{|3.4-3.41|}{3.4} \cdot 100 = 0.29 \%\)
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.
\(\text{Round up error} = 2.1\%\)
\(\text{Approximate error} = 2.0\%\)
Uncertainty and Error in Measurements - Key takeaways
- Uncertainties and errors introduce variations in measurements and their calculations.
- Uncertainties are reported so that users can know how much the measured value can vary.
- There are two types of errors, absolute errors and relative errors. An absolute error is the difference between the expected value and the measured one. A relative error is the comparison between the measured and the expected values.
- Errors and uncertainties propagate when we make calculations with data that has errors or uncertainties.
- When we use data with uncertainties or errors, the data with the largest error or uncertainty dominates the smaller ones. It is useful to calculate how the error propagates, so we know how reliable our results are.
Explanations 
Exams 
Magazine 
