What’s meaning and difference of Uncertainty and Errors (Tolerance)

Uncertainty and Errors

When measuring physical quantities like length, weight, or time, mistakes can occur during the measuring process, affecting the results. These mistakes are called errors. Errors can be caused by various factors, such as faulty measuring tools, human error in reading measurements, or issues with the measurement system. For example, if a thermometer is malfunctioning and shows an incorrect temperature, every reading taken with it will be off by a certain amount. This makes our measurements uncertain because they don’t match the true value precisely. When we’re unsure about the true value of a measurement, we consider a range of possible values, known as the uncertainty range. Understanding uncertainty and errors is crucial as it helps us make more informed decisions based on the available information.

The difference between uncertainty and error

Both errors and uncertainties are important concepts in measurement, but they have distinct meanings. An error is the numerical difference between the actual value and the measured value. On the other hand, uncertainty is an estimate of the range within which the actual value is likely to lie, based on the reliability of the measurement.

Let’s consider an example of measuring resistance. Suppose the accepted value for a material’s resistance is 3.4 ohms. When we measure it twice and get values of 3.35 ohms and 3.41 ohms, the differences between these measured values and the accepted value are errors. The range between the two measured values, which is 0.06 ohms (3.41 – 3.35), represents the uncertainty range.

Another example is measuring the gravitational constant in a laboratory. The accepted standard for gravitational acceleration is 9.81 m/s². In an experiment using a pendulum, we obtain values such as 9.76 m/s², 9.6 m/s², 9.89 m/s², and 9.9 m/s². These variations from the accepted value are errors. The mean value of these measurements is 9.78 m/s², and the uncertainty range is from 9.6 m/s² to 9.9 m/s². The absolute uncertainty is approximately half of this range, calculated as (9.9 – 9.6) / 2 = 0.15 m/s².
Understanding errors and uncertainties helps us assess the reliability of our measurements and determine the range of possible values for the actual quantity. This knowledge is essential for making well – founded decisions based on the collected data.

What is the standard error in the mean?

The standard error in the mean is a value that indicates how much the measurements deviate from the mean value. To calculate it, follow these steps:

1. Calculate the mean: Find the mean of all measurements.
2. Subtract and square: Subtract the mean from each measured value and square the results.
3. Sum the values: Add up all the subtracted values.
4. Divide by the square root of the sample size: Divide the sum by the square root of the total number of measurements taken.

For example, imagine you weigh an object four times. The object is known to weigh exactly 3.0 kg with a precision of less than one gram. Your four measurements are 3.001 kg, 2.997 kg, 3.003 kg, and 3.002 kg.

• First, calculate the mean: (3.001 kg + 2.997 kg + 3.003 kg + 3.002 kg) / 4 = 3.00075 kg. Since our measurements have only three significant figures after the decimal point, we take the value as 3.000 kg.
• Then, subtract the mean from each measurement and square the result:
(3.001 kg – 3.000 kg)² = 0.000001 kg²
(2.997 kg – 3.000 kg)² = 0.000009 kg²
(3.003 kg – 3.000 kg)² = 0.000009 kg²
(3.002 kg – 3.000 kg)² = 0.000004 kg²
Considering only three significant figures after the decimal point, we can approximate the first value as 0.
• Next, add up all the squared differences: 0 + 0.000009 kg² + 0.000009 kg² + 0.000004 kg² = 0.000022 kg²
• Finally, divide by the square root of the number of samples (√4 = 2): √(0.000022 kg² / 4) = 0.002 kg

In this case, the standard error of the mean (σx) is very small, indicating that our measurements are close to the true value of the object’s weight.

What are calibration and tolerance?

Tolerance refers to the range between the maximum and minimum allowed values for a measurement. Calibration, on the other hand, is the process of adjusting a measuring instrument to ensure that all its measurements fall within the tolerance range. To calibrate an instrument, its measurement results are compared with those of more precise and accurate instruments or with a reference object of known high – precision value.

One example is the calibration of a scale

Calibration is not a one – time task. Scales need to be recalibrated regularly to maintain their accuracy. Environmental factors like temperature, humidity, and air pressure can affect a scale’s readings. For instance, changes in temperature can cause the metal components of a scale to expand or contract, leading to inaccurate measurements. Therefore, it’s important to take these environmental factors into account during calibration.

Also, when calibrating a scale, it’s crucial to use appropriate weights. Using weights that are too heavy or too light can distort the calibration process and affect the scale’s accuracy. Overall, calibrating a scale is a vital step in obtaining accurate and reliable measurements. Following proper calibration procedures and regular recalibration are essential for maintaining the scale’s accuracy.

How is uncertainty reported?

When presenting measurement results, it’s important to report the associated uncertainty. This helps others understand the potential variation in the measurement and the level of confidence in the reported value.

For example, if we measure a resistance value of 4.5 ohms with an uncertainty of 0.1 ohms, we report it as 4.5 ± 0.1 ohms. This indicates that we are confident that the true value of the resistance lies within the range of 4.4 ohms to 4.6 ohms.

Uncertainty values are relevant in many fields, including fabrication, design, architecture, mechanics, and medicine. They play a crucial role in accurately measuring and reporting results. By reporting uncertainty values, we can minimize errors and improve the quality of our measurements, which is essential in scientific research, engineering, and healthcare.

What are absolute and relative errors?

Measurement errors can be classified as either absolute or relative. Absolute errors describe the difference between the measured value and the expected value. Relative errors, on the other hand, measure how significant this difference is in relation to the true value.

Absolute error

To calculate the absolute error, use the formula: Absolute error = measured value – expected value. For example, if the expected value is 1.4 m/s and the measured value is 1.42 m/s, then the absolute error is 1.42 m/s – 1.4 m/s = 0.02 m/s.

It’s important to note that absolute error can be positive or negative. A positive absolute error means the measured value is higher than the expected value, while a negative absolute error means the measured value is lower. In this case, since the absolute error is positive, the measured value is slightly higher than the expected value.

Although absolute error is useful for assessing the accuracy of a single measurement, it doesn’t provide information about the precision of the measurement. To evaluate precision, we need to look at the range of values obtained from multiple measurements of the same quantity.

Relative error

Relative error is a measure of the difference between the measured value and the expected value, expressed as a percentage of the expected value. It is particularly useful for comparing errors in values of different magnitudes as it takes into account the scale of the values being measured.

The formula for relative error is: Relative error = (absolute error / expected value) x 100%. Using the previous example where the absolute error was 0.02 m/s and the expected value was 1.4 m/s, the relative error is (0.02 m/s / 1.4 m/s) x 100% ≈ 1.43%.

As we can see, the relative error is smaller than the absolute error because it considers the magnitude of the values. In this case, the difference between the measured value and the expected value is only 1.43% of the expected value.

Another example to illustrate the difference in scale is an error in a satellite image. If an error in a satellite image is 10 metres, it seems large when considering human – scale distances. However, if the image measures 10 kilometres by 10 kilometres, an error of 10 metres is relatively small, as it is only 0.1% of the total area (since 10 km = 10000 m, and 10 / (10000×10000)×100% = 0.0001% in terms of area ratio). Reporting the relative error as a percentage helps readers better understand the significance of the error in relation to the expected value.

Plotting uncertainties and errors

Uncertainties are typically represented as bars in graphs and charts. These bars extend from the measured value to the maximum and minimum possible values. The range between the maximum and minimum values is the uncertainty range. See the following example of uncertainty bars:

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‍For example, consider an experiment where you measure the velocity of a ball moving a distance of 10 metres. The ball’s speed is decreasing as it moves. You mark 1 – metre divisions and use a stopwatch to measure the time it takes for the ball to move between each division. Due to the delay in your reaction to starting and stopping the stopwatch, there is an uncertainty of 0.2 m/s. Suppose you obtain velocity values of 1.4 m/s, 1.22 m/s, 1.15 m/s, and 1.01 m/s. The measurements, including the uncertainty, are reported as 1.4 ± 0.2 m/s, 1.22 ± 0.2 m/s, 1.15 ± 0.2 m/s, and 1.01 ± 0.2 m/s. The plot of the results can be reported as follows:

 

In a graph, the dots represent the actual measured values (1.4 m/s, 1.22 m/s, 1.15 m/s, and 1.01 m/s), and the bars extending from each dot represent the uncertainty of ±0.2 m/s. This visual representation helps in quickly understanding the range within which the true value might lie for each measurement.

How are uncertainties and errors propagated?

When performing calculations using values with uncertainties and errors, it’s essential to account for these uncertainties in the calculations, as they can impact the accuracy of the final result. This process is known as uncertainty propagation or error propagation, and it can lead to a deviation from the actual data, also called data deviation.

There are two common approaches to uncertainty propagation: percentage error and absolute error. In the percentage error approach, we calculate the relative error for each measurement and sum them to determine the overall percentage error propagation. In the absolute error approach, we add the absolute errors of each measurement to find the overall absolute error propagation.

For example, if we measure the gravitational acceleration as 9.91 m/s² with an uncertainty of ±0.1 m/s² and the mass of an object as 2 ± 0.001 kg, the relative error for the gravitational acceleration is (0.1 / 9.91)×100% ≈ 1% and the relative error for the mass is (0.001 / 2)×100% = 0.05%. To find the overall percentage error propagation, we add these relative errors together.

To calculate the uncertainty propagation in the result of a calculation, we need to calculate the expected value while including the uncertainties. For instance, when calculating the force produced by a falling object using the formula F = m * g (where m is the mass and g is the gravitational acceleration), we calculate the force using the measured values along with their uncertainties. The result is then expressed as ‘expected value ± uncertainty value’.

Reporting uncertainties and errors in our results is crucial so that others can evaluate the accuracy and reliability of our measurements and calculations.

Reporting uncertainties

To report a measurement result with uncertainty, we write the calculated value followed by the uncertainty. We can also enclose the quantity in parentheses for clarity. For example, if we measure a force and find that the force has an uncertainty of 0.21 Newtons and our measured value is 19.62 Newtons, we report it as 19.62 ± 0.21 Newtons or (19.62 ± 0.21) N.

Propagation of uncertainties

When propagating uncertainties in calculations, there are specific rules for different arithmetic operations:

• Addition and subtraction: When adding or subtracting values, the total uncertainty is the sum of the individual uncertainties. For example, if we have two measurements (A ± a) and (B ± b) and we add them, the result is (A + B) ± (a + b). Suppose we are adding two pieces of metal with lengths of 1.3 m and 1.2 m, with uncertainties of ±0.05 m and ±0.01 m respectively. The total length is 1.3 + 1.2 = 1.5 m, and the total uncertainty is ±(0.05 m + 0.01 m) = ±0.06 m.

• Multiplication by an exact number: When multiplying a value by an exact number, the total uncertainty is calculated by multiplying the uncertainty by that exact number. For example, if we are calculating the area of a circle with radius r = 1 ± 0.1 m, and the formula for the area of a circle is A = πr². The uncertainty in the area is 2πr×0.1. Substituting r = 1 m, we get 2×3.1415×1×0.1 = 0.6283 m² (approximate uncertainty value).

• Division by an exact number: When dividing a value by an exact number, the total uncertainty is calculated by dividing the uncertainty by that exact value. For example, if we have a length of 1.2 m with an uncertainty of ±0.03 m and divide it by 5, the uncertainty in the result is ±0.03 / 5 = ±0.006 m.

Data deviation

When performing calculations with values that have uncertainties, the resulting data will deviate from the actual data. We can calculate this deviation using the data deviation (represented by the symbol ‘δ’). The calculation of data deviation depends on the type of operation performed on the values.

• Data deviation after addition or subtraction: To calculate the data deviation of the result, we use the formula δ = √(a² + b²), where a and b are the uncertainties of the values being added or subtracted. For example, if we subtract two values, A = 10 ± 0.2 and B = 8 ± 0.3, the result is C = A – B = 2 ± 0.4. The data deviation of C is δ = √(0.2² + 0.3²) = √(0.04 + 0.09) = 0.36.

• Data deviation after multiplication or division: For multiplication or division of several measurements, we use the uncertainty – real value ratio. If we have two values A ± a and B ± b, and we multiply them, the result is C = A * B ± (A*B) * √((a/A)² + (b/B)²). If there are more than two values, we add more terms to the equation.

• Data deviation if exponents are involved: If a value has an exponent, we multiply the exponent by the uncertainty and then apply the multiplication and division formula. For example, if we have y = (A ± a)² * (B ± b)³, the data deviation is δ = √((2Aa)² + (3Bb)²). If there are more than two values, additional terms are added to the equation.

Calculating the data deviation helps us assess the impact of uncertainties on our results and determine the accuracy and reliability of our measurements and calculations.

In the process of handling errors and uncertainties, rounding numbers often becomes an essential step to render the values more tractable. This is especially true when dealing with either minuscule or extremely large uncertainties that have a negligible impact on the overall results. Rounding can involve either incrementing the value (rounding up) or decrementing it (rounding down).​

For instance, when measuring the gravitational constant on Earth, the measured value might be 9.81 m/s² with an uncertainty of ±0.10003 m/s². Here, the part of the uncertainty value after the first decimal place, 0.0003, is minuscule in comparison to the overall uncertainty of 0.1. As a result, it is reasonable to discard the digits after the first decimal place and round the uncertainty to ±0.1 m/s², as this simplification will not notably affect the integrity of the measurement.​

However, it is of utmost importance to bear in mind that rounding itself can introduce additional errors, particularly when the number of significant figures is reduced to a very low level. Thus, prior to making a decision to round or truncate values, it is crucial to carefully assess the required level of accuracy for the measurements and calculations at hand.

Rounding Integers and Decimals

The process of rounding numbers necessitates determining which values are significant, taking into account both the magnitude of the data and the desired level of accuracy for the measurements and calculations. When rounding, there are two primary approaches: rounding up and rounding down. The choice between these two depends on the digit immediately following the lowest – order significant digit.​

When rounding up, we eliminate the less – significant digits. For example, 3.25 can be rounded up to 3.3. Conversely, when rounding down, we also discard the digits that are considered less relevant. For instance, 76.24 can be rounded down to 76.2.​
As a general rule, if a number ends with a digit in the range of 1 to 4, it is rounded down. If the digit at the end is in the range of 5 to 9, it is rounded up. Notably, when the digit is 5, it is typically rounded up. For example, both 3.15 and 3.16 are rounded up to 3.2, while 3.14 is rounded down to 3.1.​

When presented with a problem, we can often infer the required number of decimal places (or significant figures) from the given data. For example, if a graph or a set of data is presented with numbers having only two decimal places, it is reasonable to expect that our answers should also be presented with two decimal places. Paying close attention to the required level of accuracy is key to determining the appropriate number of decimal places or significant figures.

Round quantities with uncertainties and errors

When working with measurements that are accompanied by errors and uncertainties, the values with larger errors and uncertainties play a dominant role in determining the overall uncertainty and error values. When answering questions that specify a particular number of decimals or significant figures, a distinct approach is required.​

For example, consider two values: (9.3 ± 0.4) and (10.2 ± 0.14). When adding these values, we must also add their uncertainties. The total uncertainty is calculated as the sum of the absolute values of the individual uncertainties, which in this case is ±(0.4 + 0.14)= ±0.54. Rounding 0.54 to the nearest tenth gives us 0.5. So, the result of adding the two numbers along with their uncertainties and rounding is 19.5 ± 0.5.

If we are tasked with multiplying two values, both having uncertainties, and need to calculate the propagated total error, we can calculate the percentage error for each value and then sum them to obtain the total error. For example, if A = 3.4 ± 0.01 and B = 5.6 ± 0.1, the percentage errors for A and B are calculated as (0.01 / 3.4)×100% ≈ 0.29% and (0.1 / 5.6)×100% ≈ 1.78% respectively. The total error is the sum of these percentage errors, which is approximately 2.07%. If we are required to approximate the answer to one decimal place, we can either simply take the first decimal digit or round the number according to the standard rounding rules.

In summary, uncertainties and errors introduce variability in measurements and their associated calculations. Reporting uncertainties is crucial as it allows users to understand the potential range of variation in the measured values. Errors and uncertainties propagate during calculations involving data with such imperfections, and it is essential to consider the error of the data with the largest error or uncertainty. Calculating how errors propagate is valuable as it enables us to assess the reliability of our results.