Uncertainty and Errors

When we measure things like length, weight, or time, we can make mistakes that affect our results. These mistakes are called errors, and they happen when something goes wrong during the measuring process. Errors can be caused by things like the tools we use, the people reading the measurements, or the system we use to measure. For example, if a thermometer is broken and shows the wrong temperature, every reading we take will be off by the same amount. This means that our measurements will always be a little bit uncertain because they're not exactly the same as the real value. So when we measure something and we're not sure what the real value is, we have to consider a range of possible values, which we call the uncertainty range. Understanding uncertainty and errors is important because it helps us make better decisions with the information we have.

The difference between uncertainty and error

Errors and uncertainties are both important concepts in measurement, but they refer to slightly different things. An error is the difference between the actual value and the measured value, while an uncertainty is an estimate of the range of possible values that the actual value could be within, based on the reliability of the measurement.

Let's look at an example of measuring resistance. We know that the accepted value for a material's resistance is 3.4 ohms, but when we measure it twice, we get slightly different values of 3.35 and 3.41 ohms. These differences are the result of errors. However, the range between these two values, which is 0.06 ohms, is the uncertainty range.

Another example is measuring the gravitational constant in a laboratory. The accepted standard for gravity acceleration is 9.81 m/s^2. In the lab, we measure the acceleration using a pendulum and get values of 9.76 m/s^2, 9.6 m/s^2, 9.89 m/s^2, and 9.9 m/s^2. These variations are the result of errors. The mean value is 9.78 m/s^2, while the uncertainty range is between 9.6 m/s^2 and 9.9 m/s^2. The absolute uncertainty is approximately half of the range, which is the difference between the maximum and minimum values divided by two.

Understanding errors and uncertainties is important because it helps us know the reliability of our measurements and the range of possible values that the actual value could be within. This knowledge is crucial in making informed decisions based on the data we collect.

What is the standard error in the mean?

The standard error in the mean is a value that tells us how much error we have in our measurements compared to the mean. To calculate this, there are a few steps we need to follow:

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

Let's look at an example. Imagine you've weighed an object four times, and you know it should weigh exactly 3.0kg with a precision of less than one gram. Your four measurements give you 3.001kg, 2.997kg, 3.003kg, and 3.002kg. To obtain the error in the mean value, we first need to calculate the mean:

(3.001kg + 2.997kg + 3.003kg + 3.002kg) / 4 = 3.000kg

Since our measurements have only three significant figures after the decimal point, we take the value as 3.000kg. Next, we need to subtract the mean from each measurement and square the result:

(3.001kg - 3.000kg)^2 = 0.000001kg^2

(2.997kg - 3.000kg)^2 = 0.000009kg^2

(3.003kg - 3.000kg)^2 = 0.000009kg^2

(3.002kg - 3.000kg)^2 = 0.000004kg^2

Since these values are so small, and we're only taking three significant figures after the decimal point, we consider the first value to be 0. Now we can add up all the squared differences:

0 + 0.000009kg^2 + 0.000009kg^2 + 0.000004kg^2 = 0.000022kg^2

When we divide this by the square root of the number of samples (which is √4), we get:

√(0.000022kg^2 / 4) = 0.002kg

In this case, the standard error of the mean (σx) is almost nothing. This means that our measurements were very close to the true value of the object's weight.

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.

However, it's important to note that calibration is not a one-time process. Scales need to be recalibrated periodically to maintain their accuracy. Environmental factors such as temperature, humidity, and air pressure can also affect the readings of a scale, so it's important to take these into account when calibrating the scale.

In addition, it's important to use weights that are appropriate for the scale being calibrated. Using weights that are too heavy or too light can affect the accuracy of the calibration.

Overall, calibrating a scale is a critical step in ensuring accurate and reliable measurements. It's important to follow proper calibration procedures and to periodically recalibrate the scale to maintain its accuracy.

How is uncertainty reported?

When taking measurements, it is important to report the uncertainty associated with the measured value. This helps readers understand the potential variation in the measurement and the level of confidence that can be placed in the reported value.

For example, let's say we measure a resistance value of 4.5 ohms with an uncertainty of 0.1 ohms. The reported value with its uncertainty would be 4.5 ± 0.1 ohms. This means that we are confident that the true value of the resistance falls within the range of 4.4 ohms to 4.6 ohms.

Uncertainty values are found in many processes, from fabrication to design and architecture, to mechanics and medicine. They are an important aspect of measuring and reporting results accurately and reliably. By reporting uncertainty values, we can reduce errors and improve the quality of our measurements, which is critical in many fields, including scientific research, engineering, and healthcare.

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

To calculate the absolute error in this example, we subtract the expected value (1.4 m/s) from the measured value (1.42 m/s):

Absolute error = measured value - expected value
Absolute error = 1.42 m/s - 1.4 m/s
Absolute error = 0.02 m/s

So the absolute error in this case is 0.02 m/s. This means that our measured value deviates from the expected value by 0.02 m/s.

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

Absolute error is a useful measure of the accuracy of a measurement, but it doesn't tell us anything 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 useful for comparing values of different magnitudes, as it takes into account the scale of the values being measured.

To calculate the relative error, we divide the absolute error by the expected value and multiply by 100 to get a percentage:

Relative error = (absolute error / expected value) x 100%

Using the previous example, the absolute error was 0.02 m/s and the expected value was 1.4 m/s. Therefore, the relative error is:

Relative error = (0.02 m/s / 1.4 m/s) x 100%
Relative error = 1.43%

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

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 because it is only 0.1% of the total area.

Reporting the relative error as a percentage can help readers understand the significance of the error and how it relates to the expected value.

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:

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:

How are uncertainties and errors propagated?

When carrying out calculations with values that have uncertainties and errors, it is important to include these uncertainties in our calculations, as they can affect the accuracy of our results. This process is called uncertainty propagation or error propagation, and it can lead to a deviation from the actual data or data deviation.

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

For example, if we measure the gravity acceleration as 9.91 m/s^2 with an uncertainty of ± 0.1 m/s^2 and the mass of an object as 2 ± 0.001 kg, we would calculate the relative error for the gravity acceleration as 1% and the relative error for the mass as 0.05%. We would then add these relative errors together to determine the overall percentage error propagation.

To calculate the uncertainty propagation in our results, we need to calculate the expected value with the uncertainties included. For example, to calculate the force produced by a falling object, we would use the formula F = m * g, where m is the mass and g is the gravity acceleration. We would then calculate the force using the measured values with their uncertainties included. The result would be expressed as 'expected value ± uncertainty value'.

It is important to report the uncertainties and errors in our results to ensure that others can understand the accuracy and reliability of our measurements and calculations.

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. Our result is 19.62 Newtons, which has a possible variation of plus or minus 0.21 Newtons.

Propagation of uncertainties

When propagating uncertainties in calculations, there are general rules that can be applied to determine the total uncertainty:

Addition and subtraction: When adding or subtracting values, the total uncertainty is the result of adding or subtracting the individual uncertainties. For example, if we have two measurements (A ± a) and (B ± b) and we add them, the result will be (A + B) ± (a + b).

For instance, if we are adding two pieces of metal with lengths of 1.3m and 1.2m, with uncertainties of ± 0.05m and ± 0.01m respectively, the total length will be 1.5m with an uncertainty of ± (0.05m + 0.01m) = ± 0.06m.

Multiplication by an exact number: When multiplying a value by an exact number, the total uncertainty is calculated by multiplying the uncertainty by the exact number. For instance, if we are calculating the area of a circle with radius r = 1 ± 0.1m, the uncertainty in the area will be 2 • 3.1415•1 ± 0.1m, giving us an uncertainty value of 0.6283m.

Division by an exact number: When dividing a value by an exact number, the total uncertainty is calculated by dividing the uncertainty by the exact value. For instance, if we have a length of 1.2m with an uncertainty of ± 0.03m and divide this by 5, the uncertainty in the result will be ± 0.03 / 5 or ±0.006.

Data deviation

When we carry out calculations using values with uncertainties, the resulting data will also have a deviation from the actual data, which we can calculate using the data deviation (symbol 'δ'). The data deviation changes depending on the type of operation that is performed on the values.

Data deviation after addition or subtraction: To calculate the data deviation of the results, we need to calculate the square root of the sum of the squares of the uncertainties:

δ = sqrt(a^2 + b^2)

For example, if we subtract two values, A = 10 ± 0.2 and B = 8 ± 0.3, the result will be C = A - B = 2 ± 0.4. The data deviation of C is δ = sqrt(0.2^2 + 0.3^2) = 0.36.

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 sum of the squared terms. For example, if we have two values A ± a and B ± b, and we multiply them, the result will be C = A * B ± (A*B) * sqrt((a/A)^2 + (b/B)^2). If we have more than two values, we need to add more terms in the equation.

Data deviation if exponents are involved: If we have a value with an exponent, we need to 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 will be:

δ = sqrt((2Aa)^2 + (3Bb)^2)

If we have more than two values, we need to add more terms in the equation.

By calculating the data deviation, we can assess the impact of uncertainties on our results and determine the accuracy and reliability of our measurements and calculations.

Rounding numbers

When dealing with errors and uncertainties, it is often necessary to round numbers to make them more manageable. This is particularly useful when dealing very small or very large uncertainties that do not significantly affect our results. Rounding numbers can involve rounding up or down.

For example, when measuring the value of the gravity constant on earth, our value is 9.81 m/s^2, with an uncertainty of ±0.10003 m/s^2. The uncertainty value after the decimal point is 0.0003, which is very small compared to the uncertainty value of 0.1. Therefore, we can remove the digits after the first decimal point and round up to ±0.1 m/s^2 as it would not significantly affect our measurement.

However, it is essential to remember that rounding can also introduce errors, particularly when we round to a low number of significant figures. Therefore, it is crucial to consider the level of accuracy required for our measurements and calculations before deciding to round or truncate our values.

Rounding integers and decimals

Rounding numbers involves deciding which values are significant based on the magnitude of the data and the level of accuracy required for our measurements and calculations. There are two options when rounding numbers: rounding up or rounding down. The option we choose depends on the number after the digit we think is the lowest value that is important.

When rounding up, we eliminate the numbers that we think are not necessary. For example, we can round up 3.25 to 3.3. When rounding down, we also eliminate the numbers that we think are not necessary. For instance, we can round down 76.24 to 76.2.

The general rule for rounding up and down is that if 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 always rounded up. For example, 3.16 and 3.15 become 3.2, while 3.14 becomes 3.1.

When given a question, we can often deduce the number of decimal places (or significant figures) required based on the given data. For example, if we are given a plot with numbers that have only two decimal places, we would be expected to include two decimal places in our answers. It is crucial to pay attention to the level of accuracy required for our measurements and calculations to determine the appropriate number of decimal places or significant figures.

Round quantities with uncertainties and errors

When dealing with measurements that have errors and uncertainties, the values with higher errors and uncertainties determine the total uncertainty and error values. When answering questions that require a specific number of decimals or significant figures, a different approach is necessary.

For example, if we have two values of (9.3 ± 0.4) and (10.2 ± 0.14) and we add them, we need to add their uncertainties as well. The total uncertainty is the sum of the absolute values of the individual uncertainties, which is ±0.54. Rounding 0.54 to the nearest integer gives us 0.5. Therefore, the result of adding both numbers and their uncertainties and rounding the result is 19.5 ± 0.5m.

If we are given two values to multiply, and both have uncertainties, and we are asked to calculate the total error propagated, we can calculate the percentage error of both values and add them up to get the total error. For example, if A = 3.4 ± 0.01 and B = 5.6 ± 0.1, the percentage errors are 0.29% and 1.78%, respectively. The total error is the sum of the percentage errors, which is 2.07%. If we are asked to approximate the answer to one decimal place, we can either take the first decimal or round up the number.

In summary, uncertainties and errors introduce variations in measurements and their calculations, and it is crucial to report uncertainties so that users can know how much the measured value can vary. Errors and uncertainties propagate when we make calculations with data that has errors or uncertainties, and we must consider the error of the data with the largest error or uncertainty. It is useful to calculate how the error propagates so that we can determine how reliable our results are.

Uncertainty and Errors

What is the difference between error and uncertainty in measurement?

Errors are the difference between the measured value and the real or expected value; uncertainty is the range of variation between the measured value and the expected or real value.

How do you calculate uncertainties in physics?

To calculate uncertainty, we take the accepted or expected value and subtract the furthest value from the expected one. The uncertainty is the absolute value of this result.