Uncertainty: Videos & Practice Problems

In any scientific calculation, understanding the concept of experimental error is crucial, as it reflects the inherent uncertainty in measurements. This uncertainty can be categorized into three main types: absolute uncertainty, relative uncertainty, and percent relative uncertainty.

Absolute uncertainty is the plus or minus value that accompanies a measurement, indicating the range within which the true value is expected to lie. For instance, if a student measures 25.00 ml of water with an uncertainty of ±0.02 ml, the absolute uncertainty here is 0.02 ml. This value provides a direct indication of the precision of the measurement.

Relative uncertainty, on the other hand, is calculated by dividing the absolute uncertainty by the measured value. In our example, the relative uncertainty would be calculated as follows:

Relative Uncertainty = \(\frac{\text{Absolute Uncertainty}}{\text{Measured Value}} = \frac{0.02 \, \text{ml}}{25.00 \, \text{ml}} = 0.0008\).

To express this relative uncertainty as a percentage, we multiply it by 100, yielding the percent relative uncertainty:

Percent Relative Uncertainty = Relative Uncertainty × 100 = \(0.0008 \times 100 = 0.08\%\).

These calculations are essential for accurately conveying the reliability of measurements in scientific work. As you engage with various problems involving uncertainties, it is important to remember these definitions and their applications, as they will enhance your understanding and accuracy in experimental contexts.

To calculate relative and percent relative uncertainty, we start with a measurement expressed as \(3.25 \pm 0.03\), where \(0.03\) is the absolute uncertainty. The relative uncertainty is determined by dividing the absolute uncertainty by the measurement itself. This can be expressed mathematically as:

\[\text{Relative Uncertainty} = \frac{\text{Absolute Uncertainty}}{\text{Measurement}} = \frac{0.03}{3.25} \approx 0.009230769\]

For practical purposes, we round this value to one significant figure, resulting in a relative uncertainty of \(0.009\). Thus, we can rewrite the measurement with its relative uncertainty as:

\[3.25 \pm 0.009\]

Next, to find the percent relative uncertainty, we multiply the relative uncertainty by \(100\):

\[\text{Percent Relative Uncertainty} = 0.009 \times 100 = 0.9\%\]

Therefore, the percent relative uncertainty can be expressed as:

\[3.25 \pm 0.9\%\]

This indicates that the true value of the measurement is expected to fall within a range defined by \(3.25\) plus or minus \(0.9\%\) of that value. To clarify, this means the actual value is likely to be between:

\[3.25 - (0.009 \times 3.25) \quad \text{and} \quad 3.25 + (0.009 \times 3.25)\]

In a practice scenario where you are given the percent relative uncertainty and need to find the absolute uncertainty, you would reverse the steps taken. Start by determining the absolute uncertainty using the formula:

\[\text{Absolute Uncertainty} = \text{Percent Relative Uncertainty} \times \text{Measurement} \div 100\]

By applying this method, you can effectively work backwards to find the absolute uncertainty from the percent relative uncertainty provided.

Understanding how to manage uncertainties in measurements is crucial in scientific calculations, particularly when performing operations like addition, subtraction, multiplication, and division. These operations often involve random errors, and applying the correct rules for uncertainty propagation is essential for accurate results.

One key principle to remember is the real rule, which states that the first digit of the absolute uncertainty corresponds to the last significant digit in the final answer. For instance, if you have an absolute uncertainty of 0.0005 and a measurement of 7.95876, the significant digit in the uncertainty is 5, which is in the fourth decimal place. This indicates that the measurement should also be expressed to four decimal places. Since the digit following the last significant digit (6) rounds the last digit up, the final measurement would be reported as 7.9588 ± 0.0005.

In another example, if the absolute uncertainty is 0.1 and the measurement is 13.235, the significant digit is 1, which is in the first decimal place. Therefore, the measurement should be rounded to one decimal place. Since the digit following the last significant digit (3) does not require rounding, the final result would be 13.2 ± 0.1.

When performing addition and subtraction, it is vital to apply the real rule consistently to ensure that the final results reflect the appropriate level of precision. This approach helps maintain the integrity of the data and provides a clearer understanding of the uncertainties involved in the measurements.

When performing addition and subtraction in measurements, the uncertainty in the final result is derived from the individual absolute uncertainties of the measurements involved. The final answer's precision is determined by the measurement with the least number of decimal places. For instance, if three measurements each have two decimal places, the final answer will also be expressed with two decimal places.

To calculate the absolute uncertainty for the final result, one must use the formula for combining uncertainties in addition and subtraction. This involves taking the square root of the sum of the squares of each individual uncertainty. For example, if the absolute uncertainties are 0.04, 0.03, and 0.03, the calculation would be:

$$\text{Uncertainty} = \sqrt{(0.04)^2 + (0.03)^2 + (0.03)^2}$$

This results in:

$$\text{Uncertainty} = \sqrt{0.0016 + 0.0009 + 0.0009} = \sqrt{0.0034} \approx 0.058$$

In reporting the absolute uncertainty, it is essential to apply the "real rule," which states that the first significant digit in the absolute uncertainty dictates the number of decimal places in the final measurement. In this case, the 0.058 indicates that the first significant digit is 5, but the presence of the 8 means it rounds up to 6. Therefore, the final measurement is reported as 6.85, with the absolute uncertainty rounded to 0.06, maintaining two decimal places as required.

For addition and subtraction, the process is relatively straightforward: square the uncertainties, sum them, take the square root, and apply the real rule to determine the appropriate number of decimal places. However, when it comes to multiplication and division, the method for determining uncertainty becomes more complex, requiring a different approach. Understanding these principles is crucial for accurate scientific measurement and reporting.

When performing multiplication and division with measurements, it is essential to convert absolute uncertainties into percent relative uncertainties. For example, consider the values 3.68 ± 0.05, 1.15 ± 0.06, and 0.92 ± 0.06. Each of these measurements has a specific number of significant figures: 3.68 and 1.15 have three significant figures, while 0.92 has two. Initially, the calculated result of the operation may be 4.6, but it is crucial to retain the full answer before rounding according to significant figures.

To convert absolute uncertainties into percent relative uncertainties, divide each absolute uncertainty by its corresponding measurement and multiply by 100. For instance, the relative uncertainty for 3.68 is calculated as:

$$\text{Relative Uncertainty} = \left(\frac{0.05}{3.68}\right) \times 100$$

Repeat this process for the other measurements to obtain their percent relative uncertainties. After calculating these values, square each percent relative uncertainty, sum them, and take the square root of the total. This process yields a combined percent relative uncertainty, which can be expressed as:

$$\text{Combined Percent Relative Uncertainty} = \sqrt{(u_1^2 + u_2^2 + u_3^2)}$$

where \(u_1\), \(u_2\), and \(u_3\) are the individual percent relative uncertainties.

Once the combined percent relative uncertainty is determined, convert it back to absolute uncertainty by dividing by 100 and multiplying by the measurement. This step is crucial as the first significant digit in the absolute uncertainty dictates the last significant figure in the final measurement. For example, if the absolute uncertainty results in a value that rounds the measurement to one decimal place, the final answer should reflect this precision.

In summary, when dealing with multiplication and division, the process involves converting absolute uncertainties to percent relative uncertainties, performing the calculations, and then converting back to absolute uncertainties. This method ensures accurate propagation of uncertainties in measurements, which is vital for maintaining precision in scientific calculations. As you progress to more complex problems, remember to apply these rules consistently based on the mathematical operations involved.

When performing addition or subtraction with measurements, it is essential to determine both absolute and relative uncertainties to ensure accuracy. In this case, we start with three measurements that, when added together, yield a total of 5.028. The significant figures in the final result depend on the least number of decimal places in the measurements involved. Here, we have measurements with 3 decimal places, 2 decimal places, and 3 decimal places, which means our final result should ideally reflect 2 decimal places.

To find the absolute uncertainty, we first square each individual absolute uncertainty, sum these squares, and then take the square root of that sum. For example, if the absolute uncertainties are represented as \( u_1, u_2, \) and \( u_3 \), the formula is:

\[ u = \sqrt{u_1^2 + u_2^2 + u_3^2} \]

After calculating, we find that the square root of the sum of the squared uncertainties results in approximately 0.024495. According to the rules of significant figures, the first significant digit in the absolute uncertainty indicates the last significant figure in the measurement. Here, the first significant digit is in the second decimal place, which means we round the last digit of our measurement from 8 to 3. Thus, the final measurement with its absolute uncertainty is:

\[ 5.03 \pm 0.02 \]

Next, we calculate the relative uncertainty, which is defined as the absolute uncertainty divided by the measurement itself. The formula for relative uncertainty is:

\[ \text{Relative Uncertainty} = \frac{u}{\text{Measurement}} \]

Substituting our values, we have:

\[ \text{Relative Uncertainty} = \frac{0.02}{5.03} \approx 0.003976 \]

Rounding this to the appropriate significant figures gives us 0.004. Therefore, the final result for the measurement with its relative uncertainty is:

\[ 5.03 \pm 0.004 \]

In summary, the absolute uncertainty is 0.02, and the relative uncertainty is 0.004, providing a comprehensive understanding of the precision of the measurement.

When dealing with multiplication and division in measurements, it is essential to determine both absolute and relative uncertainties to ensure accuracy in results. To begin, calculate the overall value by multiplying the relevant measurements and then dividing by any constants involved. For instance, if you multiply two numbers and divide by 9.17, you might arrive at a result like 7.5201. However, the focus should be on the uncertainties associated with each measurement.

To find the percent relative uncertainty for each measurement, divide the absolute uncertainty by the measurement itself and multiply by 100. For example, if you have an uncertainty of 0.03 for a measurement of 9.17, the calculation would be:

Percent Relative Uncertainty = \(\frac{0.03}{9.17} \times 100 \approx 0.327\%\), which can be rounded to 0.33%.

Once you have the percent relative uncertainties for all measurements, the next step is to combine them to find the overall percent relative uncertainty. This is done by squaring each percent relative uncertainty, summing them, and then taking the square root of that sum:

Overall Percent Relative Uncertainty = \(\sqrt{(0.2587)^2 + (0.2242)^2 + (0.3272)^2} \approx 0.4742\%\), which rounds to 0.47%.

To convert this percentage into a relative uncertainty, divide by 100, yielding a relative uncertainty of 0.0047. To find the absolute uncertainty, multiply this relative uncertainty by the initial measurement (7.5201), resulting in an absolute uncertainty of approximately 0.035344, which rounds to 0.04 when following significant figure rules.

Thus, the final result can be expressed as:

Measurement = \(7.52 \pm 0.04\)

Additionally, the relative uncertainty can be expressed as:

Relative Uncertainty = \(7.52 \pm 0.005\)

These calculations illustrate the importance of understanding how to manage uncertainties in measurements, particularly when performing operations like multiplication and division. Practicing these steps will enhance your proficiency in handling uncertainties in various scientific contexts.

In a laboratory setting, two students are tasked with preparing a stock solution, but they use different types of pipettes, which affects the precision of their measurements. Student A employs an uncalibrated pipette that delivers 50.00 ± 0.02 mL, while Student B uses a calibrated pipette that delivers 40.00 ± 0.01 mL. Their goal is to deliver a total of 200 mL to a container.

To determine the absolute uncertainty in their deliveries, we first analyze Student A's approach. Since they need to reach 200 mL using the uncalibrated pipette, they must perform the measurement four times (4 × 50.00 mL). The total volume delivered is 200.00 mL, and because the pipette is uncalibrated, the uncertainties are additive. Thus, the total uncertainty is calculated as:

Absolute Uncertainty = 4 × 0.02 mL = ±0.08 mL.

For Student B, who uses a calibrated pipette, they need to deliver 200 mL in five increments of 40.00 mL. The total volume is again 200.00 mL, but the method for calculating uncertainty differs. When adding uncertainties from calibrated measurements, we use the square root of the sum of the squares of the individual uncertainties:

Absolute Uncertainty = √(5 × (0.01 mL)²) = √(5 × 0.0001 mL²) = √(0.0005 mL²) ≈ 0.02236 mL.

Rounding this to two decimal places (to match the precision of the measurements), we find the absolute uncertainty for Student B is ±0.02 mL. Therefore, the final result for Student B is:

200.00 ± 0.02 mL.

This comparison illustrates that using a calibrated pipette significantly reduces the absolute uncertainty, leading to a more accurate final volume closer to the desired 200 mL. In contrast, Student A's uncalibrated pipette results in a larger uncertainty of ±0.08 mL, indicating a greater potential variation in the final volume.

Next, to calculate the molarity of the solution prepared by each student, we use the formula for molarity (M), which is defined as:

Molarity (M) = Moles of solute / Volume of solution in liters.

Since the volumes are initially given in milliliters, they must be converted to liters by dividing by 1000. This conversion applies to both the measured volume and the associated uncertainty. For example, if Student A's final volume is 200.00 mL, it converts to 0.20000 L, and the uncertainty becomes ±0.00008 L. Similarly, for Student B, the volume of 200.00 mL converts to 0.20000 L with an uncertainty of ±0.00002 L.

By applying these calculations, students can determine the molarity of their solutions, reinforcing the importance of accurate measurements and the impact of uncertainty in laboratory experiments.

A Class A 50 milliliter burette is designed to deliver precise volumes, with a manufacturer-certified uncertainty of ±0.05 milliliters. The smallest graduation on the burette is 0.1 milliliters. When using this burette to titrate a solution, five successive volumes were measured, resulting in a total volume of 30.80 milliliters.

To determine the uncertainty associated with this final volume, it is essential to understand the difference between calibrated and uncalibrated burettes. In this case, the manufacturer’s certification indicates a level of uncertainty, but it does not confirm that the burette is calibrated. An uncalibrated burette introduces systematic error, which affects how we calculate uncertainty.

For each measurement, the uncertainty is ±0.05 milliliters. Since there are five measurements, the total uncertainty is calculated by summing the individual uncertainties:

Uncertainty = 5 × ±0.05 mL = ±0.25 mL

Thus, the final volume added is expressed as:

30.80 mL ± 0.25 mL

If the burette had been calibrated, the uncertainty would be treated differently, involving random error calculations. In such cases, the total absolute uncertainty would be determined by taking the square root of the sum of the squares of the individual uncertainties. However, since this burette is uncalibrated, we simply add the uncertainties to arrive at the final result.

In summary, the total volume added is 30.80 milliliters, with an associated uncertainty of ±0.25 milliliters, reflecting the systematic error inherent in using an uncalibrated burette.

In a volumetric analysis, understanding the concept of uncertainty is crucial for accurate measurements. When using a Class A 250 milliliter volumetric flask with an uncertainty of ±0.15 milliliters and a 50 milliliter volumetric pipette with an uncertainty of ±0.05 milliliters, we can determine the remaining volume after removing aliquots.

Initially, the total volume in the flask is 250 ± 0.15 mL. If four 50 mL aliquots are removed, the total volume subtracted is 200 mL (4 × 50 mL), with an associated uncertainty of ±0.05 mL for each aliquot. The total uncertainty for the four aliquots is calculated by taking the square root of the sum of the squares of the individual uncertainties:

Let the uncertainty for the total volume subtracted be:

Uncertainty = √((0.05)² + (0.05)² + (0.05)² + (0.05)²) = √(4 × (0.05)²) = √(0.01) = 0.1 mL

Now, we combine this with the uncertainty of the initial volume:

Total uncertainty = √((0.15)² + (0.1)²) = √(0.0225 + 0.01) = √(0.0325) ≈ 0.181 mL

Rounding this to one significant figure gives us an absolute uncertainty of ±0.2 mL. Therefore, the remaining volume in the flask is:

Remaining volume = 50.0 ± 0.2 mL

To find the relative uncertainty, we divide the absolute uncertainty by the measured volume:

Relative uncertainty = (0.2 mL) / (50.0 mL) = 0.004

Thus, the final expression for the remaining volume, including both absolute and relative uncertainties, is:

50.0 ± 0.2 mL (absolute uncertainty) and 0.004 (relative uncertainty).

This example illustrates the importance of applying the correct rules for uncertainty calculations, particularly distinguishing between operations involving addition/subtraction and those involving multiplication/division.