11.1.4. Quantifying Uncertainty in Reported Measurements

This is the uncertainty in the measurement expressed in the same units as the measurement itself.

For Analog Instruments (e.g., ruler, thermometer, burette): The absolute uncertainty is typically taken as ± half of the smallest scale division. For instance, a thermometer marked in 1 °C increments would have an uncertainty of ± 0.5 °C. A burette, read to two decimal places (e.g., 23.45 mL), means the smallest division is 0.1 mL, so the uncertainty is ± 0.05 mL for a single reading. Since a burette measurement involves two readings (initial and final), the total uncertainty for a delivered volume is often taken as the sum of uncertainties for two readings, i.e., ± 0.10 mL.

For Digital Instruments (e.g., electronic balance, pH meter): The absolute uncertainty is usually taken as ± the smallest scale division (the precision of the last displayed digit). For example, a balance reading to two decimal places (e.g., 2.50 g) has an uncertainty of ± 0.01 g.

Absolute uncertainty is a way of conveying how precise a measurement is by including a range of possible values around the reported measurement. For analog instruments, it's often calculated as half of the smallest division an instrument can measure. For example, if a thermometer can measure in increments of 1 °C, the uncertainty is ±0.5 °C. In the case of a burette, which has a resolution of 0.1 mL, the uncertainty becomes ±0.05 mL for a single reading. When using the burette for a volume measurement that includes two readings (like initial and final liquid levels), the uncertainties are added together, leading to a total uncertainty of ±0.10 mL. Digital instruments typically present a clearer uncertainty based on the last digit shown, which represents the smallest increment the device can measure, like ±0.01 g for a balance showing 2.50 g.

Think of measuring ingredients while cooking. If you're using a measuring cup marked in 1 mL increments, you might estimate a liquid at 50 mL, but really, it could be anywhere between 49.5 mL and 50.5 mL. Just like that, when a scientist measures something, they know the measurement isn't perfect, so they indicate a range (or uncertainty) to express how accurate they believe their reading is.

This expresses the absolute uncertainty as a percentage of the measured value. It provides a useful way to compare the relative precision of different measurements, regardless of their magnitude.

Formula: Percentage Uncertainty=(Measured Value−Absolute Uncertainty)×100%

Example: A mass of 2.50 g measured with an uncertainty of ± 0.01 g has a percentage uncertainty of (2.50 g/0.01 g)×100%=0.4%. A volume of 25.00 mL measured with an uncertainty of ± 0.05 mL has a percentage uncertainty of (25.00 mL/0.05 mL)×100%=0.2%. In this example, the volume measurement is relatively more precise than the mass measurement.

Percentage uncertainty is a valuable metric for expressing how significant the uncertainty is compared to the actual measurement. It's calculated by taking the absolute uncertainty of a measurement, dividing it by the measured value, and then multiplying by 100 to get a percentage. This helps to compare the uncertainties of different measurements regardless of their magnitudes. For instance, if you have a mass of 2.50 g with an uncertainty of ±0.01 g, you find the percentage uncertainty by dividing the uncertainty (0.01 g) by the measured value (2.50 g) and multiplying by 100, which results in a 0.4% uncertainty. In contrast, a volume measurement of 25.00 mL with an uncertainty of ±0.05 mL yields a percentage uncertainty of 0.2%, indicating that this measurement is relatively more precise than the mass measurement.

Imagine you have two jars of money. One jar contains $100 with a possible counting error of$1 (which is a 1% uncertainty). The second jar has $1,000 but also has a possible counting error of$10 (which works out to 1% too). If you counted the first jar as being more precise because the absolute uncertainty is smaller, you would be mistaken; both jars are equally precise in percentage terms. Knowing this allows you to judge the reliability of each amount of money better.

When experimental values, each with its own uncertainty, are used in calculations to derive a final result, the uncertainties must be combined or "propagated" to determine the overall uncertainty in that final result.

1. For Addition and Subtraction:

• When quantities are added or subtracted, their absolute uncertainties are added.
• If C=A+B or C=A−B, and A has absolute uncertainty ΔA and B has absolute uncertainty ΔB, then the absolute uncertainty in C is: ΔC=ΔA+ΔB
• Example: If a beaker weighs 50.15 ± 0.01 g and the beaker plus contents weighs 52.34 ± 0.01 g, then the mass of the contents is 52.34−50.15=2.19 g. The absolute uncertainty is 0.01+0.01=0.02 g. So, the mass of contents is 2.19 ± 0.02 g.

2. For Multiplication and Division:

• When quantities are multiplied or divided, their percentage uncertainties are added.
• If C=A×B or C=A/B, then the percentage uncertainty in C is: Percentage Uncertainty in C=Percentage Uncertainty in A+Percentage Uncertainty in B
• Once the total percentage uncertainty for the result is calculated, convert it back to an absolute uncertainty for the final answer.

3. For Powers (and Roots):

• If a quantity is raised to a power, C=An, then the percentage uncertainty in C is the absolute value of the power multiplied by the percentage uncertainty in A.
• Percentage Uncertainty in C=|n|×Percentage Uncertainty in A.
• Example: If a volume is 2.0 ± 0.1 L (5% uncertainty), and you calculate its square for some reason, the percentage uncertainty in the squared volume would be 2×5%=10%.

When doing calculations that involve measurements with uncertainties, it's essential to propagate these uncertainties to understand how they affect your final result. There are specific rules for different operations. When adding or subtracting, you simply add the absolute uncertainties. For instance, if you weigh a beaker and find it to be 50.15 g ± 0.01 g and then add the contents weighing 52.34 g ± 0.01 g, the mass of the contents would be calculated as 2.19 g ± 0.02 g, since you add the uncertainties together. For multiplication or division, you add the percentage uncertainties instead. Lastly, when dealing with a measurement raised to a power (like squaring a quantity), the percentage uncertainty for that result is calculated by multiplying the percentage uncertainty of the base measurement by the exponent. For example, squaring a volume measurement of 2.0 L with a 5% uncertainty gives you a 10% uncertainty for the squared volume.

Imagine you're baking and need to combine flour and sugar. If you know that you have 200 g of flour with a +/- 5 g uncertainty, and you're adding 150 g of sugar with a +/- 3 g uncertainty, you can think of their uncertainties in terms of the total weight of your ingredients. In the end, when you calculate your final dough's weight, you need to account for how much error there could have been in measuring both ingredients combined. It’s like measuring with a tape measure: every time you pull it out, it could be slightly off due to how you read it; likewise, the same concept applies when measuring weights or lengths in any recipe.

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