Sig figs/certainty in percent error calculation

• Derrick Palmiter
In summary, the conversation discusses the application of significant figures in percent error problems. It is determined that leading zeros are never significant and significant digits are about relative precision. The precision of the result is of equal degree as the precision of the measurement. It is recommended to do proper error analysis for a more advanced understanding.
Derrick Palmiter
Hello, all. Quick question about how to apply sig figs in percent error problems. Eg. If the actual/target value is 1.95 g and we measure 1.87 g, then should the percent error of our measurement be reported as 4.10% or 4%? Normally, at least abstracting from the problem, after subtracting 1.95 from 1.87 for a difference of -0.08, the zeros would not be considered significant and we would consider this result to have one significant digit, but really, we have certainty about those zeros as values, correct? (So, I'm leaning towards saying three sig figs for the purpose of the subsequent division problem by 1.95. -0.08/1.95 = -4.10% and not -0.08/1.95 = -4%.) Does this seem like an appropriate interpretation of the situation to any of you, or am I forgetting something?

I know the rule technically says to count "placeholding" zeros as non-significant, but here it seems to me they are reporting data about the measurements used in the initial calculation, not just placeholding. Many thanks in advance for your help.

Derrick Palmiter said:
Normally, at least abstracting from the problem, after subtracting 1.95 from 1.87 for a difference of -0.08, the zeros would not be considered significant and we would consider this result to have one significant digit, but really, we have certainty about those zeros as values, correct?
Incorrect. Leading zeros are never significant. Only trailing zeros may be. Significant digits are not really about how many of the digits you have confidence in, it is about relative precision. If you have three significant digits and change the last by one, then the result should not differ more than 1% from your original number.

It is easier to think about significant digits in scientific notation ##x\cdot 10^y##.

As another example, you can write 143 as 0000000000143. It does not change the fact that there are three significant digits even if you know that the number is not 1000000000.

Please help me understand how this applies to this situation, for both the initial subtraction of 1.95-1.87 and the subsequent division by 1.95.

1.95-1.87 = 0.08, which is one significant digit.

Orodruin said:
1.95-1.87 = 0.08, which is one significant digit.
And that precision is relative to? The precision of the original measurement which is to the hundredths place? Is that your meaning? That the precision of the result is of equal degree as the precision of the measurement?

These are absolute numbers. Your initial measurements have an uncertainty of 0.01, the difference has a similar uncertainty.

0.01 is 1/8 of 0.08, so the relative uncertainty of your difference is of the order of 10%.

Derrick Palmiter
Look at the max and min for the difference:
$$\begin{array}{|c|c|c|} \hline nom&max&min \\ \hline 1.950&1.955&1.945 \\ \hline 1.870&1.865&1.875\\ \hline 0.080&0.090&0.070 \\ \hline \end{array}$$

WWGD and Derrick Palmiter
Can anyone recommend a good text/reading material that explains this concept in a bit more detail than a standard high school physics/chemistry textbook would? Many thanks to all involved for your assistance.

Significant digits is essentially poor-man’s error analysis. You will typically not find an in depth analysis of this as the more advanced thing would be to actually do proper error analysis.

gmax137

What are significant figures and why are they important in scientific calculations?

Significant figures, also known as sig figs, are the digits in a number that carry meaning and contribute to the precision of a measurement. They are important in scientific calculations because they help to ensure the accuracy and reliability of the results. Sig figs also allow for consistency when reporting and comparing data among different scientists and experiments.

How do you determine the number of significant figures in a measurement?

The general rule for determining the number of significant figures in a measurement is to count all non-zero digits and any zeros between them. For example, the number 123.45 has five significant figures. However, there are some additional rules for specific cases, such as trailing zeros after a decimal point or leading zeros before a non-zero digit. It is important to follow these rules carefully to ensure the correct number of significant figures is used in calculations.

What is the purpose of calculating percent error in scientific experiments?

Percent error is a measure of the accuracy of an experimental result compared to the accepted or expected value. It allows scientists to evaluate the reliability of their data and identify any sources of error in their experiment. By calculating percent error, scientists can make adjustments to their methods and improve the accuracy of their results.

How do you calculate percent error and what does a positive or negative result indicate?

The formula for percent error is: (|experimental value - accepted value| / accepted value) x 100%. A positive percent error indicates that the experimental value is higher than the accepted value, while a negative percent error indicates that the experimental value is lower. Ideally, the percent error should be as close to 0% as possible, indicating a high level of accuracy in the experiment.

What is the relationship between significant figures and percent error calculation?

The number of significant figures used in a measurement or calculation can affect the percent error. If the number of significant figures is too low, it can result in a higher percent error and vice versa. Therefore, it is important to use the correct number of significant figures in both the experimental value and the accepted value when calculating percent error.

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