I was recently asked this question:

An experiment is done 3 times, with a different measuring tool used for each time. The results of the experiment are 1.39, 0.9, and 1.0 (all centimeters). Which measurement is the most precise, and how do you know?

On my paper, I put that the answer was 1.39 because it had the most digits past the decimal point and submitted. However, I've read online and found since then that my explanation actually referred to accuracy, not precision. My question is, would saying "1.39 is the most precise because it has the most significant digits" be a correct answer?

  • 5
    $\begingroup$ The question on the test is poorly worded and of the answer you should have written is: Nobody can answer this question with the given information. If your teacher, disagrees, explain with respect that his/her answers are not correct. $\endgroup$ – M. Farooq Sep 1 '20 at 0:20

The single measured values provided by 3 different tools gives us just a hint about possible accuracy or precision of the tools. The tool providing the value 1.39 is probably, but not necessarily more precise and accurate than the other two.

What these values really say is just the tool resolution.

There are 4 terms:

  • resolution
  • precision
  • accuracy(1), also defined by ISO as trueness
  • accuracy(2), defined by ISO as combined trueness and precision

  • Resolution relates to the least distinguishable difference of values provided by a tool. A tool able to tell ..1.38, 1.39, 1.40.... has resolution 0.01. It is not the same as the tick mark distance, as even the decimal fraction between the marks can be estimated for analogue output ( not digital one ).

  • Precision relates to the random errors during repeated measuring of the same value by the same tool or measurement method at the same conditions. It is usually expressed as a estimation of the standard deviation or confidence interval. It can be absolute, or relative wrt the measured value. It is usually worse than resolution, unless the latter is not fine enough. E.g. the resolution may be 0.001, but precission +- 0.02

  • Accuracy(trueness) relates to the systematic error of the tool or method aka the bias. It is the absolute or relative error of the mean measured value wrt the true or known value. The acccuracy(trueness) error is usually, but not necesserily bigger than precission error, e.g. the precission can be +- 0.05, but the accuracy(trueness) can have bias +0.1.

As formulated by Wikipedia in above links:

In measurement of a set, accuracy is closeness of the measurements to a specific value, while precision is the closeness of the measurements to each other.

Accuracy has two definitions:

  • More commonly, it is a description of systematic errors, a measure of statistical bias; low accuracy causes a difference between a result and a "true" value. ISO calls this trueness.
  • Alternatively, ISO defines accuracy as describing a combination of both types of observational error above (random and systematic), so high accuracy requires both high precision and high trueness.

Precision and accuracy(trueness) are mutually independent. The former is related to random errors, while the latter is related to the bias of the measuring tool a/o method. The accuracy(ISO) can be expressed as error $+0.1 \pm 0.05$.


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