While reporting data along with standard error, e.g. 11.2 ± 0.33 g/L should we be more concerned with number of significant figures or the decimal places? Is there a format or rule of thumb?
I found an article stating, " round to one more decimal place than your original data. If your original data were mixed, round to one decimal place more than the least precise." Its according to suggestion by Sullivan, Michael, Fundamentals of Statistics 3/e (Pearson Prentice Hall, 2011) page 118. However, it comes with a disclaimer, " It won’t always be strictly correct according to the rules of significant figures, but it’s usually right for data sets of size roughly 10–30." http://www.tc3.edu/instruct/sbrown/stat/rounding.htm
I have a set of triplicate data. Is it okay to use this rule of thumb?
There is a detailed explanation by Mr.Jochen Wilhelm. The number of decimals should be logically linked to population size. As the size of the population increases by a jump of ten on th right side one can add a decimal point on the right. The degree of absolute precision increases.
Standard error is reference to the sample observations.Given a finite population of 'N', a sample of size n-statistically large or small- will help in the estimation of precision of sample estimates to population size. Standard error(s.e.) is a measure of precision rather than of accuracy.
As such, it is advisable to use up to two decimal places & use round off principle as has been normally done. I do not understand mixing of data of original one into the new ones.If there is a definite connotation of data mixing please let me know. One caution is s.e. is not free from units of measurement. However, we expect s.e. to be as small as possible for sample to be precise.
I'm afraid my answer isn't going to provide clear definitive answers, but hopefully explain the background to that rule of thumb to help you work out how to apply it.
Standard practice would be to round to one level lower than needed to get a reasonable estimate of precision (which is the purpose of the rule of thumb you cite). The purpose is to show clearly that you are not trying to mask the performance of your analysis, but allow readers to see that you are presenting it to a level of precision that allows them to evaluate if it performs sufficiently well.
On the flip side, providing an error estimate to too high a level of precision can be misleading if your certainty in that error is low. If you have only 3 samples and you cite your error to many decimal points it can mislead the casual reader into thinking it was better defined than it is.
You mention triplicates, but are these triplicates of multiple samples so that you are in fact estimating your sample error multiple times and not just 3 samples? Or are there indeed only 3 measurements contributing to the error estimate? If the latter then you estimate of error is going to be highly uncertain so the presented value should not appear too precise.
If a difference of 0.1 matters but 0.05 doesn't, round to 2 decimal places. For values greater than 10 if the difference that matters is 1000 only round the last two digits to zero.
If you want to try applying these rules to your data I suggest trying to imagine you are critically reviewing your arch rival's work. Would you feel that the data was presented in such a way as to allow realistic interpretation of performance?
The standard error (SE) is itself a (kind of) measure of precision. There don't need to be more than 2 digits of the SE be considered significant. The second digit is already completely within the margin of error, and (roughly) 5-10 times smaller than the precision of the estimate, as supported by the observed data. I can't imagine a reasonable argument why more digits should be provided.
Starting from this point (the number of significant digits of the SE), one can find how many significant digits the estimate should have - and this should be one digit less.
Examples (there are always 2 significant digits of the SE given):
0 ± 330 (huuge SE, the estimate of about 11 is irrelvant)
10 ± 33 (SE in the same magnitude of the estimate; only magnitude matters)
11 ± 3.3
11.2 ± 0.33 (correct as given in the question)
11.24 ± 0.033
11.237 ± 0.0033
...
There is a detailed explanation by Mr.Jochen Wilhelm. The number of decimals should be logically linked to population size. As the size of the population increases by a jump of ten on th right side one can add a decimal point on the right. The degree of absolute precision increases.
Hello, Saumya! :-) The reference you cite is correct. All results reported should be based on # of significant digits, not # of decimal places (neither of which is associated with sample size nor population size). The reason for using significant digits is to *not* imply more precision ("exactness") in the calculated results than is associated with the *original measurements.* If the original measurements have two digits, results should *not* be reported to more than three digits (no matter the number of decimal places that results in - if any). If the sample (or population) is "large" reporting results with four significant digits would be as if you somehow *increased* the precision of the original measurements just by performing calculations!
I also found two articles claimning that in multiplication of divisions, the final results can be the same or adding one more digits than the minimum number of significant figures. One example in the appendix A in the book "The Uncertainty in Physical Measurements" by Paolo Fornasini, 83.642/72 = 1.1616944 should be written as 1.16 rather than 1.2. It said that the reason is to keep the uncertainty at similar levels. In another book written in Chinese, it recommend that when the most significant digit is greater then 3, results should keep at the same number of digits, but when it is 1 or 2, the results should have one more digits. But the reason of those two methods don't match. I'm confuse too!
The general rule is to use no more significant digits in a (calculated) result than the number of significant digits in the datum with the least number of significant digits. Some people "relax" this rule by using one more digit in the calculated result than in the datum with the least number of significant digits.
https://www.researchgate.net/post/What_is_the_acceptable_expression_for_the_uncertainty_of_a_measurement
The reliability of the sample mean as an estimate of the population mean is not affected at all by the number of significant figures in the data from which the mean was calculated. The mean's reliability as an estimate (it's only purpose) is determined by the standard error. A good method to produce consistent results in reports, at least for life science data (Simpson, Roe & Lewontin, 1960, Quantitaive Zoology) is to first divide the standard error by 3 and note the decimal place of the first non-zero digit. Then, round the reported mean to that place, and the reported standard error to one more place.
The number of digits has nothing to do with the *reliability* of any reported results...it has everything to do with conveying the precision and accuracy of the result. Reporting results with more digits than the original measurements implies greater precision and accuracy that does not exist.
Wayne Fischer : Sir can we use z-scores to "zero decimal place (integer)" for estimation in model as factor scores?
I'm not sure I understand your question, Noman. Can you give a specific example?
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