# The more decimal places, the better?

#### Wrichik Basu

Gold Member
2018 Award
A paper has been published in Phys. Rev. Educ. Res.:
Karel Kok et al. Phys. Rev. Phys. Educ. Res. 15, 010103 – Published 7 January 2019
The Abstract says:
In this study with 153 middle school students, we investigate the influence of the number of decimal places from the reading of a measurement device on students’ decisions to change or keep an initial hypothesis about falling objects. Participants were divided into three groups, introduced to two experiments—the time it takes a free falling object with a zero, and a nonzero initial horizontal velocity to fall a certain distance—and asked to state a hypothesis that compares the falling times of the two experiments. We asked the participants whether they wanted to change or keep their initial hypothesis after they were provided with data sets. Members of each group were given the same number of measurements but with a different number of decimal places. Results show that for an increase in the number of decimal places, the number of participants switching from a false to a correct hypothesis decreases, and at the same time the number of students switching from a correct to a false hypothesis increases. These results indicate that showing more exact data to students—given through different resolutions of the measurement device—may hinder students’ ability to compare data sets and may lead them to incorrect conclusions. We argue that this is due to students’ lack of knowledge about measurement uncertainties and the concept of variance.
The paper is open Access, so it's worth to read through the sections Discussion and Implications for Teachers in the article text.

I remember that my classmates back in high schools had similar notions - often, when the reading of the optical bench would be between two marked places, they would take it as 0.05cm. That is, if the reading was between 13.6 cm and 13.7 cm, they considered it as 13.65 cm.

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#### kuruman

Homework Helper
Gold Member
A favorite student answer to the question "Suggest ways to improve this experiment" is "More accurate measuring equipment" as if that would compensate for poor technique or inability to identify sources of systematic errors not to mention misunderstanding the scope and goals of the experiment. When I taught advanced laboratory on the first meeting I would present this hypothetical picture to the students.

Suppose you had a magic lamp that controlled a genie who can quote the numerical value of any measurable quantity to infinite accuracy. Would you use it? Be careful what you wish for. For one thing, infinite accuracy means infinitely many numbers to write down which will consume the rest of your life. OK, then let's consider stopping at some number. What's a good place to stop and why?

That started the conversation going and cast doubts on their perceived need for "more accuracy". They were guided to the realization that measuring instruments are like genies programmed to stop after a given number of figures; all they had to do is guess that number.

#### Nik_2213

IIRC, we had 'significant figures' and 'error margins' hammered into our hapless little student heads...

Mind you, when many of our calculations were done with 4-figure log tables and linear slide-rules, getting too many DPs in the answer was easy to spot !!

Given serious pocket calculators, long-FP software and data-acquisition instruments that could record eg an HPLC eluent peak to dozen-digit precision, more thought was required...

Accuracy of said dozen digit peak ? Well, that's why we ran a bunch of standards before, then pairs between our samples, did statistics to be sure, to be sure they were 'good enough'.

#### 256bits

Gold Member
Why was "Graph the Results" not a given option for the of the analysis of the data?
Is that method being discouraged?
I would have thought that with any number ( at least to a reasonable degree that can be discerned on graph paper ) of decimal places, the evidence would have been more obvious to accept or reject a hypothesis for those individuals with limited statistical analytical tools and comprehension. ( grades 8 to 10 in the study ).

Since most individuals can remember not more than 3 numbers at a time ( or numbers in groups of three ), initial visual inspection of the tabulated data could initially lead to erroneous bias ( as mentioned in the study limited decimal places has its own problems with perceived variance and acceptability ). More than 3 digits, especially when several digits may change between trials would just lead to confusion for most individuals as to what the tabulated data set represents.

As mentioned also in the study, students of this age are familiar with point data much more than set data.
It would be worth testing college students and see what turns up there.

#### rsk

I find (high school) students (1) have far too much faith in digital readings and (2) are scared to discount extra decimal places or significant figures in numbers - in calculations as well as in their own experimental data. They think more sf means more accurate and it takes time for teaching to counter that. In the context of this study I'm not in the least surprised at the outcome - the vertical fall vs the horizontal launch is something they find difficult to accept anyway.

I recently had a good class of 14-15 year olds calculate the speed of light in some kind of material, given the speed of light as 3.0 x 10^8 and a RI of 1.3 - at least half of them gave me an answer of 230769230 rather than 2.3 x 10^8

Introducing the idea of % difference, % uncertainty etc can help - our curriculum introduces this from age 16 but it's helpful to do it earlier so that they begin to see that those later figures really are less significant. I think my younger students might have said the same as those in the study but the older ones, who've looked more at experimental uncertainty, would have made a better interpretation of the data.

#### mathwonk

Homework Helper
In this vein, I now prefer to think π = (roughly) 3.141592653589793..... (I hope), and am greatly awed by my granddaughter who knows a version with some 50 more decimal places. Next I aspire to proving it. My feeble attempts so far have seared a profound respect for the prowess of the illustrious Euler into my consciousness.

#### James Pelezo

Gold Member
For what it's worth, I find dividing measurement accuracy & precision into two general areas, experimental and theoretical. Experimental accuracy & precision is limited by the devices used to obtain the measurements. Students are asked to report their concluding values to the accuracy of the least accurate device used in the laboratory experiment. Such is established in the opening question-answer session for the lab plan and removes the ambiguity in deciding how to report ones final results. Also makes the conclusion section of the lab report easier to grade as all students of the class know the limitations and application of the equipment they are using. For theoretical accuracy & precision students are asked to apply the definition & rules of significant figures; i.e., all digits know with certainty plus one uncertain digit when computing theoretical results.

"The more decimal places, the better?"

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