[Compaq]

I'm writing a lab report in an analytical chemistry course I'm taking, and I'm having troubles with significant numbers. I understand its extreme importance, so please make this clear to me!

Okay, so we've done an experiment in which the goal was to compare the the effect of doing a solid phase extraction before HPLC as opposed to just doing HPLC. A total of six equal samples were analysed, three of which underwent solid phase extraction before the HPLC analysis. We should compare the estimate on concentrations in these two groups. Please note that we're not using formal, statistical analyses, just mean values and standard deviations.

So, calibration curve is found, area from the samples is inserted to the formula and the average values from the two groups (n=3), is as follows:

(mean)[Cafestol]_SPE-->HPLC = 9,1473 ug/mL
sd = 0,028

Estimate = 9,1 ± 0,028

(mean)[Cafestol]_{just HPLC} = 5,926 ug/mL
sd = 3,030
comment on sd: two concentrations were similar, one deviated very much.

This introduces the problem I'm having with significant numbers. How to I report the estimate on the concentration when the first digit in sd is the first digit in the mean value? Do I estimate the concentrations to be 5 ± 3,0? Do I round down to 0??

It seems to me that there have been some error here, as the six samples were initially equal. Maybe we've diluted incorrectly. Something to report in possible sources of error..

Anyway, could anyone please help me concerning the significant number situation? Our professor is pretty nazi in this, and as this is the fifth lab report in the course, I possibly should have understood it already.

Why don't I ask my professor? The report is due tomorrow 23:59

Thanks in advance, guys!
Anders

[Borek]

Significant digits are only an approximation, poor cousin of the real thing. I would list results for the first case - with so low standard deviation - as 9.147±0.028, and (to be consistent) for the other as 5.926±3.030. But from your description it is obvious that the second result is not worth much.

Sadly, chemistry students are told significant digits are verrry important, while in reality they are not. Physicist - that have to report their experimental data as often as chemist do - don't pay special attention to significant digits, instead they pay much more attention to the statistic and error propagation in their calculations.

[Arkcon]

Why are you rounding to two significant figures in the first place?  What is it, in this experiment, that makes the other digits not significant?  Generally, for real samples, a measurement lacks significant figures because the instrument has a limitation.  But I don't see that here.  You report average and  S.D to to sig figs, why did you do that?  You see, if you had a good reason to do that, the explanation would tell you how many sig. figs to report.

[Compaq]

Hmm, not sure I'm completely following. I'll provide more data.

Standard curve: x = y/0,57

Sample 1 Sample 2  Sample 3 Sample 1 SPE Sample 2 SPE Sample 3 SPE HPLC graph area [mV]    5,203     5,233         5,206         4,323         4,427            1,384        Concentration  [ug/mL]  9,128     9,181         9,133         7,584         7,767            2,428

Average concentration for HPLC only = 9,1473
Standard deviation = 0,0289

Average concentration for SPE + HPLC = 5,9263
Standard deviation = 3,0309

NOTE: originally, I got ten decimals from the chromatograph, but shortened for space issues here.

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As for the significant digits, maybe the focus on them is to make the students aware of experimental error always exists, and that no number can be trusted fully. In either way, I need to understand how it works.

The way I've understood it, is that the first non-zero digit in the SD tells us the first "un-significant" digit in the average. So, including any other digit than the the secure ones, is "wrong" and un-professional and mis-leading. My secure digits in the mean(HPLC), is 9 and 1, because there are zeroes in the 1-position and the 10th position in the SD. Therefore, reporting 4, 7 or 3 isn't correct. Instead I report 9,1 ± SD, as all digits in the SD are significant.

This is where the logic fails me when I'm to report mean(SPE+PLC), because, at least in my mind, my argument implies that I should report the concentration as 0 ± SD, if I'm to be consistent in my notation.

[Borek]

The way I've understood it, is that the first non-zero digit in the SD tells us the first "un-significant" digit in the average.

Yes - but you know how "unsignificant" these digits are, showing them with their uncertainty doesn't mean you suggest they are accurate, as you are defining the range in which to expect the correct value. Sure, reporting something like 2.67896789234±0.006798236321 doesn't make sense - I would report standard deviation with 2 or 3 digits and the average in such a way that these digits from the standard deviation are all "anchored" in the average, so perhaps 2.6790±0.0068 or 2.67897±0.00680 (note rounding).

To be honest I don't know how many digits to show in the standard deviation, I don't think there are any serious rules. 2 or 3 just sounds reasonable and most physics constants are reported this way (apart from those defined as being exactly equal to something).