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unionised
International Hazard
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Quote: Originally posted by sbreheny  |
About pH - someone asked if I had measured the pH of my distilled water. I just did that. If I do it with just distilled water, I find the pH meter
very slow to respond and it is unclear whether the result is accurate. I get about 6.26.
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It is difficult to measure the pH of unbuffered solutions (and it's impossible with an ordinary pH meter)
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sbreheny
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Hi everyone,
I received my ultra-pure $25 per liter (!) water and repeated the experiment a few times. I put together a spreadsheet which automatically applies
several corrections and tells you the error from the ideal value. I have attached the spreadsheet (LibreOffice format) and here is an image of the
sheet:
<a href="http://s13.postimg.org/bwp92sl6v/table.png" target="_blank"><img src="http://s13.postimg.org/bwp92sl6v/table.png" width="800"
/></a>
The spreadsheet doesn't handle significant figures very well (or at least I don't know an easy way to get it to do so), so some of the numbers given
are ridiculously long.
I have 5 instances of the experiment recorded, three with high purity water and two with Target distilled water. There is no noticeable difference
with my equipment between the high purity and simple distilled (as expected). All 5 readings, now that I am much more careful about drying out the
flask and I add in the air buoyancy correction, are within the tolerances of my flasks.
Temperature corrected volume is the volume of the flask assuming that it is at the same temperature as the water, using a standard correction formula.
Balance correction factor is the ratio of 50 grams to the reading of my balance when I weigh a class 1 50 gram weight. Balance discrepancy and volume
discrepancy (at the far right) are values of balance error and flask volume error, respectively, which would explain the entire observed error if that
error were solely due to error from that instrument.
Thanks again for your help and for listening!
Sean
Attachment: water density.ods (15kB)
This file has been downloaded 411 times
<!-- bfesser_edit_tag -->[<a href="u2u.php?action=send&username=bfesser">bfesser</a>: reduced
image width, linked to full-res.]
[Edited on 21.2.14 by bfesser]
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bfesser
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Is this more like what you were going for?
<table><tr><td> Attachment: water_density_2.pdf (25kB)
This file has been downloaded 542 times</td><td> Attachment: water_density_2.ods (15kB)
This file has been downloaded 409 times</td></tr></table>
By the way, simply changing the extension from .jpg to .png doesn't make a PNG—they're totally different compression schemes.
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sbreheny
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| Quote: Originally posted by bfesser | Is this more like what you were going for?
<table><tr><td> </td><td> </td></tr></table>
By the way, simply changing the extension from .jpg to .png doesn't make a PNG—they're totally different compression schemes.
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Hi bfesser,
Thanks for your suggestion. No, what you did is not exactly what I am looking for. For one thing, the percent error is off by a factor of 100 (100x
too high). Also, some of the significant digits are actually hidden now. I know that I could manually adjust the format of each cell to reflect the
number of sig figures but what I meant is that I am not aware of a way to make a typical spreadsheet automatically show the correct number of sig
figures of a computed value given the number of sig figures in the input values.
JPG vs PNG - yes, I know what you mean. The original image is indeed a PNG file. When I uploaded it to that image hosting site for some reason it
converted it to a JPEG format but left the extension as PNG. Sorry for the confusion.
Sean
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AJKOER
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| Quote: Originally posted by AJKOER |
A few comments from the point of view of statistical analysis. In statistical sampling theory for ratio estimators (the density is a ratio estimate),
the common measure employed is the ratio of the means of two variate and not the average of individual ratios as the sampling variance of the latter
is much greater. It is, however, a bias estimate which can be adjusted. For an in depth discussion of various estimators and their respective standard
error see "Advances in Sampling Theory-Ratio Method of Estimation" by Hulya Cingi, Cem Kadilar at http://books.google.com/books?id=ORy83SaeWqgC&printsec=f... .
Now, in the case of your experiments, I will assume that the sample ratio (r), calculated as the sum (or average) of the respective water weights
divided by the respective volume measures, would be .9936. This statistic is, however, bias and needs to be corrected (see good discussion at
Wikipedia at on Ratio Estimator at http://en.wikipedia.org/wiki/Ratio_estimator ).
If mean of the weights and volumes employed are both greater than 10 (that is, the volume of water measured each time is over 10 cc), I would
recommend using the bias correction formula specified in Wikipedia (using a somewhat larger sample size) where the density estimate should have an
error in the order of at most 1/n cubed.
Any error observed in excess of what is expected per sampling theory should then be attributed to experimental design.
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Based on my prior statement (above) and the fact that your sample size is 5, my suggested density sampling estimate would still have a statistical
error in the order of (1/5)^3 or .008, which is very large.
As your current perceived error is less than this, I continue to recommend an increased sample size if you intend to present the results of
experiments formally. For example, with 10 samples, the error with respect to purely statistical sampling error (employing the estimator and bias
correction factor I suggested) would be in the order of (1/10)^3 or .001.
If, with an increased sample size and your indicated error is too large for sampling error alone, there could be potentially a systematic experimental
measuring error (to be determined) accounting for the difference. Else, your done, your estimate is statistically within tolerance levels.
---------------------------------------------
The simple average of the observed ratio, the so called index method, is generally bias and subject to large variability. Source, please see "Ratios:
A short guide to confidence limits and proper use", by V.H. Franz, October, 2007, available at http://arxiv.org/pdf/0710.2024.pdf). To quote from page 12:
"Also, if the mean ratio r-bar is used as a point estimate for rho it shows systematic biases and can be much more variable than the ratio of the
means"
The author also notes on pages 23 and 24, to quote:
"The index method is used very often (almost all of the example studies in the supplementary material provided with this article used this method). We
can justify the method in the context of a linear model if the denominator [the volume in our case] is bounded away from zero and if the data have a
specifc heteroscedastic structure, such that the numerator [measure of mass] has larger variability at larger values of the denominator [or volume, in
our case] .....
Because the method is used so often and because it seems unlikely that the data in all these cases show the specific heteroscedastic structure."
Now, in the current context, I am not sure if this is indeed completely correct. I expect the error in measuring the volume to be independent of the
magnitude of the volume. The error in weighting may,or may not, be proportional to the weight (and its volume as required by the modeling assumption).
Repeated weighting of the same set of water volumes varying in size, with different scales, may give insight as to if and how the error variance in
determining mass is related to volume.
----------------------------------------------------------------
If you continue to believe in small sample based quotients as a reliable indicator of a population mean, try a spreadsheet simulation. The model would
be a regression through the origin where x is a selected from a distribution of possible masses, and y is constructed from the product of the x with
the known population density plus, say, a Normal distribution error term with specified mean of zero and standard deviation based on your data.
Tabulate the generated densities (y/x ratios) and store to compute the mean bias, associated standard error, the median absolute deviation,...over
your simulation run (several thousand).
If you need to do some research on constructing such a simulation, the topic is Monte Carlo simulation techniques.
[Edited on 26-2-2014 by AJKOER]
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