# Calculating errors from given graphs

In a recent experiment, I have found the viscosity of glycerol by measuring the time it took for steel balls of measured density and radius to fall a certain distance in the liquid.

Now, I need to compare this data with a theoretical value given by a certain graph of glycerol viscosity by temperature. I have measured the temperature in the lab and know the average temperature throughout the experiment and its error.

In order to compare the values, I need to know how to properly take data from a given (printed) graph and calculate the error of the value.

I don't really know how to approach it, and it is not something that has been explained previously (nor did I find much data online about lab statistics).

A possible method I thought about is to add the resolution errors of the axes and add the error in measured temperature, but that would give an answer with inconsistent units.

• There isn't a generally reliable method of extrating error data from somebody else's graph unless they indicate it explicitly on the graph. You have to scour the caption and text relating to the figure for whatever hints you've been given. Commented Jan 13, 2020 at 19:12
• Also, the printing of books and journal articles is notorious for distorting such graphs. So, unless the thing you are doing is explicitly accounted for, you should be prepared for it to be skewed. The graph may be offset (in either coordinate), or even tilted relative to what it should be. Commented Jan 13, 2020 at 19:19
• So I have no way of evaluating the accuracy of data? The graph is from the Handbook of Chemistry and Physics if that means something. Commented Jan 13, 2020 at 19:26
• I have to say, that you are using a CRC Handbook is information you should have included in the post. That's not some random source that no one will have heard of. It's one that every particpiant on the site over the age of ::mumble mumble:: will have made heavy use of at one point in their career. Commented Jan 13, 2020 at 20:04

From a comment.

So I have no way of evaluating the accuracy of data? The graph is from the Handbook of Chemistry and Physics if that means something

I said that you should look in the source material for hints. The CRC handbooks are authoritative sources (well, for the year they were printed, anyway) and are prepared by metrologists to a very high professional standard in reporting.

You can feel confident that the precision of their reported numeric values accurately represents the uncertainty of the measurements (and they will often give you explicit error estimates as well).

In this case you are not looking at error data in the graph, but at least one point in the graph will also appear in a table. Find it, and it will tell you the scale of the uncertainty associated with the graphed data.

If this is for homework, your best bet is to do the old "manual digitization" and then use the produced data itself as a source of error in the result. Scan the image into your favorite electronic device. Then put the graph into your favorite spread sheet program. Then build a polynomial graph that fits over the scan as closely as possible. Then use how much "tolerance" there is in the graph you create to estimate the error in the graph. That is, if you bump the polynomial parameters, how much can you bump them and still think the graph is fitted well by your polynomial? Then use the polynomial to predict the values where you need them, along with the error.

But as I said in my comment, be wary of the possibility the graph is distorted in printing.

If this is for research, probably you want to get better data for your standard. Depending on exactly what it is, you may be able to get some well-known standard that tabulates the viscosity for the material you are using. Maybe you can do some searching on the net, or on Google Scholar or some such. Maybe the CRC data handbook has some useful values.

https://en.wikipedia.org/wiki/CRC_Handbook_of_Chemistry_and_Physics

Maybe your prof has a CRC handbook in his office? Or your school library?

• Thank you. This is for homework, and the method you've shown is clearly beyond what's expected of us. Commented Jan 13, 2020 at 19:29
• The data itself is from the CRC handbook, but from previous experience with it I didn't find there references for the errors of their data (which is what I'm looking for the comparison) Commented Jan 13, 2020 at 19:30
• Well, in that case, you could do the old ruler-and-scale thing. Measure the full scale. Measure the elevation of the curve at 2 or 3 points. Make a linear or quadratic fit to those. Use the usual half-the-smallest-division rule for errors on the ruler. Commented Jan 13, 2020 at 19:31
• And how would the error of the measured temperature play in it? As the choice of x-axis value for which to look is itself inaccurate. Commented Jan 13, 2020 at 19:33