# How do I analyse this data? (optimization experiment)

I conducted an experiment to find the optimal concentration for sugar in yeast respiration, and this is the data I found:

What do I do from here? Do I just fit a linear model on the increasing portion, another one on the decreasing portion, and see where they meet and say that's the optimal value? If not, do I try to fit another type of model? How do I justify that model?

• From the looks of it, you can try to fit a quadratic $-ax^2 + bx +c$ using least square or some such fit. The vertex of such a quadratic is $\frac{b}{2a}$ which is your optimal value – Shailesh Feb 15 '16 at 9:07
• Short answer: It depends on how accurately need that value and what do you use it for. Someone longer: Fitting this data is almost meaningless. You need at least 1-2 points between 50 and 100 to have at least a guess what you should fit. – Greg Feb 15 '16 at 9:58
• Since you say 'average' concentration of sugar solution and 'average' volume of water displaced, do you have more data? As @Greg said, having more points will make it clearer where the optimal value is. – N A Feb 15 '16 at 14:50
• @Greg I don't have any values for between 50 and 100. And yes, I can see what you mean by almost meaningless. I previously just had a linear line of best fit through the two sections (marked with different colours) and calculated where they met. I'm now considering instead to run a one-way ANOVA and then Tukey, but I'm not sure if that's correct. I'll ask in math stack exchange as well I suppose – hm527 Feb 16 '16 at 2:03
• Thank you all for answering! I will try those things and check with my teacher – hm527 Feb 16 '16 at 2:07

## 2 Answers

Some materials (e.g. ionic salts) in certain solvents (e.g. water) actually do decrease volume of the solution in certain concentrations (obviously at some point it inevitably increases).

So you certainly can end up with non-linear functions of displacement from $\Delta_{conc}$.

That said, a negative quadratic may fit your data well, e.g. $ax^2 + bx + c$, with $a < 0$. Maybe even a damped exponential, e.g. $ae^{-bx}$. You can use Microsoft Excel etc. for a quadratic fit or fancy math skills for other functional forms.

Another Ben describes in more detail here but the image he used is worth repeating. Here the concentration of an added substance actually shrinks/increases volume displacement "oddly".

As you add sugar the displacement will only increase. In other words, if you add 300 grams of sugar to a liter of water, the volume will have increased (or been displaced) by about 150 ml (guesstimated)... the displacement is certainly not 25 millileters!

For what it's worth, you should have two (practically) straight lines. One relatively straight line before the saturation point (at around 200g sugar per liter at room temperature) and another relatively straight line after the saturation point. But believe me, the more sugar you add, the greater the volume will be (until the volumetric cylinder is full).

Displacement in a nutshell:

• Isn't the water displacement a separate measure of the yeast respiration and not referring to the displacement of water due to sugar addition i. e. Not related to the density increase? – Beerhunter Jun 25 '16 at 22:02
• @Beerhunter At first, I wondered if he added the yeast and allowed the sugar to be fermented before measuring... then I figured that if he added 300g of sugar, it should be converted to around 180ml of alcohol... so however he came up with ~25ml it's simply a mistake. Part 2- Technically, displacement is related to density (see the story behind Eureka). Displacement is the measure of volume-increase when stuff is added to a liquid. So, I'm not sure what he did, but J.Ma should (re)measure of how much the water rose. – Ben Welborn Jun 27 '16 at 12:36
• It does not say anything about the volume to which the sugar is added to. Your statement in the first paragraph is maybe based on a litre of water. But if I add 30 g sugar to 100 mL water, I could very well have a displacement of 25 mL. – Martin - マーチン Jul 5 '16 at 17:51
• @Martin-マーチン okay, + for thinking outside the beaker... but the chart says 300 grams... so that's gonna add more than 25ml. 50g of sugar displacing 25 ml makes sense... 100g displacing 50ml makes sense... then you add more, and viola it collapsed? – Ben Welborn Jul 5 '16 at 17:55
• I feel like you are mocking me now and I can't really appreciate this. I am simply saying that you are basing your first statement on an assumption, you are over interpreting the given data. There's really no need to involve ninjas here. – Martin - マーチン Jul 5 '16 at 19:04