How to find a formula relating three values?

Question

I haven't studied maths since high-school (20 years ago) and would like to find a formula to relate these values:

 x     y      z
-----|------|-------  
   1 | 0.25 | 0.25
   2 | 0.25 | 0.625
   3 | 0.25 | 0.75
   4 | 0.25 | 0.8125
   5 | 0.25 | 0.85

 1.5 | 0.25 | 0.5
  10 | 0.25 | 0.925
   2 | 0.1  | 0.55

The formula would find z from known x and y values.

e.g. if x = 6 and y = 0.25 what is z?

Thanks for any help solving the problem or pointing me in the right direction. I can try to explain the context if that helps, but perhaps it's easier just to work with the numbers.

Answer 1

Welcome back to maths. I hope the the re-introduction will not be too fearful!

Visualizing 3-dimensional data is very hard.

Whenever possible, try to start with the simplest model possible to describe your situation and build upwards. Thus, as Gerry suggested, given that for all but one data point, the y-value is constant, consider the scenario if you temporarily ignore it and consider this a 2-dimensional question.

In this case, graphing $x$ versus $z$ on graph paper or via Excel should always be your first step.

In my experience as a mathematician and statistician, I always find it useful to mentally think what do I expect the answer to be, before I plug my data into any formulae or models.

Once you do this you will immediately realize how much your point $x=10, z=0.925$ is separated from all your other data points.

The second thing to notice is that if you ignore this point, too, then a line fits the rest of your data quite well. Using something like Excel, with its built-in capabilities of line-of-best-fit, you would get something like $z=0.15x+0.22$. This would give value of $z=1.12$ if $x=6$.

However, you will notice that the if you include that far-right point, the data does not really look like a straight line at all, but rather a curve (maybe a logarithmic graph). In this case you would probably get a $z$ vaue of $\simeq 0.9$ if $x=6$.

This is where you have to balance up two questions:

1. Based on how I collected my data, would I expect to be a simple linear relationship or a more complex curve? If so, would I expect it to curve upwards or downwards. Other questions like the following may also help: "Do i expect the line-of-best fit to go through the origin?", "Can we have negative $x$ values, and/or negative $z$ values?, "What might I expect for very large $x$?"

2. How much confidence do I have that the point x=10, which lies much further away than the other points, is correct? Can we obtain more data in between $x=5$ and $x=10$ to strengthen our confidence?

Good luck.