Why is it important to keep all variables the same except the independent variable?

Answer 1

In fact you can vary several variables. This, however, must be planned carefully in order to get results that can be unambiguously interpreted. For this reason the term DoE, Design of Experiments is used.

The term "independent variable" has a precise mathematical meaning: it means that the selected independent variables in an experiment are orthogonal, i.e. do not depend on each other. For example "up" and "right" are orthogonal, as you can go 1 m right but stay 0 m up (at the same level), but "up" and "60 degrees tilted downwards" are not, since you go about -0.32 m up (0.32 m down) for each 1 m directly downwards.

If we find 2 true independent variables, we're looking for a model where for a phenomenon A there is a function A = f(X, Y) where X and Y are variables. If A is a scalar this defines a "mountainy" surface of A values on the X-Y coordinate plane. Now a typical strategy is to find A for each pair of X and Y. For example A is altitude and X and Y are latitude and longitude. Then the best way to give a good general idea is to sequentially traverse each latitude, and record the altitude for each longitude. If the latitude is not kept constant, we gather inconsistently measured data that looks more like a shotgun target than a regular lattice. There are clusters where a lot of points randomly hit, and correspondingly and more seriously large voids about which there is no data. If latitutde goes unmeasured, then each altitude data point has little meaning despite accurate longitude readings, since the actual location of the data point isn't known.

In science it's more a rule than an exception that the choice of possible variables is large and correspondingly the space of possibilities defined by them is intractably large. Furthermore it is often not known which variables are independent and which are not. For this reason, better algorithms than "try out every possible option" are mandatory. Often this means selecting the most promising variables and "discarding" the rest. Alas, all of these possible variables may affect the result just as badly as a latitude left unrecorded in the previous example. Thus all variables that may affect the result must be either set constant or varied systematically.

If the variables are not limited, the phenomenon of combinatorial explosion occurs - the number of possible choices is the arithmetic product of the sizes of the ranges of the variables. In other words, the orders of magnitude are additive, adding 10 more data points from a new variable increases the data set size by 10-fold. That is, if the latitude varies by 180 degrees and longitude 360 degrees, there are 64800 distinct pairs of latitude and longitude one degree apart. But if we include for example local temperature, which varies by 140 C, then we have 9072000 possible latitude-longitude-temperature points one degree apart and temperatures differing at least by 1 C. As you can understand, if we start including things like pressure, solar radiation intensity, wind direction, humidity, etc., this weather simulation soon requires a supercomputer. It would be practically impossible to fit any simple function to this data (unless the variables weren't independent in the first place).

In summary, you will have to control all the data you put into your experimental subject in order to conveniently interpret the data that comes out of it, that is, isolate the effect of the experimental subject vs. the input data on the output data. If input data is inconsistent or unknown, you'll see the convolution of the bad input data and the effect of the subject in the output data, not just the effect of the subject, which you're interested in. Even more succintly, GIGO: Garbage In, Garbage Out.

Answer 2

You need to have only one independent variable so that your results are somewhat connected to that one variable. Does that help?