I'm a high school math teacher who is a bit stumped. A Biology student came to me with his experiment wanting to know what kind of statistical analysis he can do with his data (yes, he should have decided that BEFORE the experiment, but I wasn't consulted until after).
He is trying to determine what effect insulin has on the concentration of glucose in a cell culture. There are six culture grouped into three pairs (one with insulin and one without) each under slightly different conditions.
The problem is that he only took one sample from each so the there is no standard deviation (or the standard deviation is 0 since the value varies from itself by 0).
Is there any statistical analysis he can perform with this data? What advice should I give him other than to redo the experiment?
Best Answer
Unfortunately, your student has a problem.
The idea of any (inferential) statistical analysis is to understand whether a pattern of observations can be simply due to natural variation or chance, or whether there is something systematic there. If the natural variation is large, then the observed difference may be simply due to chance. If the natural variation is small, then it may be indicative of a true underlying effect.
With only a single pair of observations, we have no idea of the natural variation in the data we observe. So we are missing half of the information we need.
You note that your student has three pairs of observations. Unfortunately, they were collected under different conditions. So the variability we observe between these three pairs may simply be due to the varying conditions, and won't help us for the underlying question about a possible effect of insulin.
One straw to grasp at would be to get an idea of the natural variation through other channels. Maybe similar observations under similar conditions have been made before and reported in the literature. If so, we could compare our observations to these published data. (This would still be problematic, because the protocols will almost certainly have been slightly different, but it might be better than nothing.)
EDIT: note that my explanation here applies to the case where the condition has a potential impact on the effect of insulin, an interaction. If we can disregard this possibility and expect only main effects (i.e., the condition will have an additive effect on glucose that is independent of the additional effect of insulin), then we can at least formally run an ANOVA as per BruceET's answer. This may be the best the student can do. (And they at least get to practice writing up the limitations of their study, which is also an important skill!)
Failing that, I am afraid the only possibility would be to go back to the lab bench and collect more data.
In any case, this is a (probably painful, but still) great learning opportunity! I am sure this student will in the future always think about the statistical analysis before planning their study, which is how it should be. Better to learn this in high school rather than only in college.
Let me close with a relevant quote attributed to Ronald Fisher: