Say I have multiple samples and I want to calculate the average amount of protein X present in each of them. However, to measure the quantity of protein X in each sample, I took three readings and averaged them. If I then average my measured quantities across each sample, each value is now an average itself and has its own standard deviation. What is the appropriate way to calculate the overall standard deviation?
This is a very common situation in experimental science because it's often a good practice to take multiple readings when making measurements. In the past, I've often ignored the in-sample standard deviations as they're often small enough to be irrelevant, but sometimes they have a lot of error and I haven't been able to figure out how to deal with that.
Your aim is to give a measure of precision/variability between (independent) specimen that can be expected when someone repreats your experiment. Unfortunately, there is not only (biological) variation between specimen but there is also some technical variation ("measurement error"). When you take a measurement of two (similar, but independent) specimen, your recorded values will differ by the sum of both sources of variation (biological + technical). But you are actually interested only in the biological variation.
Sometimes, the contribution of the technical variation is negligible and can simply be ignored. But sometimes, the technical variation is considerable and possibly disturbing. In such cases, one can take several measurements from the very same specimen, so that there is only technical variation between these measurements. By averaging, you "average out" a part of the measurement error. The rule is: with averaging n technical replicates you decrease the measurement error by a factor of sqrt(n) (e.g. averaging 4 replicates reduces the measurement error in the averages to one half, compared to single measurements).
You take the entire process of meauring a specimen several times and averaging these values to provide a single (averaged) value as one step. You are actually not interested in their variance (unless your aim is to compare different processes/technologies/assays) so all you need is this one averaged value per specimen. This is your read-out. The whole process, including the information of how many technical reps were averaged, should be described in the methods, so that this is replicable. Note that this process can comprise more than just taking a measurement. It can (and should) include further preparation steps that are likely to give variable results.
You then proceed with these values in your analysis. As simple as that.
It is important to use only the averages in hypothesis tests. The test expect that each value provided is independent, and only these averages are independent (because these are from independent specimen), but the technical reps are not. Technical reps in tests are called "pseudo-replications". The test falsely "thinks" you have much more independent information that you actually have and will overestimate the statistical significance of your data. So this is a no-go.
The story becomes more complicated when you are really interested in dissecting the different sources of variability, e.g. to optimize an assay (treatment, sample prep, measurement). You will need (rather many) replications at each level, and then you can model the different sources of variability in an appropriate hierarchical or multilevel model (including random terms to account for the variation introduced bin the different levels). But this is advanced stuff.
- Badges
- Science topic
- Similar topics
- Biological Science
- Molecular Cell Biology
- Culture Cells
- Cell Line