Hi all,
Suppose I have measured the same thing a few times, and each result already has an error associated with it. How do I average the results?
e.g. say I made 2 different dilutions of the same solution, and each dilution I made in triplicates. I first averaged the analysis results of the 3 triplicates of each dilution (multiplied the average by the dilution factor) and calculated the error.
say I got 14±2 and 12.5±1
(± being the confidence interval for an α that I chose)
The results are within error from each other, and I want to average them. What is the confidence interval on the global average going to be?
Should I just ignore the errors I already have, and look at it as a set of 2 measurements? (where in reality I made 6 measurements, so I'm much more confident in the average value, and would not like to have a large ± as I might get for n=2).
I went over several explanation in the net, but none referred to such case, except for one that mentioned that the error should go down by the square root of n. However, it was pretty vague and I'm not sure that errors and confidence intervals are the same in that sense.
Thanks!
Here are some ideas.
1. If you have all the raw/initial measurements separately, that you used to calculate the confidence intervals, you can just take all those measurements and find the overall confidence interval out of it.
2. If you measure everything using same instrument and expect the error will have the same distribution, rather same standard deviation, you can find the pooled standard deviation of the estimates. You have the $\alpha% confidence intervals, from there you can find your initial estimates of $\sigma$. Now you can use them to find the pooled variance and hence standard deviation, using the following formula,
https://en.wikipedia.org/wiki/Pooled_variance
Now your estimate of pooled mean is the mean of the means and you have the estimate of the pooled variance, you can use those things to find the confidence interval using the usual methods.
3. If all the means are computed with the same sample size, and the number of means or the number of confidence intervals you computed are moderate, you can just find the standard deviations of those means and use it to get the confidence interval.
Let me elaborate. For confidence intervals, we have
$\bar{X} \sim N(0, \sigma^2/n)$
and the confidence interval is $\bar{X}+-1.96*(\sigma/\sqrt{n})$.
Usually we have one sample, so we use the sample standard deviation as an estimate of sigma. However, in this case you have multiple samples. So you have multiple $\bar{X}$'s. So if you take the standard deviations of $\bar{X}$, call it s_{mean}, you can have the new confidence interval as
mean of $\bar{X}$+- (appropriate z value)*(s_mean).
It sounds like you are looking for the Standard Error of the Mean
https://en.wikipedia.org/wiki/Standard_error
If your experimental conditions are the same, then for a simpler approach I would suggest pool all the data together and calculate your statistics that are of interest.
The triplicates of each dilution should probably be treated as subsamples and not true replicates. Subsamples are valuable in that they provide us with a better average for that dilution, but the error amongst the triplicates is not the error of interest. You would then calculate your error (SD, CI) based on the two average values.
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