Hello,
I have the following problem. I have made three measurements of the same event under the same measurement conditions.
Each measurement has a unique probability distribution. I have already calculated the mean and standard deviation for each measurement.
My goal is to combine my three measurements to get a general result of my experiment.
I know how to calculate the combined mean: (x_comb = (x1_mean+x2_mean+x3_mean)/3)
I don't know how to calculate the combined standard deviation.
Please let me know if you can help me. If you have any other questions, don't hesitate to ask me.
Thank you very much! :)
Assuming that these three measurements are independent, you can combine the three distributions based on the law of combination of distributions (LCD). Please refer to: Article Two simple and practical methods for combining prior informa...
Create a new variable y=x1+x2+x3. Then calulate the mean and standard deviation of the y values.
Henry Ra
Under the assumption of normality, the law of combination of distributions (LCD) will give the same results for the combined mean and variance as the weighted mean and weighted variance.
variance, var(...), has such formula (for sum of random variables):
var(x_comb) = var(x1+x2+x3)
= var(x1) + var(x2) + var(x3) + 2cov(x1,x2) + 2cov(x1,x3) + 2cov(x2,x3)
where cov(a,b) = covariance of a-and-b.
https://www.google.com/search?q=variance+of+sum+of+random+variables
Finally,
the desired standard deviation (in your question) is the sqrt( var(x_comb) ) = i.e. = square root from the combined variance.
What is the pooled standard deviation?
The pooled standard deviation is a method for estimating a single standard deviation to represent all independent samples or groups in your study when they are assumed to come from populations with a common standard deviation. The pooled standard deviation is the average spread of all data points about their group mean (not the overall mean). It is a weighted average of each group's standard deviation.
Attached is the formula.
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