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I have the mean & SD of several different time points, I want to combine them into one mean and SD. Is it possible? Example;

At 1 hour; mean=5, SD=1, n=42

At 2 hours; mean=6, SD=3, n=42 and so on.

Any equation to do that?

Note: at some time points, I have different group numbers (n), due to missing one or two patients. I really appreciate your help, thanks!

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  • $\begingroup$ The covariance or correlation coefficient is needed for combining the SDs. $\endgroup$ – user158565 Dec 30 '18 at 5:23
  • $\begingroup$ And what about the mean? $\endgroup$ – Mohamed Gomaa Kamel Dec 30 '18 at 5:44
  • $\begingroup$ You can combine means given you have sample size n if you think it is meaningful. $\endgroup$ – user158565 Dec 30 '18 at 5:46
  • $\begingroup$ Thanks for your answer What is the formula? $\endgroup$ – Mohamed Gomaa Kamel Dec 30 '18 at 6:08
  • $\begingroup$ $(N_1\bar X_1 + N_2\bar X_2 +...+N_k\bar X_k )/(N_1 + N_2+...,+N_k)$ $\endgroup$ – user158565 Dec 30 '18 at 6:35
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Rules to pool mean and variance of two groups can be found in O'Neill (2014) (Result 1). The formulas are:

$$\begin{equation} \begin{aligned} \bar{x}_\text{pooled} &= \frac{1}{n_1+n_2} \Bigg[ n_1 \bar{x}_1 + n_2 \bar{x}_2 \Bigg], \\[10pt] s_\text{pooled}^2 &= \frac{1}{n_1+n_2-1} \Bigg[ (n_1-1) s_1^2 + (n_2-1) s_2^2 + \frac{n_1 n_2}{n_1+n_2} (\bar{x}_1 - \bar{x}_2)^2 \Bigg]. \\[10pt] \end{aligned} \end{equation}$$

To pool a larger number of groups you can proceed recursively using these formulae.

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Unless you truly believe there is a time difference-driven discrepancy, just pool them.

Imagine you are measuring the heights of students in 12th grade. Does it matter if one class comes at 1 PM and the next at 2 PM for measurement? No. You just deal with it as one sample.

On the other hand, if you are doing the measurements in Q1 (January) and Q2 (April), you should likely assume the kids grew. If that is the case, it is a different problem - and a harder one.

More info on what you are trying to find would help. But, based on your limited description, simply pool the data and calculate the mean and SD for the whole.

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  • $\begingroup$ Thanks for your kind comment, Yes, it will not affect (not a big difference in time) So, how to pool them, please? $\endgroup$ – Mohamed Gomaa Kamel Dec 30 '18 at 6:36
  • $\begingroup$ Take the 84 data points and ignore the time. calculate the mean and SD for the 84 combined. Let me assume, for example, you want average weight and height (perhaps to calculate average BMI and its SD?) based on these two groups given weight and height. Treat them as one group and get the average weigh, and average height, over the 84 +/- 2 or 3 difference in count. $\endgroup$ – eSurfsnake Dec 30 '18 at 6:48

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