Estimating the variance of a computation from descriptive statistics

#1
Hello,

I am a Ph.D. candidate in the behavioural sciences and I am a bit at a loss with respect to finding a solution to the problem to follow:

I have the means and standard deviations of three time points in a sample of twenty people.

Mean (SD)
10.3 (2.4)
9.2 (1.3)
9.5 (1.8)

I want to calculate the area under the curve for these points. Each point is separated by 20 minutes.

((10.3+9.2)*20)/2+((9.2+9.5)*20)/2 = Area under the curve

Simple enough. The thing I'm struggling with, though, is how to calculate the variance, or the standard deviation, of area under the curve with only the SD of each time point and the sample size. I'm interested not only in the value, but in the formula used to estimate the variance (if it is statistical, is there a name for the equation? If it is mathematical, could you present the formula?)

This may or may not be an elementary question, but any help on this would be greatly appreciated.
 

BGM

TS Contributor
#2
Denote \( X_1, X_2, X_3 \) be the value at the 3 time points respectively.

I assume that you want to calculate the area of two trapezium, which can be simplified as

\( A = 10(X_1 + 2X_2 + X_3) \)

The variance of this random variable is very standard if you know the covariance between the \( X \). This fact may depends on how you model the values at those time points, say time series model or other. If they are independent it is even easier.
 
#3
Dear BGM,

Thanks for your quick reply! To be more specific, these values are time series data (they are not independent). I am under the impression that the covariance between the X cannot be computed given the provided data:

Mean (SD)
10.3 (2.4)
9.2 (1.3)
9.5 (1.8)

Is that correct? If I can compute the covariance without the raw data (I can estimate it from an F-test, for example), what would then be the formula to estimate the variance of A?

Thanks in advance!
 

BGM

TS Contributor
#4
Yes from the given statistics alone one cannot obtain the covariance. You need other source of information (of course raw data would be the best, but some other statistics maybe sufficient depends on your model)


For the standard formula, you can find

http://en.wikipedia.org/wiki/Variance#Basic_properties

which tells you how to compute the variance of a linear combination of random variables.
 
#5
Dear BGM,

Thanks again for a quick reply. I will get to work on plugging in the formulas described and I will post an update on my analyses.

Much obliged.