Calculating Overall Variance & Standard Deviation for 3 Sets of Data

[Excom]

Hi

If I have measured the resonance frequency of three sets of resonators and calculated the mean, variance and standard deviation for each set. How do I add the three variances and standard deviations to get an overall variance and standard deviation?

Well, I know that the standard deviation is the square root of the variance and therefore I only need to know how to calculate the overall variance.

I hope that someone can help me with my problem. Thanks in advance.

Best regards
Tom

 

[micromass]

If your three resonance frequencies are independent, then

$$Var(X_1+X_2+X_3)=Var(X_1)+Var(X_2)+Var(X_3)$$

If not, then

$$Var(X_1+X_2+X_3)=Var(X_1)+Var(X_2)+Var(X_3)+2Cov(X_1,X_2)+2Cos(X_2,X_3)+2Cov(X_1,X_2)$$

 

[chiro]

If your three resonance frequencies are independent, then

$$Var(X_1+X_2+X_3)=Var(X_1)+Var(X_2)+Var(X_3)$$

If not, then

$$Var(X_1+X_2+X_3)=Var(X_1)+Var(X_2)+Var(X_3)+2Cov(X_1,X_2)+2Cos(X_2,X_3)+2Cov(X_1,X_2)$$

Is your last term meant to be Cov(X_1,X_3)?

 

[Excom]

Thanks

One clarifying question. If I have an infinite number of sets and I then add the variances for all the sets then the variance and hence the standard deviation will also be infinite?