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?

 

[blue_raver22]

Hello Tom,

To calculate the overall variance and standard deviation for three sets of data, you will need to use the formula for pooled variance. This formula takes into account the individual variances and sample sizes of each set of data.

The formula for pooled variance is:

s^2p = ((n1-1)s1^2 + (n2-1)s2^2 + (n3-1)s3^2) / (n1 + n2 + n3 - 3)

Where:
s^2p = pooled variance
n1, n2, n3 = sample sizes for each set of data
s1, s2, s3 = individual variances for each set of data

To calculate the overall standard deviation, simply take the square root of the overall variance.

I hope this helps and good luck with your calculations!

Best regards,