Number of degrees of freedom in the sigma estimate

[Reid]

Homework Statement

I have estimated the standard deviation of the population of my samples from the standard deviations from each of the samples with the equation found below. And I am to construct a confidence interval for a contrast, thus I will need the number of degrees of freedom for which the estimate of the standard deviation is based on. And I really can't tell!

Homework Equations

The estimation of the standard deviation is given by
$$\sigma=\sqrt{\frac{N_{X}(\sigma_{X}^{2}+\mu_{X}^{2})+N_{Y}(\sigma_{Y}^{2}+\mu_{Y}^{2})}{N_{X}+N_{Y}}-\mu^{2}_{XY}},$$
where $$N_{X}, N_{Y}, \mu_{X}, \mu_{Y}, \mu_{XY}$$ are the sample populations of samples X and Y, the means of samples X, Y and the mean of the entire population XY.

The Attempt at a Solution

For every estimate of a population one looses one degree of freedom but then the standard deviation would be based on $$N-1=25-1=24$$ degrees of freedom... is this correct?

 

[nate808]

Hello,

Thank you for sharing your equation and approach to estimating the standard deviation of your population. It appears that your calculation is correct, assuming that your sample size (N) is 25. When estimating the standard deviation of a population, it is common to use the sample standard deviation formula, which divides by N-1 instead of N. This is because using N-1 accounts for the fact that the sample standard deviation is an estimate of the population standard deviation, and using N would result in a biased estimate. Therefore, in your case, using N-1 would result in 24 degrees of freedom for your estimate of the standard deviation.

However, it is important to note that the number of degrees of freedom may vary depending on the specific statistical test you are conducting. For example, if you are using a t-test to compare the means of two samples, the degrees of freedom would be calculated differently. It is always best to consult with a statistician or refer to a statistical textbook for the appropriate degrees of freedom for your specific analysis.

I hope this helps clarify the issue for you. Best of luck with your research!