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2. Bessel's correction - Wikipedia

   en.wikipedia.org/wiki/Bessel's_correction

   In statistics, Bessel's correction is the use of n − 1 instead of n in the formula for the sample variance and sample standard deviation, [1] where n is the number of observations in a sample. This method corrects the bias in the estimation of the population variance. It also partially corrects the bias in the estimation of the population ...

3. Variance - Wikipedia

   en.wikipedia.org/wiki/Variance

   If the set is a sample from the whole population, then the unbiased sample variance can be calculated as 1017.538 that is the sum of the squared deviations about the mean of the sample, divided by 11 instead of 12. A function VAR.S in Microsoft Excel gives the unbiased sample variance while VAR.P is for population variance.

4. Standard error - Wikipedia

   en.wikipedia.org/wiki/Standard_error

   Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.

5. Popoviciu's inequality on variances - Wikipedia

   en.wikipedia.org/wiki/Popoviciu's_inequality_on...

   Popoviciu's inequality on variances. In probability theory, Popoviciu's inequality, named after Tiberiu Popoviciu, is an upper bound on the variance σ2 of any bounded probability distribution. Let M and m be upper and lower bounds on the values of any random variable with a particular probability distribution.

6. Lehmann–Scheffé theorem - Wikipedia

   en.wikipedia.org/wiki/Lehmann–Scheffé_theorem

   The theorem states that any estimator that is unbiased for a given unknown quantity and that depends on the data only through a complete, sufficient statistic is the unique best unbiased estimator of that quantity. The Lehmann–Scheffé theorem is named after Erich Leo Lehmann and Henry Scheffé, given their two early papers. [2][3] If T is a ...

7. Estimation of covariance matrices - Wikipedia

   en.wikipedia.org/wiki/Estimation_of_covariance...

   The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in Rp×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. [1]

8. Variance of the mean and predicted responses - Wikipedia

   en.wikipedia.org/wiki/Variance_of_the_mean_and...

   The predicted response distribution is the predicted distribution of the residuals at the given point xd. So the variance is given by. The second line follows from the fact that is zero because the new prediction point is independent of the data used to fit the model. Additionally, the term was calculated earlier for the mean response.

9. Cochran's theorem - Wikipedia

   en.wikipedia.org/wiki/Cochran's_theorem

   Cochran's theorem then states that Q1 and Q2 are independent, with chi-squared distributions with n − 1 and 1 degree of freedom respectively. This shows that the sample mean and sample variance are independent. This can also be shown by Basu's theorem, and in fact this property characterizes the normal distribution – for no other ...