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### Covariance Matrix vs. Pairwise Covariance Matrix?

I found this equation here to calculate a covariance matrix of any number of variables using matrix algebra. $$\frac1{N} (X - 1\bar{x})^T(X - 1\bar{x}^T)$$ For a given matrix $X$ with $N$ samples. ...
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### Confused when to use Population vs Sample standard deviation in engineering testing

When I run an test for something (say 10 trials) and want to find the standard deviation of all 10 trials, I am getting confused if I should use the sample or population standard deviation. My initial ...
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### “… because sample mean gets different values from sample to sample and it is a random variable with mean $\mu$ and variance $\frac{\sigma^2}{n}$.”

This answer by user "sevenkul" says the following: The sample mean $\overline{X}$ also deviates from $\mu$ with variance $\frac{\sigma^2}{n}$ because sample mean gets different values from ...
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### Alternate (?) definition of sample variance [duplicate]

The variance of a sample can be defined as $$s^2 = \frac{1}{2}\frac{1}{n(n-1)}\sum_{i}\sum_{j\ne i}\left(x_i - x_j\right)^2$$ Apart from the factor of $1/2$, this can be paraphrased verbally as ...
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### Standardizing dependent and independent variable formula for ridge regression

For ridge I now that I have to standardize the the predictors before applying ridge regression. I want to standardize them like in the glmnet function. And I don't know wich formula is used for ...
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### What is the story of “$n-1$” in the denominator of an estimated variance? [duplicate]

In the book "Numerical Recipes in C" there is said that there is a long story about why the denominator of variance is $N-1$ instead of $N$. If you have never heard that story, you may consult any ...
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### How exactly did statisticians agree to using (n-1) as the unbiased estimator for population variance without simulation?

The formula for computing variance has $(n-1)$ in the denominator: $s^2 = \frac{\sum_{i=1}^N (x_i - \bar{x})^2}{n-1}$ I've always wondered why. However, reading and watching a few good videos about "...
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### A/B test and SRS

I was reading about Simple Random Sampling (SRS) from this lecture notes. It says that SRS, which actually is a sampling without replacement makes $x_i$ not independent. Then I thought about A/B ...
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### Which are the expressions for the variance and standard deviation of a sample?

I have checked several offline and online resources, and they are conflictive. Some define the sample variance as $$s_n^2 = \frac{\sum_{i}^{n}(x_i-\overline{x})^2}{n}$$ That is just the second ...
We have two estimators for samples dispersion: $\frac{1}{n}\sum_1^n (X_i - \overline{X_n})^2$ and $\frac{1}{n - 1}\sum_1^n (X_i - \overline{X_n})^2$. The second estimator unbiased - it has ...