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### why sample variance has has n-1 in the denominator? [duplicate]

Sample variance is calculated according to: $s^2=\frac{\sum{(x-\bar{x})^2}}{n-1}$ Population variance is calculated according to: $\sigma^2=\frac{\sum{(x-\mu)^2}}{n}$ Why denominator for sample ...
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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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### The derivation of standard deviation [duplicate]

So, if there are $N$ data points in my sample space of any randomly distributed variable $X$. The standard deviation, $\sigma$, (from my understanding) is the root mean squared of the error (from the ...
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### Estimates of variance from an iid sample [duplicate]

There are two kinds of estimates of variance from an iid sample $X1, \dots, X_n$ $1/n * \sum_i (X_i - \bar{X})^2$, which is MLE $1/(n-1) * \sum_i (X_i - \bar{X})^2$, which is unbiased. The unbiased ...
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### Intuition behind Besel's correction for an unbiased estimate of the variance [duplicate]

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 ...
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### Why Standard deviation formula has n-1 in the denominator but Variance only n? [duplicate]

If is true that SQRT(Variance) = SD then one cannot have n-1 in the denominator and the other not but should be same? SD formula: Variance formula:
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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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### Explanation for Bessel's correction [duplicate]

In textbooks and online the explanation for Bessel's correction is that the sample points are closer to the sample mean than the population mean.This is true for sampling with replacement as well. So, ...
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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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### Sample Variance and Dividing by $n-1$

In this video... https://www.youtube.com/watch?v=sHRBg6BhKjI ...and in many others, the explanation for why when calculating the sample variance we divide by $n-1$ instead of by $n$ is the following:...
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### Estimation of variance: How to bring Bessel's correction together with degrees of freedom?

I have been considering multiple textbooks to find out the reason that the denominator of the estimation of the population variance is n-1 rather than n. Depending on the book, two reasons are given: ...
### Why is $\frac{\sum^n_{i=1}(X_i-\bar{X})^2}{\sigma^2}$chi-square distributed with $n-1$ degrees of freedom?
On Wikipedia, it says If $X_{i};i=1,\ldots ,n$ are independent normal $(\mu ,\sigma ^{2})$ random variables, the statistic $$\frac{\sum\limits^n_{i=1}(X_i-\bar{X})^2}{\sigma^2}$$ ...