Questions tagged [bessels-correction]

Bessel's correction lies in dividing by $n-1$ instead of $n$ in computing the sample variance.

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Intuitive explanation for dividing by $n-1$ when calculating standard deviation?

I was asked today in class why you divide the sum of square error by $n-1$ instead of with $n$, when calculating the standard deviation. I said I am not going to answer it in class (since I didn't ...
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Why do we divide by $n-1$ when calculating sample correlation?

I understand the rationale for dividing by $n-1$ when calculating the sample variance, i.e. that if we divide by $n$ we will have an estimate of population variance that is biased to be too low. ...
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Why is binomial variance calculated as $p(1-p) / (n -1)$?

I had to translate several given statistics equations into code, and I came across this formula: Variance of a simple random sample $= \frac{p(1-p)}{n-1}$ The sample in question are test letters ...
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Degrees of Freedom In Sample Variance

Recall the formula for sample variance $$s_{n - 1}^2 = \dfrac{1}{n -1} \sum_{i = 1}^n (\bar{x} - x_i)^2,$$ where $\bar{x}$ is the sample mean. There are many proofs for why $s_{n - 1}^2$ is an ...
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Is Bessel's correction required when calculating mean?

Bessel's correction is used for the variance calculation from samples. In my understanding the reason is because the mean of samples has an error or bias from the true mean, and dividing by (n-1) ...
It is well known that Bessel's correction creates an unbiased estimator of variance. What it basically does is divide by $n-1$ instead of $n$. Now what I did is that I chose a few number, like 1,2,3,... 1answer 305 views Don't understand the proof that unbiased sample variance is unbiased Wikipedia gives the following proof why to use Bessel's correction for the unbiased sample variance: \begin{align} E[\sigma_y^2] & = E\left[ \frac 1n \sum_{i=1}^n \left(y_i - \frac 1n \sum_{j=1}^... 1answer 43 views Why is there no Bessel's correction equivalent for\sigma^2 = \sum [(x - \mu)^2 P(x)]$? Why is sample variance and discrete variable variance different in this sense? Is it because Bessel's correction exists for a better reflection of the population variance whereas$\sigma ^2already ... 1answer 203 views Bessel's correction demonstration I am currently trying to understand the proof of the Bessel's correction Proof of correctness 2 and there is one step in the demonstration that I do not understand: \operatorname{Var}(\bar x) = \... 0answers 27 views Tail of the CDF of noncentral chi-squared RV The pdf and cdf of the non-central chi squared RV (under the scenario I am studying) is given as follows: \begin{align} &f(x)=\frac{1}{v} \exp\left(\frac{-(a+x)}{v}\right)I_{0}\left(\frac{\sqrt{xa}... 1answer 1k views formula for sample covariance: Bessel's correction Two options for the sample covariance between X and Y: 1)(with Bessel's) COV(X,Y) = 1/(n-1) * \Sigma (Xi - mean(X))*(Yi - mean(Y)) 2)(without) COV(X,Y) = 1/n * \Sigma (Xi - mean(X))*(... 1answer 83 views Where to find a citation for the n/(n+1) sample variance correction? In my Master's thesis project I could not show my (normally-distributed) samples to have a common population variance (through Levene's test or otherwise), so I could not use the n/(n-1) Bessel's ... 2answers 130 views Why do we prefer unbiased estimators instead of minimizing MSE? I was thinking about why, usually, \hat{\sigma}^2=\hat{p}(1-\hat{p}) is used to estimate the variance in a Bernoulli population instead of s^2=\hat{p}(1-\hat{p})\frac{n}{n-1}. s^2 is unbiased, ... 0answers 244 views Generalization of Bessel's correction to higher order models? [duplicate] Multiplying sample variance (i.e. variance from sample mean) by \frac{n}{n-1} to obtain an unbiased estimate of the population variance (i.e. variance from population mean) is called Bessel's ... 0answers 31 views Is there ever a right time to not use Bessel's Correction? [duplicate] I can not find a clear explanation for when NOT to use Bessel's correction and use N instead of N-1. As I understand it, Bessel's correction would apply to things such as clinical trials, sampling ... 0answers 42 views 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, ... 1answer 103 views Why are the MLE and MMSE corrections for sample variances different? I have a number of samples of sample size 2 and a number of sample of sample size 3. If my samples are all samples from populations with a shared population variances, I wish to estimate population ... 0answers 71 views 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 ... 0answers 839 views 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: ... 1answer 91 views How can we write the sample variance's formal definition of a continuous random variable considering Bessel's correction? [closed] I am trying to find the formal way of writing the sample variance of a continuous random variable considering Bessel's correction. I ask because the sample variance is usually written this way: S^2 ... 0answers 38 views Why use n-1 in the sample variance when random tests show little differences between a "n-1 standard deviation" and a "n standard deviation"? [duplicate] I generated random tests to calculate standard deviations of populations and samples using both n-1 and n for the denominator of the variance formula with datasets like [5,20,25... numbers between 5-... 0answers 63 views Difficulty with averaging corrected sample variances of different degrees of freedom: I have a number of measurement samples of which some have 2 measurements and some have 3. I wish to make the most accurate estimation of population variance I can, and understand that ignoring data ... 1answer 539 views When to use Bessel correction and and does it alter the standard error? is there any condition about when to use the Bessel correction as Python uses it to find the Standard Deviation by default. The standard error of the mean is\...
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 ...