Questions tagged [bessels-correction]
Bessel's correction lies in dividing by $n-1$ instead of $n$ in computing the sample variance.
28
questions
2
votes
2
answers
65
views
When calculating sample variance around a sample mean can I take degrees of freedom to be a fraction
Normally if you have $N$ samples of a random variable, from which you estimate both the sample mean and sample variance, you need to account for the missing degree of freedom in the sample variance:
$$...
- 323
0
votes
0
answers
31
views
Bessel correction for the variance of dependent sample
Assuming a sample $X_1, X_2, ..., X_n$, the sample variance is calculated as
$s^2 = \frac{1}{n-1} \sum (X_i-\bar{X})^2$
The fact that there is $n-1$ in the denominator instead of $n$ is called the ...
- 217
0
votes
0
answers
69
views
Can variance be calculated with a sample size of n=1?
I am currently analysing inter-rater reliability/agreement data for a single case with multiple raters. For that I am using Gwet's $AC_2$ (as described here) using the irrCAC package in ...
- 85
5
votes
2
answers
115
views
Finite population correction for the variance
Just when I thought I was starting to understand Bessel's correction, I noticed that it is not valid when the sample size equals the population size and so likely not valid for sample sizes close to ...
- 253
2
votes
1
answer
86
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 ^2$ already ...
- 137
0
votes
0
answers
46
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-...
- 1
2
votes
2
answers
714
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, ...
- 580
7
votes
4
answers
1k
views
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) ...
- 1,374
1
vote
1
answer
125
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 ...
- 67
1
vote
0
answers
376
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 ...
- 375
1
vote
0
answers
40
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 ...
- 11
1
vote
0
answers
57
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, ...
- 11
1
vote
1
answer
244
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 ...
- 67
0
votes
0
answers
78
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 ...
- 67
1
vote
1
answer
1k
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 $\...
- 244
0
votes
1
answer
144
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 ...
- 113
2
votes
0
answers
33
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}...
- 21
12
votes
2
answers
3k
views
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 ...
1
vote
1
answer
2k
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) = $1/n$ * $\Sigma$ $(Xi - mean(X))$*$(...
- 33
1
vote
0
answers
109
views
What is the story of "$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 ...
- 111
0
votes
0
answers
209
views
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 ...
- 101
2
votes
1
answer
228
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) = \...
- 373
13
votes
1
answer
14k
views
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.
...
1
vote
0
answers
1k
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:
...
- 1,342
3
votes
1
answer
414
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}^...
- 6,096
11
votes
2
answers
17k
views
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 ...
- 213
5
votes
2
answers
713
views
Could Bessel's correction make sample variance estimation even more biased?
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,...
- 6,096
235
votes
19
answers
265k
views
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 ...
- 21.1k