Linked Questions

1
vote
1answer
8k views

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 ...
8
votes
1answer
143 views

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 ...
1
vote
1answer
634 views

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 ...
0
votes
1answer
223 views

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 ...
0
votes
0answers
134 views

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 ...
0
votes
1answer
80 views

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:
1
vote
0answers
69 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 ...
1
vote
0answers
28 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, ...
1
vote
0answers
20 views

Different sample covariance formulae (conventions) [duplicate]

Page 358 of Introduction to probability, second edition, by Blitzstein and Hwang, defines the sample covariance as $$r = \dfrac{1}{n} \sum_{i = 1}^n (x_i - \bar{x})(y_i - \bar{y}),$$ where $\bar{x} = \...
1
vote
0answers
19 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 ...
0
votes
0answers
17 views

Show estimator is unbiased [duplicate]

Consider the estimator of the variance given by the formula: $(S')^2 = \frac{1}{n} \sum_{i=1}^{n}(Y_i − µ)^2$ Is this a biased or unbiased estimator? I'm not sure if it is possible to prove ...
0
votes
0answers
15 views

Degree of freedom [duplicate]

Why do we use degree of freedom while dealing with samples(e.g. while calculating t statistic and sample variance, we divide by $ n-1 $ degree of freedom, but while calculating z score and population ...
72
votes
5answers
39k views

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 "...
16
votes
3answers
4k views

Why does one have to use REML (instead of ML) for choosing among nested var-covar models?

Various descriptions on model selection on random effects of Linear Mixed Models instruct to use REML. I know difference between REML and ML at some level, but I don't understand why REML should be ...
8
votes
3answers
2k views

When estimating variance, why do unbiased estimators divide by n-1 yet maximum likelihood estimates divide by n?

I am totally confused: On the one hand you can read all kinds of explanations why you have to divide by n-1 to get an unbiased estimator for the (unknown) population variance (degrees of freedom, not ...
3
votes
1answer
8k views

What is the difference between Mean Squared Deviation and Variance?

I am doing some tutoring for an AS-Level maths student and unfortunately for me they are doing statistics. This is not my strong point, mainly from the point of view of remembering all of the ...
4
votes
1answer
6k views

Estimating the covariance matrix in linear discriminant analysis

I was trying to derive the equations from page 109 in "elements of statistical learning" (image below) To be honest, I am not sure how the covariance $\Sigma$ is estimated (the third bullet point in ...
0
votes
3answers
7k views

Measure the fluctuation of data

I have 2 series of data, for example: s1: 0.3 0.3 0.4 0.8 0.6 0.5 0.7 s2: 0.7 0.7 0.6 0.8 0.7 0.5 0.6 It's easy to see that ...
6
votes
2answers
162 views

What value of $\alpha$ makes $\sum_{i=0}^n (x_i-\alpha)^2$ minimal?

I know that $\bar{x}$ makes absolute result of $\sum_{i=0}^n (x_i-\alpha)$ minimum. In fact it makes it zero. But how to find what value of $\alpha$ makes $\sum_{i=0}^n (x_i-\alpha)^2$ minimal? What ...
2
votes
2answers
2k views

Expected number of successes from $N$ Bernoulli trials with different $p$

Suppose I have N probabilities $(p_1, p_2,...,p_N)$ that represent the chance that each that a corresponding test was passed. How do I apply the Bernoulli distribution to determine the expected number ...
11
votes
2answers
1k views

Why does Restricted maximum likelihood yield a better (unbiased) estimate of the variance?

I'm reading Doug Bates' theory paper on R's lme4 package to better understand the nitty-gritty of mixed models, and came across an intriguing result that I'd like to understand better, about using ...
4
votes
1answer
223 views

An unbiased estimator of σ³

As it was suggested in the linked answer, $s_n = \sqrt{\frac{\sum_{i = 1}^n (x_i - \bar{x})^2}{n - 1}}$ is not an unbiased estimator of $\sigma$. I suspect neither $s_n^3 = \sqrt{\frac{\sum_{i = 1}^n ...
1
vote
1answer
1k views

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. ...
4
votes
2answers
213 views

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:...
1
vote
1answer
844 views

Variance estimator of Bernoulli RV (in CI)

Assume n samples from Bernoulli distribution with unknown parameter p, i. e. $x_{1},....,x_{n}\underset{iid}{\sim}Ber\left(p\right)$ It is known that the confidence interval is given by: $CI=\hat{...
3
votes
1answer
492 views

Dividing by degrees of freedom [duplicate]

When estimating parameters such as (I don't care about this specific instance particularly) Variance of a random variable X, one usually adopts Bessel's correction, i.e. using the formula $\hat{Var}{(...
1
vote
0answers
650 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: ...
2
votes
1answer
137 views

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 $$\frac{\sum\limits^n_{i=1}(X_i-\bar{X})^2}{\sigma^2}$$ ...
0
votes
3answers
159 views

Variance (different definitions)

I'm new to statistics and reading stats books I found different definitions for Variance. Definition1: $ s^2 = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2 $ Definition2: $ s^2 = \frac{1}{n} \sum_{i=...
5
votes
2answers
132 views

Definition of a sample: can it include the same object twice?

Wikipedia defines a sample as: a subset of a population. While exploring the reason why we divide by $(n-1)$ instead of $n$ when calculating standard deviation (discussed in this question), I came ...

15 30 50 per page