Linked Questions

0
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1answer
6k 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
112 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 ...
0
votes
1answer
200 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
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0answers
113 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 ...
1
vote
0answers
62 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 ...
0
votes
1answer
44 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 ...
68
votes
5answers
35k 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 "...
14
votes
3answers
3k 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 ...
7
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 ...
6
votes
2answers
153 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 ...
3
votes
1answer
6k 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 ...
0
votes
3answers
5k 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 ...
2
votes
2answers
1k 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 ...
3
votes
1answer
4k 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 ...
11
votes
2answers
886 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 ...

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