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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: ...
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### 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 $$\frac{\sum\limits^n_{i=1}(X_i-\bar{X})^2}{\sigma^2}$$ ...
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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: $s^2 = \frac{1}{n} \sum_{i=... 2answers 132 views ### Definition of a sample: can it include the same object twice? Wikipedia defines a sample as: a subset of a population. While exploring the reason why we divide by$(n-1)$instead of$n$when calculating standard deviation (discussed in this question), I came ... 1answer 92 views ### Why do I need to rounding x and y values to the nearest 0.5 when manually calculating correlation? I'm trying to calculate correlation using a formula in Statistics 4th Edition by Freedman: r = average of (x in standard units) * (y in standard units) If I try this out ... ... 1answer 146 views ### A/B test and SRS I was reading about Simple Random Sampling (SRS) from this lecture notes. It says that SRS, which actually is a sampling without replacement makes$x_i$not independent. Then I thought about A/B ... 1answer 111 views ### Statistical analysis for replication in very small experimental datasets A colleague does replication on a quite coslty experiments. There are four different conditions, each one duplicated. The outcome with$4\times 2 = 8$points is illustrated below: The analysis is ... 0answers 141 views ### What is the expected value of sample variance in this problem? Let$x_1,x_2,..x_{21}$be a random sample from a distribution having variance 5. Let $$\bar X = \frac1{21}\sum_{i=1}^{21}X_i\quad\text{and}\quad S=\frac1{21}\sum_{i=1}^{21}(X_i-\bar X)^2$$ What is ... 0answers 136 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 ... 0answers 123 views ### What are biased and inefficient estimators? I’m studying statistics from Schaum’s Outline, which gives the following: ... 2answers 40 views ### “… because sample mean gets different values from sample to sample and it is a random variable with mean$\mu$and variance$\frac{\sigma^2}{n}$.” This answer by user "sevenkul" says the following: The sample mean$\overline{X}$also deviates from$\mu$with variance$\frac{\sigma^2}{n}$because sample mean gets different values from ... 0answers 106 views ### Standardizing dependent and independent variable formula for ridge regression For ridge I now that I have to standardize the the predictors before applying ridge regression. I want to standardize them like in the glmnet function. And I don't know wich formula is used for ... 1answer 55 views ### Trouble understanding some r code I have this code here (it's out of a book called "An Introduction to Bootstrap Methods with Applications to R") We are working with the estimator:$S_n^2=\frac{\sum_{i=1}^{n}(X_i-X_b)^2}{n}$where$...
The r.v. $X$ represents the time taken by a computer in company $1$ in order to perform a certain job, and $Y$ represents the same thing but for company $2$. A sample of $n_X = 12$ computers are taken ...