Linked Questions

2
votes
1answer
3k views

What is the expected value and the mean of sample standard deviation? [duplicate]

What is the expected value and the mean of sample standard deviation? I know that I can derive the expectation and variance of sample variance using the $\chi^2$ pdf. But I don't know how to start ...
3
votes
2answers
1k views

Unbiased estimator of standard deviation of a normal distribution, using gamma function [duplicate]

According to the Wikipedia article, the following estimator of the standard deviation $$s=\sqrt{\frac{1}{n-1}\sum_{k=1}^n(x_i-\bar{x})^2}$$ for a normal variable, verifies $E[s]=C_4(n) \sigma$, where ...
2
votes
1answer
2k 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 ...
1
vote
0answers
136 views

Estimating process standard deviation for a quality control chart. What is $s/(c4)$ and how to calculate it in R? [duplicate]

I recently came across quality control charts in my studies and found out that the estimator $$\hat \sigma =\frac{s}{c_4}$$ is preferred for UCL and LCL calculations in an x-bar chart as it is ...
0
votes
1answer
82 views

why is standard deviation a biased estimator [duplicate]

In this post: Why is sample standard deviation a biased estimator of $\sigma$?, I am having difficulty understanding some of the steps. We have : (a) $s^2=\frac{1}{n-1}\sum_{i=1}^{\infty}(x_i-x\bar)^2$...
74
votes
5answers
42k 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 "...
54
votes
3answers
34k views

Standard deviation of standard deviation

What is an estimator of standard deviation of standard deviation if normality of data can be assumed?
22
votes
5answers
6k views

Why are we using a biased and misleading standard deviation formula for $\sigma$ of a normal distribution?

It came as a bit of a shock to me the first time I did a normal distribution Monte Carlo simulation and discovered that the mean of $100$ standard deviations from $100$ samples, all having a sample ...
21
votes
4answers
17k views

Calculating required sample size, precision of variance estimate?

Background I have a variable with an unknown distribution. I have 500 samples, but I would like demonstrate the precision with which I can calculate variance, e.g. to argue that a sample size of 500 ...
11
votes
3answers
7k views

How can I find the standard deviation of the sample standard deviation from a normal distribution?

Forgive me if I've missed something rather obvious. I'm a physicist with what is essentially a (histogram) distribution centered about a mean value that approximates to a Normal distribution. The ...
11
votes
4answers
20k views

How to calculate 2D standard deviation, with 0 mean, bounded by limits

My problem is as follows: I drop 40 balls at once from a certain point, a few meters over the floor. The balls roll, and comes to a rest. Using computer vision, I calculate the center of mass in the X-...
18
votes
4answers
2k views

Is variance a more fundamental concept than standard deviation?

On this psychometrics website I read that [A]t a deep level variance is a more fundamental concept than the standard deviation. The site doesn't really explain further why variance is meant to ...
17
votes
2answers
832 views

For which distributions is there a closed-form unbiased estimator for the standard deviation?

For the normal distribution, there is an unbiased estimator of the standard deviation given by: $$\hat{\sigma}_\text{unbiased} = \frac{\Gamma(\frac{n-1}{2})}{\Gamma(\frac{n}{2})} \sqrt{\frac{1}{2}\...
9
votes
1answer
6k views

Unbiased estimators of skewness and kurtosis

The skewness and kurtosis are defined as: $$\zeta_3 = \frac{E[(X-\mu)^3]}{E[(X-\mu)^2]^{3/2}} = \frac{\mu_3}{\sigma^3}$$ $$\zeta_4 = \frac{E[(X-\mu)^4]}{E[(X-\mu)^2]^2} = \frac{\mu_4}{\sigma^4}$$ The ...
6
votes
1answer
7k views

Degrees of freedom for standard deviation of sample

would someone please explain why the degrees of freedom for a random sample is n-1 instead of n ? I'm looking for an explanation that is intuitive and easily understood by a high school student.

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