The expected squared deviation of a random variable from its mean; or, the average squared deviation of data about their mean.

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12 views

R: result of Levene test correct?

I want to use the Levene test to quantify the homo/heterogeneity of the variances of two samples. The density plot looks like this: But the Levene test for the data [...
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2answers
16 views

analysing difference in variability between multiple related time series

I have constant time series (yearly from 1950 to 2010) of the abundance of several species that were captured by different groups. These series are somewhat related because the quantity of all species ...
0
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0answers
14 views

Mean of a function of one random variable

I've read this post and it could help me a lot if I would understand some Points: I would really appreciate if you could explain me some steps (I'm quite a beginner..) in the answer of mpiktas: I'm ...
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2answers
43 views

How to get F-test p-value

Suppose we can choose from two different catalysers. 10 observations are taken from the first one and 12 from the other one. If $s_1 = 14$ and $s_2 = 28$, can we reject at $\alpha = 5\%$ the ...
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0answers
28 views

Simple proof that variance of variance estimate is $\sigma^4 \cdot \frac{2}{n-1}$ given normal iid sample

Let $X_1,\dots X_n$ be i.i.d. and $N(\mu,\sigma^2)$ distributed. Let $$\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i$$ and $$S^2=\frac{1}{n-1}\sum_{i=1}^n (X_i -\overline{X})^2$$ Then I know that ...
0
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1answer
24 views

Confusion related to variance and mse

I was reading this wikipedia article and it states that MSE of a predictor is equivalent to variance of the error. To test it I did something like this ...
1
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0answers
31 views

Calculation of standard variance $s^2$

First picture is the question and its answer key: The second picture is my solution: I know that $S^2=\frac{\sum(x_i-\bar x)^2}{n-1}$ But in the answer key, $s^2$ cannot be found in that way. ...
0
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0answers
15 views

test for variance by using chi square test in R

how can i know if these data -> 93, 91, 93, 150, 80, 104, 128, 83, 88, 95, 94, 97 58, 139, 91 provide evidence that the pop variance is greater than 0.05 (a=0.05) by chi square test in R? please ...
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1answer
37 views

What is the difference between finite and infinite variance

What is the difference between finite and infinite variance ? My stats knowledge is rather basic; Wikipedia / Google wasn't much help here.
2
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1answer
40 views

Which error is displayed in an error ellipse?

I have some bivariate data and I have calculated the error ellipse in the following way: I have first calculated the covariance matrix and then to obtain the radii of the ellipse I have taken the ...
0
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0answers
11 views

Robust variance estimators

Given $N$ data points in $\mathbb{R}^p$ - some of which are outliers (drawn from a different distribution from the inliers) - what sorts of algorithms have been designed to estimate the robust ...
1
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0answers
30 views

How remove variations in a time series X caused by another time series Y?

I have a time series on a monthly basis (a commodity) of which much variation is caused by the weather. I want to adjust this commodity for weather changes. I use Heating degree day as a proxy for ...
2
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0answers
32 views

Which observation has the largest variance of the residual?

a) For which of the following observations (obs1, obs2, obs3) is the variance of the residual the largest? Which observation has the highest leverage? And which one the smallest? Explain why. ...
1
vote
1answer
43 views

Variance and covariance notation: $\sigma^2 V_1$, $\sigma^2 V_2$

I am reading an internal paper that says: Let $\sigma^2 V_1$ equal the variance of $\sum_{m\in M}Z_m - Z_0$ and $\sigma^2V_2$ equal the covariance of $||M||^{-1}\sum_{m\in M}Z_m - Z_0$ and $Z_m ...
2
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0answers
27 views

$\sigma^2 \le (\mu-a)(b-\mu)$ for all probability distributions bounded on $[a,b]$? [duplicate]

Let $\mu$ be the mean and $\sigma$ the standard deviation of a probability distribution defined on the bounded interval $[a,b]$ (that is, the probability that the random variable lies outside $[a,b]$ ...
1
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0answers
20 views

Sample variance and error using Monte Carlo

Asked to compute estimator for the following function, $\theta = \int_0^\infty e^{-x^2}$ which can be solved by transforming the limits to 0 to 1 and solving the following expectation using Monte ...
4
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0answers
38 views

Variance inflation factor for generalized additive models

In the usual VIF calculation for a linear regression, each independent/explanatory variable $X_j$ is treated as the dependent variable in an ordinary least squares regression. i.e. $$ X_j = \beta_0 + ...
4
votes
1answer
84 views

Covariance term in simple linear regression

I am trying to derive the expression for the variance of $\hat{\beta_0}$ in simple linear regression. I substitute $\bar{y} - \hat{\beta_1} \bar{x}$ for $\hat \beta_0$, but in the intermediate steps ...
3
votes
2answers
193 views

Why does sphericity diagnosed by Bartlett's Test mean a PCA is inappropriate?

I understand that Bartlett's Test is concerned with determining if your samples are from populations with equal variances. If the samples are from populations with equal variances, then we fail to ...
1
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1answer
43 views

How to make sense of 6mo worth of weight loss data?

-I have ~6mo of data that contains daily weight/caloric intake & #carbs/fats/proteins. I'm trying to figure out a way to make sense of the data. It's not a perfect experiment so there are other ...
0
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1answer
33 views

pairwise variance ratio in Stata

Suppose, I want to compute the pairwise correlation in Stata, then I do the following: sysuse auto pwcorr Now, I want to compute the variance for each variables ...
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0answers
20 views

Posterior variance reduction

As detailed on its Wikipedia page, Mutual information, $I(X,Y)$, can be bounded by the Jensen inequality to show that it is always positive. Also, one can show that $$ I(X,Y) = H(X) - H(X|Y). $$ ...
2
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0answers
35 views

Variance of the sum of correlated random variables

i'm trying to compute the variance of the random variable $$X = \frac{1}{N}\sum_{i=1}^N x_i$$ where $x_i$ are correlated identical random variables (mean and variance defined) obtained from a ...
3
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2answers
140 views

Overdispersion in logistic regression

I'm trying to get a handle on the concept of overdispersion in logistic regression. I've read that overdispersion is when observed variance of a response variable is greater than would be expected ...
11
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3answers
443 views

Deduce variance from boxplot

I was wondering how to deduce the variance of a variable using a boxplot. Is it at least possible to deduce if two variables have the same variance observing their boxplot?
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1answer
97 views

$E(\overline{x})$ vs $E(x)$?

If I have a problem down to $$k(E(x^2) - E(x)^2) = E(\overline{x}^2) - E(\overline{x})^2$$ ...can I say $k = 1$? $X$ has an unbiased mean and a variance. Each $x$ in $X$ is independent. Or is it ...
6
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1answer
531 views

How can a distribution have infinite mean and variance?

It would be appreciated if the following examples could be given: A distribution with infinite mean and infinite variance. A distribution with infinite mean and finite variance. A distribution with ...
1
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0answers
16 views

Why did fixing the covariance value cause AMOS to mis-estimate variance?

I have 100 data points on two variables, a and b. The correlation between the two is .3 and the SD is 1. When I run the ...
4
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0answers
58 views

Avoiding large variances when taking the logs of small values

I have two random variables $(X$ and $Y)$ that are always positive. The assumption I'm making is that their logs follow normal distributions (i.e., $N(\overline{\log(X)},s^2_{\log(X)})$ and ...
1
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1answer
68 views

P-value of F-test to compare two variances (var.test in R)

I am trying to understand where the p-value of a F-test comparing two variances comes from. More specifically, the p-value given by R's var.test function does not ...
0
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0answers
25 views

Forecast mean and variance for group data

Apologies if this is a bit of a simple question, but I haven't been able to find any answer to this over the past week and it's driving me crazy. Background Info: I have a dataset that tracks the ...
0
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1answer
25 views

How to compare distributions of two groups where each group has multiple observations from a small group?

I have data on continuous measurements (length of time of a behaviour) from two groups of individuals which differ in their phenotype (phenotype A or B, the variable used to group them). Each group is ...
7
votes
1answer
105 views

Do the mean and the variance always exist for exponential family distributions?

Assume a scalar random variable $X$ belongs to a vector-parameter exponential family with p.d.f. $$ f_X(x|\boldsymbol \theta) = h(x) \exp\left(\sum_{i=1}^s \eta_i({\boldsymbol \theta}) T_i(x) - ...
0
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0answers
10 views

How do I test for equal variances in a nested design?

I've got a dataset where stations (3) are nested within beaches (5). Every station is replicated 6 times per beach, so that I have a total of 90 observations. I now want to test for equal variances, ...
2
votes
1answer
30 views

Dealing with poorly estimated/missing explanatory variable values in GLMs

Context I am using generalised linear models to analyse some ecological data looking at the relationship between the population density of moth larvae and the prevalence (%) of viral mortality in the ...
0
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0answers
9 views

Estimating variance of prediction error in bootstrapped training sets with clustered data

I have C clusters with m elements each. I split the C clusters into a large training set D and a test set T. Hence, each element in D and T has m related elements, so its a cluster. I want to ...
1
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1answer
45 views

delta method with higher order terms to improve variance estimation accuracy

I need to apply the delta method principle using a Taylor expansion that retains higher order terms (i.e. to second or third order) in order to improve the accuracy of variance estimation. The ...
0
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0answers
9 views

comparing variation of a blended product from populations of different sizes (dilutions)

Original question: If population 1 consists of blended piles of size X_1 and population 2 consists of blended piles of size 4*X_1. Should I expect the standard deviation of X_1 to be equal to 4*the ...
3
votes
1answer
37 views

Variance of absolute value of a rv

Suppose that $X \backsim iid (\mu, \sigma^2)$. We are interested in $E (|X|)$ and ${\rm Var}(|X|)$. Can you suggest a way to proceed? I thought of rewriting $|X|$ as : $|X| = Xd - X(1-d)$, ...
1
vote
1answer
58 views

Basic statistical equation

How would I go about showing that: $$\sum_{i=1}^n (X_i-\bar{X})^2=\sum_{i=1}^n X_i^2-n\bar{X}^2\,,$$ (where $\bar{X}=\frac{1}{n}\sum_{i=1}^n X_i$)? Thank you for your help!
4
votes
1answer
116 views

Variance of Z for Z = X + Y, when X and Y correlated

So I'm trying to show that ${\rm Var}(Z) \le 2({\rm Var}(X)+{\rm Var}(Y))$ for $Z = X + Y$. This seems to be pretty easy to show given that $X$ and $Y$ are uncorrelated. But I'm running into trouble ...
0
votes
1answer
63 views

Expectation / minimizing variance of weighted sum of means

(I'm assuming the second $\bar x$ should be a $\bar y$), but I'm mostly confused how to solve this problem because it seems like since $\bar x$ and $\bar y$ are values, not random variables, $W$ is ...
1
vote
1answer
33 views

Is variance replacing square of standard deviation in a t test?

I am writing a t test using the code here : How to perform two-sample t-tests in R by inputting sample statistics rather than the raw data? But I would like to use the root of the variance as an ...
-3
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1answer
50 views

Self-study question

I'm currently working on a self study worksheet. I understand most parts of the solution for part III, but I can't seem to make out how this comes about: QUESTION: ANSWER:
2
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0answers
60 views

Estimating variance for identically non independent data

Let $X_{ij}$ with $1\leq i<j\leq n$ (that are $X_{12},\dots, X_{1n},\dots,X_{(n-1)n}$) be ${n \choose 2}$ identically normal distributed $N(0,\sigma^2)$ such that $ \text{corr}(X_{ij},X_{rs})=\rho ...
3
votes
3answers
108 views

Looking for a proof that overfitting a model leads to greater variance estimates (under OLS)

So I've been trying to algebraically prove that overfitting a model leads to greater variance values for the parameter estimates. I've gotten close (reduced the problem to showing a certain matrix is ...
1
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0answers
15 views

Calculating variation of one variable within one experiment

I am conducting a preliminary study, where people assess (7-point likert scala) different pictures. Besides asking for my 2 Independent variables (which are the manipulations for my main study, an ...
4
votes
1answer
129 views

Derive Variance of regression coefficient in simple linear regression

In simple linear regression, we have $y = \beta_0 + \beta_1 x + u$, where $u \sim iid\;\mathcal N(0,\sigma^2)$. I derived the estimator: $$ \hat{\beta_1} = \frac{\sum_i (x_i - \bar{x})(y_i - ...
2
votes
1answer
50 views

Taylor expansion to contain sample mean, sample variance, sample skewness, and sample kurtosis

I have the following expression: $$\frac{1}{p} \ln\left(1+\frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n} \sum_{i=1}^n x_i^2 + \frac{p^3}{3!n} \sum_{i=1}^n x_i^3 + \frac{p^4}{4!n} \sum_{i=1}^n ...
8
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2answers
324 views

Is variation the same as variance?

This is my first question on Cross Validated here, so please help me out even if it seems trivial :-) First of all, the question might be an outcome of language differences or perhaps me having real ...