Questions tagged [variance]
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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When would one need to be able differentiate Var(X)?
Reading Blitzstein and Hwang’s introduction to probability http://probabilitybook.net (really good so far!).
On page 172, they discuss why $E[|X-E[X]|$ isn’t used as the definition of variance. They ...
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Structural equation modeling vs. mixed effects model
I have a question about the difference between structural equation modeling and mixed effects models.
What I'm trying to do is to use anatomical data (e.g. volume of a certain brain region) to explain ...
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Estimating the white noise variance
This is a mock-exam question:
The following measurements have been made
$\hat{\gamma}(0) = 570, \quad \hat{\gamma}(1) = 505, \quad \hat{\gamma}(2) = 460, \quad \hat{\gamma}(3) = 420$
Suppose that the ...
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Practical use for heritability value at sub-population level?
For me as a layperson, heritability is something that much smarter people than me calculate, and there are several questions already on the forum about how to do that (eg, this and this). I ...
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How to calculate errors of best value of parameters that obtained from MCMC method and observational data
I had a model and some observational data. I used MCMC method to obtain the best value of free parameters and used some coding to plot contours of 1 to 3 sigma confidence levels (as you see in the ...
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Proportion of explained variance for a probability model(binary logistic regression)
in the article written by Frank Harell ,Statistically Efficient Ways to Quantify Added Predictive Value of New Measurements,(https://www.fharrell.com/post/addvalue/)
Harell is writing:
For a ...
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Calculating the variance of a function which depends on the standard deviation and mean of a random variable
I am a bit new to statistics and had some conceptual questions regarding the calculation of variance.
I want to calculate the variance of a function $y=\frac{\sigma_{X}}{g(\overline{X})}$. As seen in ...
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Which is correct $Var(\theta) \ge (F^{-1})_{ii} $ or $Var(\theta) \ge 1/(F_{ii}) $?
With $\textbf{F}(\theta)$ denoting the Fisher matrix, $\textbf{V}(\theta)$ the variance matrix, $\textbf{C}(\theta)$ the covariance matrix, and parameter vector $\theta = [\theta_1, \theta_2, \dots,...
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Estimating the $\chi^2$-divergence with Monte Carlo: which distribution to sample from?
Notation: let the $\chi^2$-divergence between $p, q$ be defined as
$$\chi^2 (p||q) := \int \left ( \frac{p(x)}{q(x)} \right )^2 q(x)\mathrm{d}x -1 = \int \frac{p(x)}{q(x)} p(x)\mathrm{d}x - 1. $$
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Pooling variance of two samples with sample means and sample variance [duplicate]
I am new to this so I don't have enough reputation to comment on the original post here. I have a similar problem with equal weighting, and have n = m = 3.
I believe this means I need to use sample ...
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Distribution of Sample Variances for Half Normal Distributions
If $X$ is distributed according to a normal distribution with zero-mean $\mathcal{N}(0, \sigma_N^2)$, $Y:=\vert X\vert$ is said to be distributed according to a half-normal distribution, cf. 2.
I am ...
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What low bias has to do with "good fit"
Look at the bias-variance decomposition below:
In pratice we often consider low bias as good fit to train data but i dont understand the why this by bias-variance decomposition.Why if i got "...
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Significance test testing whether one variance is larger than the other
I feel like I am missing a key link and this tasks is quite easy:
I am given the following averages and standard deviations:
$\bar{x}_1 = 971$, $s_1 = 42$, $n_1 = 1000$
$\bar{x}_2 = 983.9$, $s_2 = 60....
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Estimate confidence interval for the variance of whatever population distribution
Given a sample of the variable $X$ = ($X_1$, .., $X_n$).
If the $X$ follows a normal distribution, I know that we can estimate the confidence interval of the variance of $X$ by using the $\chi^{square}...
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Measures of variability that do not depend on the sample size
I want to compare the range of scores across two samples with differing sample sizes. The usual measures, standard deviation (SE), standard error (SE), and variance all depend on sample size, with n ...
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How to estimate the variance of correlated observations?
Assume we have n observations $x_i$ (i from 1 to n), each from the a normal distribution with mean 0 and some variance component: $X_i \sim N(0, \sigma^2)$. The random variables $X_i$s have some (let'...
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Intuition for hypergeometric variance?
I'm trying to learn the major facts about a bunch of probability distributions, hypergeometric included. I can use the commonalities between it and a binomial to my advantage for thinking through some ...
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Removing variables from a linear regression improves $R^2_{adj.}$
I am working on a linear regression model.
The complete model with 11 variables in total has a quite low adjusted R-squared ($R^2_{adj.}$) of 0.11.
4 variables have a significant influence on the DV....
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How to estimate the effect of sample size on the variance of a statistic (using a given sample)?
Question
Let's say I have some sample, of size N, with observations ($y_i$ with $i=1,...,N$). And I have some statistic based on the observations (say $S = f(y_1, ..., y_N)$).
I would now like to ...
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How to interpret a PCA Biplot? [duplicate]
I constructed a PCA plot from a very high-dimensional dataset that contains features relating to fraud. After creating the PCA plot, I created a biplot with the features to see how they interact. The ...
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Expectation and Variance of Sum of dependent discrete variables
Q. Let X be a discrete random variable such that X = 0 with probability 0.5 and X = 1 with probability 0.5. Let Y be a discrete random variable such that Y = 1 when X = 1 and Y = 0 when X = 0. What is ...
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Distribution of solution to linear system
I have a random symmetric matrix $ A \in \mathbb{R}^{M \times M}$, and random vector $b \in \mathbb{R}^M$. I also have access to expressions for the mean and variance of each element of $A$ and $b$ (...
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Factor Analysis in R with FAMD / PCA - Explained variance very low
I used a the FAMD algorithm of R on my own dataset.
My dataset contains about 10 columns of numerical(float) data, 6 columns of one-hot-encoded data and 1 column with categorical data (zip codes) with ...
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When is subsetting survey data prior to analysis NOT a problem?
Survey researchers are typically advised to not subset their data prior to analysis because it will produce incorrect variance estimates. My understanding of the inferential issue is that removing ...
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Variance of number of unique elements drawn with replacement
This question is similar to the one asking for the expected number of unique elements drawn from a uniform distribution but with the difference I am interested in calculating (or at least ...
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Is there a way to quantify the amount of spread of a plot of mean values over time?
I have a plot that is a mean of multiple measures of a motion over time with error bars 1 standard deviation away on the positive and negative side, which are calculated for each time point. I am ...
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Can bootstrapping be used to find the standard error of variance?
I am a bit new to the field and had a question regarding bootstrapping. As bootstrapping gives an idea of the standard deviation of the sampling distribution of the mean, does it also give an idea of ...
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Why do we prefer unbiased estimators instead of minimizing MSE?
I was thinking about why, usually, $\hat{\sigma}^2=\hat{p}(1-\hat{p})$ is used to estimate the variance in a Bernoulli population instead of $s^2=\hat{p}(1-\hat{p})\frac{n}{n-1}$.
$s^2$ is unbiased, ...
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How does minimizing MSE maximize variance?
I was reading about PCA and they had this interesting line that "A line or plane that is the least squares approximation of a set of data points makes the variance of the coordinates on the line ...
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Given the range $R$ of $n$ data points, what is the range of these points' standard deviation? [duplicate]
Given the range $R$ ($= max - min$) of $n$ data points ($x_1, x_2, ..., x_n$), what is the range of these points' standard deviation $\sigma$? It's easy to see that the min value of $\sigma$ can be $0$...
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What can be deduced about the covariance between f(x) and f(x + K) for constant K?
The question concerns transformations of a random variable $X$ by a function $f$. I’d like to understand whether there are conditions that could make:
$$\operatorname{covar}(f(X), f(X + K) ) \geq \...
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Pooled Variance from two Variables' SD, but no correlation?
Is there any way to calculate a pooled variance in a within subjects design when I have the SDs and sample size, but no correlation between X and Y?
I'm trying to calculate an effect size using single ...
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Stratified sampling ratio with srswor
The variable X under study have rectangular distribution with interval (a, a+ d ) the interval is divided into k equal subintervals which form k equal strata of equals sizes . From each stratum ...
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How can I calculate variance of log Odds Ratio if Odds Ratio extracted from the study is pre-calculated?
I want to conduct a meta-analysis using odds ratio as my metric. The studies I am using provided pre-calculated odds ratios only, along with their confidence intervals and/or standard error. I need ...
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Hypothesis Test of 2 binomial distributions with unknown and unequal variances
I want to double check if my math and logic is correct in performing a hypothesis test of two sample means from binomial distributions with the assumption of unknown and unequal variances from the ...
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How many times should training data be resampled to reduce variance?
Is there a general rule for the number of times training data for a model should be resampled to reduce variance?
Here is an example
Run 1:
Sample 1) 0.8431
Sample 2) 0.8430
Sample 3) 0.8431
Sample 4) ...
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Estimate population mean from "best of N" samples
If I have a data set for which I know all measurements represent the largest of N observations, is there a good method for estimating the mean of all observations? So for example if N=10 and I have 3 ...
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Variance introduced by dropout
I just read this paper Sampling-free Epistemic Uncertainty Estimation Using Approximated
Variance Propagation and stumbled upon two points at the end of section 3.1 that I don't understand. Here, the ...
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Variance of difference in mean estimator for Average Treatment Effects [duplicate]
Consider $n$ Patients each of whom is treated with probability $p$ and given control with $1-p$ independent of everything else. Let $W_i$ be $1$ if the patient $i$ is treated and $0$ otherwise.
Let $...
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Random walk analysis of a univariate timeseries
As the title describes, I want to conduct a random walk analysis of a univariate time series $Y_t$. What are the tests and steps that you guys would suggest for this purpose?
My current thinking:
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How to calculate $Var(X)$ for the union size?
Let's say there is a set of $4$ bags $\{{a,b,c,d\}}$ containing balls of colors $\{{red,blue,green,orange,black\}}$. Balls are assigned to bags in an arrangement that allows a variable number of ...
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Why a variance of a sample mean is the population variance divided by a sample size?
On the wikipedia page about variance I have found the following equality
$$ Var(\overline{X})=Var(\frac{1}{n}\sum_{i=1}^{n}X_{i})=\frac{1}{n^{2}}\sum_{i=1}^{n}Var(X_{i})=\frac{1}{n^{2}}n\sigma^{2}=\...
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Variance of coefficients in linear regression when responses are correlated
I have a linear regression problem given by:
$Y = X\beta+\epsilon$, where $Y\in R^n, X\in R^{n\times k}, \text{and }\beta \in R^k$. Rather than assuming that the error is independently distributed, i....
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what is the relationship between the two standard errors
I have two variables $X$ and $Y$.
Consider that there is one sample with 1000 observations, we can get the standard error of coefficient by this equation:
$se(\hat\beta) = \sqrt{\sigma^2(X^TX)^{-1}}$.
...
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Is partitioning variance of random and fixed effects in mixed models more sensitive to missing data (rank deficiency)?
I have a follow-up question to my question about modeling the nested crossed design here: (Crossed random effects: how do we model multiple reciprocal transplants in lme4?). We want to partition ...
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Calculating a variance term for AUC using data points extracted from a figure
I am currently performing a meta-analysis where I am pooling area under the curve values.
For some papers I only have access to the figure. For example:
Using WebPlotDigitizer I can extract the mean ...
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Using variance of y variable as weights for weighted linear regression when both x and y variables contain negative values?
I am currently dealing with a weighted linear regression problem in the context of an instrument calibration in analytical chemistry. Let's assume I have a response variable y and a predictor variable ...
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What is the variance of n standard normal distributions? [duplicate]
I have $X_1, X_2, .., X_n$ i.i.d. standard normal random variables. If $\overline{X}_n$:
$$\overline{X}_n = \frac{X_1 + X_2 + .. + X_n}{n}$$
What is $Var(\overline{X}_n)$ in terms of n?
The solution ...
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Unexpected Zero Variance for an Unbiased Estimator: Is the Estimator Consistent?
$\newcommand{\szdb}[1]{\!\left[#1\right]}\newcommand{\szdp}[1]{\!\left(#1\right)}$
Problem Statement: Let $Y_1, Y_2,\dots,Y_n$ denote a random sample from the probability
density function
$$f(y)=
\...
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Forecasting Incoming Payments - Increasing Variance in a Forecast
I am forecasting daily incoming deposits for a bank for the next 3-6 months. This is going to be used to forecast the bank's liquidity going forward. I have used both Holt-Winters and a time series ...