# What is Variance: Definition and 356 Discussions

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. In other words, it measures how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by

σ

2

{\displaystyle \sigma ^{2}}
,

s

2

{\displaystyle s^{2}}
, or

Var

(
X
)

{\displaystyle \operatorname {Var} (X)}
.

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1. ### Understanding Conditional Expectation, Variance, and Precision Matrices

My question relates to subsection 2.2.1 of [this article][1]. This subsection recalls the work of Lindgren, Rue, and Lindström (2011) on Gaussian Markov Random Fields (GMRFs). The subsection starts with a two-dimensional regular lattice where the 4 first-order neighbours of $u_{i,j}$ are...
2. ### Expected value of variance of Hamiltonian in coherent states

I am trying to find the expected value of the variance of energy in coherent states. But since the lowering and raising operators are non-hermitian and non-commutative, I am not sure if I am doing it right. I'm pretty sure my <H>2 calculation is right, but I'm not sure about <H2> calculation...
3. ### MHB TEST OF HYPOTHESIS INVOLVING THE POPULATION MEAN 𝝁 WHEN THE VARIANCE IS UNKNOWN

1. You are working in a company facing attrition problems of the customer service representatives for the past five years. The company president proposed that if the attrition rate is at least 10 per month, then the salary scale, compensation package, and professional development programs for...
4. ### MHB Test of hypothesis involving the population mean 𝝁 when the variance is known

1. The Head of the Mathematics Department announced that the mean score of Grade 11 students in the second periodical test in Statistics was 89, and the standard deviation was 12. One student believed that the mean score was less than this, randomly selected 34 students, computed their mean...
5. ### Calculate the pooled estimate of variance

OK, Let me attempt part (i), first, Here we have; ##s^2_p ##=##\dfrac{ (n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}## ##s^2_p ##=##\dfrac{ (7-1)0.63953+(7-1)0.6148}{7+7-2}## ##s^2_p ##=##\dfrac{3.83718+3.6888}{12}## ##s^2_p ##=##\dfrac{3.83718+3.6888}{12}## ##s^2_p ##=##\dfrac{7.52598}{12}##...
6. ### I Finding Momentum Mean & Variance from Wavefunction

I've a Gaussian momentum space wavefunction as ##\phi(p)=\left(\frac{1}{2 \pi \beta^{2}}\right)^{1 / 4} e^{-\left(p-p_{0}\right)^{2} / 4 \beta^{2}}## So that ##|\phi(p)|^{2}=\frac{e^{-\left(p-p_{0}\right)^{2} / 2 \beta^{2}}}{\beta \sqrt{2 \pi}}## Also then ##\psi(x, t)=\frac{1}{\sqrt{2 \pi...
7. ### A Need help about a demo with inverse weighted variance average

I have a problem of understanding in the following demo : In a cosmology context with 2 probes (spectroscopic and photometric), let notice ##a_{\ell m, s p}## the spectroscopic and ##a_{\ell m, p h}## the photometric coefficients of the decomposition in spherical harmonics of the distributions...
8. ### MHB Why Is Standard Deviation Calculated as 10√2 Instead of 14?

Hi all - I wonder if you can help please. Watching a video on youtube to help me understand about the mean, variance and standard deviation but last part of video left me confused. The speaker said the following for the formula for standard deviation: Consider if the variance is 200 for the...
9. ### I Demonstration of inequality between 2 variance expressions

Just to remind, ##C_\ell## is the variance of random variables ##a_{\ell m}## following a Gaussian PDF (in spherical harmonics of Legendre) : ##C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\operatorname{Var}\left(a_{l m}\right)## 1)...
10. ### A Calculating the variance of integrated Poisson noise on a defined quantity

It is in cosmology context but actually, but it is also a mathematics/statistical issue. From spherical harmonics with Legendre deccomposition, I have the following definition of the standard deviation of a ##C_\ell## noised with a Poisson Noise ##N_p## : ## ...
11. ### Solve the variance problem below - statistics

The question is below: below is my own working; the mark scheme for the question is below here; i am seeking for any other approach that may be there...am now trying to refresh on stats...bingo!
12. ### A Photometric Galaxy Clustering Error and Poisson Noise

The error on photometric galaxy clustering under the form of covariance which is actually a standard deviation expression for a fixed multipole ##\ell## : ## \sigma_{C, i j}^{A B}(\ell)=\Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A...
13. ### A Add factor Δℓ or Δℓ^2 in the variance for integrated Shot noise

I have the following expression for an error on a Cℓ : ##\sigma(C_{\ell,X})=\sqrt{\dfrac{2}{(2 \ell+1)\Delta\ell \,f_{\mathrm{sky}}}}\left[C_{\ell,X}+N_{X}(\ell)\right]## where ##X## corresponds to spectroscopic/photometric shot noise and with ##\Delta\ell## is the bin width between 2 values...
14. ### How do I calculate variance for volume, 𝑉 (i.e, ⟨Δ𝑉2⟩=⟨𝑉2⟩−⟨𝑉⟩2)?

This is not actually a homework problem. I'm old but having trouble with something that's probably at student level because it's so long since I learned this stuff. I would be grateful if someone would please take pity on me and help me out! I am trying to calculate something that includes...
15. ### Matching couples at a party (mean and variance)

Hi, I was attempting the following problem and I didn’t know how to start it off correctly. Question: At a party there are ##n## couples. When the last song comes on, each person randomly picks a dance partner. What is the: (a) mean, (b) variance of the number of couples that are paired...
16. ### A Introduction of a factor Δℓ when summing equal distants 𝐶ℓ

Hello, In the context of Legendre expansion with ##C_\ell## quantities, below the following formula which is the error on a ##C_\ell## : ##\sigma_(C_{\ell})=\sqrt{\frac{2}{(2 \ell+1)\Delta\ell}}\,C_{\ell}\quad(1)## where ##\Delta\ell## is the width of the multipoles bins used when computing...
17. ### MHB Likelihood Ratio Test for Common Variance from Two Normal Distribution Samples

$\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Let $S_1^2$ and $S_2^2$ denote, respectively, the variances of independent random samples of sizes $n$ and $m$ selected from normal distributions with means $\mu_1$ and $\mu_2$ and common...
18. ### Variance of a point chosen at random on the circumference of a circle

Hi, I was looking at this problem and just having a go at it. Question: Let us randomly generate points ##(x,y)## on the circumference of a circle (two dimensions). (a) What is ##\text{Var}(x)##? (b) What if you randomly generate points on the surface of a sphere instead? Attempt: In terms of...
19. ### Force pressure variance after hole is plugged?

There is a tall cylinder filled with water. And there is a 3 in diameter hole near the bottom and water is gushing out. (assume the cylinder is continually being re-filled from the top) You work to plug the hole with a 10 inch long cylinder that is exactly the perfect diameter fit to plug the...
20. ### I How to calculate expectation and variance of kernel density estimator?

This is a question from a mathematical statistics textbook, used at the first and most basic mathematical statistics course for undergraduate students. This exercise follows the chapter on nonparametric inference. An attempt at a solution is given. Any help is appreciated. Exercise: Suppose...
21. ### A Analytical expression of Cosmic Variance - Poisson distribution?

I have an expression of Matter Angular power spectrum which can be computed numerically by a simple rectangular integration method (see below). I make appear in this expression the spectroscopic bias ##b_{s p}^{2}## and the Cosmic variance ##N^{C}##. ## \begin{aligned}...
22. ### B Why is V(1/5X) equal to 1/25*V(X)?

In my book, when calculating the variance of X = (x_1 + x_2 + x_3 + x_4 + x_5)/5 in an example it says: V(X) = V(1/5(X_1 + X_2 + X_3 + X_4 + X_5)) = 1/25*V(X_1) + 1/25*V(X_2) + 1/25*V(X_3) + 1/25*V(X_4) + 1/25*V(X_5) = 1/5Ф I don't understand how V(1/5X) can be turned into 1/25*V(X), shouldn't...
23. ### A Expression of the mean and variance of spectroscopic Shot Noise

Hello, I would like to know the right expression for the expression of variance of Shot noise in spectroscopic probe. Sometimes, I saw ##\sigma_{SN,sp}^{2} = 1/n_{sp}## with ##n_{sp}## the average density of galaxies, whereas my tutor tells me that ##\sigma_{SN,sp}^{2} = 1/n_{sp}^{2}## , so I...
24. ### A Fourier transform and Cosmic variance - a few precisions

I cite an original report of a colleague : 1) I can't manage to proove that the statistical error is formulated like : ##\dfrac{\sigma (P (k))}{P(k)} = \sqrt{\dfrac {2}{N_{k} -1}}_{\text{with}} N_{k} \approx 4\pi \left(\dfrac{k}{dk}\right)^{2}## and why it is considered like a relative error ...
25. ### A Calculate variance on the ratio of 2 angular power spectra

In the context of Survey of Dark energy stage IV, I need to evaluate the error on a new observable called "O" which is equal to : O=\left(\frac{C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}}{C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}}\right)=\left(\frac{b_{s p}}{b_{p...
26. ### A Computing a variance in astrophysics context

Below the error on photometric galaxy clustering under the form of covariance : $$\Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A B}(\ell)+N_{i j}^{A B}(\ell)\right]$$ where ##_{\text {sky }}## is the fraction of surveyed sky and ##A, B##...
27. ### MHB Distribution, expected value, variance, covariance and correlation

Hey! :giggle: Let $X$, $Y$ and $Z$ be independent random variables. Let $X$ be Bernoulli distributed on $\{0,1\}$ with success parameter $p_0$ and let $Y$ be Poisson distributed with parameter $\lambda$ and let $Z$ be Poisson distributed with parameter $\mu$. (a) Calculate the distribution...

43. ### Mean, variance and correlation function of Langevin equation

Homework Statement I have simulated Langevin equation (numerically in Matlab) for some specific conditions, so I have obtained the solution ##X(t)##. But now, with the solution I have obtained, I have to calculate ## <X(t)|x_0>, <X^2(t)|x_0>-(<X(t)|x_o>)^2 ## and the conditional correlation...
44. ### Calculating Expected Value and Variance of Coin Toss Results

A coin had tossed three times. Let ##X##- number of tails and ##Y##- number of heads. Find the expected value and variance ##Z=XY##. My solution: We know, that ##Y=3-X##, so ##Z=(3-X)X## for ##X=0,1,2,3##. ##Z=2## for ##X=1,2## and ##Z=0## for ##X=3,0## So, ##E(Z)=E((3-X)X))= 2 \cdot ⅜ +2 \cdot...
45. ### Showing Rejection Region Equality with Fisher Distribution

Homework Statement [/B] For reference: Book: Mathematical Statistics with Applications, 7th Ed., by Wackerly, Mendenhall, and Scheaffer. Problem: 10.81 From two normal populations with respective variances ##\sigma_1^2## and ##\sigma_2^2##, we observe independent sample variances ##S_1^2## and...
46. ### B Random Variable - Mean and Variance

Problem: We play roulette in a casino. We watch 100 rounds that result in a number between 1 and 36. and count the number of rounds for which the result is odd. assuming that the roulette is fair, calculate the mean and deviation Solution: I understand that the probability - Pr = 0.5. and...
47. ### Prove that ##\psi## is a solution to Schrödinger equation

Homework Statement For a wavefunction ##\psi##, the variance of the Hamiltonian operator ##\hat{H}## is defined as: $$\sigma^2 = \big \langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)^2 \psi \big\rangle$$ I want to prove that if ##\sigma^2 = 0##, then ##\psi## is a solution to the...
48. ### Variance of binomial distribution

Homework Statement Random variable Y has a binomial distribution with n trials and success probability X, where n is a given constant and X is a random variable with uniform (0,1) distribution. What is Var[Y]? Homework Equations E[Y] = np Var(Y) = np(1-p) for variance of a binomial...
49. ### A What is the variance of a Gaussian RV

Hi, Let y = x + z, where x and z are mutually independent RVs. Also, z is a complex gaussian RV with zero mean and variance sigma^2. My question is as follows: For x = y - z, what is the variance of (-z) ? Any help could be useful. Thanks in advance.
50. ### Variance with Poisson distribution

<Moderator's note: Moved from a technical forum and thus no template.> So, I have this problem and I am stuck on a sum. The problem I was given is the following: The probability of a given number n of events (0 ≤ n < ∞) in a counting experiment per time (e.g. radioactive decay events per...