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. 2

    Variance in position for the infinite square potential well?

    [Note from mentor: this thread originated in a non-homework forum, therefore it doesn't use the standard homework template] ------------------------------------------ This exercise pops up in the Cavendish Quantum Mechanics Primer (M. Warner and A. Cheung) but I can't seem to figure it out. So...
  2. D

    Expected value and variance of these sums

    Hi guys, Suppose I have the function x = a + b -1 where a, b have expected values of 0.5 each. What is the expected value of x? is it 0.5 + 0.5 -1 = 0? or is it just 0.5 + 0.5? Secondly, suppose the same equation as above, x = a + b -1. If the variance of both a and b is 1/12, what is the...
  3. B

    Mean and variance of loggamma distribution

    The loggamma distribution is defined by $$ g(x) = \frac{1}{ \Gamma ( α) θ^{ α} } \frac{(ln( x))^{ α - 1}}{x^{1+\frac{1}{θ}}} $$, for $$ 1 < x < ∞ $$ where α is a positive integer. I've been trying to find the mean and variance of this distribution. It's been somewhat frustrating because the...
  4. D

    Find mean and variance of a random vector

    Homework Statement The components of a random vector ##\mathbf{X} = [X_1, X_2, \ldots, X_N]^{\intercal}## all have the same mean ##E_X[X]## and the same variance ##var(X)##. The "sample mean" random variable $$ \bar{X} = \frac{1}{N}\sum_{i = 1}^NX_i $$ is formed. If the ##X_i##'s are...
  5. H

    Calculating Variance for Y=3x^2+3x+3

    Homework Statement Find the variance of Y=3x^2+3x+3The Attempt at a Solution Let Y = 3x^2 +3x +3 Var(Y) = Var(3x^2 +3x +3) = 9Var(x^2) +9Var(x) = 9 [E[X^4] - E[X^2]^2 +E[X^2] - E[X]^2] This is wrong.
  6. Mogarrr

    Expected Value and Variance for Wilcoxon Signed-Rank Test

    Using a normal approximation method for the Wilcoxon Signed-Rank Test, I've seen that the expected value is \mu = \frac {n(n+1)}2 and the variance is \sigma^2 = \frac {n(n+1)(2n+1)}{24} . I'm wondering why these are the expected value and variance. I do recognize the formula for the sum of...
  7. G

    What is the Significance of Testing Population Variance in a Sample?

    Homework Statement Given that 8 person vote for Mr Tan , i can't understand why the probability is sum of 0 people vote for him until 8 people vote for him. Why not the probability = 22c8 ( (0.6)^8 )( (0.4)^14) ? Homework EquationsThe Attempt at a Solution
  8. D

    MHB Mean and variance of difference operators on a time series process

    \text{Consider the following decomposition of the time series }{Y}_{t}\text{ where }{Y}_{t}={m}_{t}+{\varepsilon}_{t},\text{ where }{\varepsilon}_{t}\text{ is a sequence of i.i.d }\left(0,{\sigma}^{2}\right)\text{ process. Compute the mean and variance of the process }{\nabla}_{2}{Y}_{t}\text{...
  9. P

    MHB Confused about law of total variance

    Ok, so I got this question on an exam some time ago and I still don't understand why I didn't get it (I can't remember the exact question, but this is very similar): "A lottery winning amount is determined in the following manner: first a die is thrown. If the result is 1 or 2, the lottery...
  10. W

    Radioactivite decay variance due to distance from the sun?

    Hello! I'm not entirely sure this is very "scientific", but I have an idea that I would like some feedback on. I remember reading an article related to radioactive decay varying with the rotation and distance from the sun on Phys.org a few years back (http://phys.org/news202456660.html) and was...
  11. A

    Comparing the variance of two samples with differing measurement error

    Homework Statement This is a statistics as applied to astronomy problem. My stats knowledge is horrible. Anyway, the problem: For several hundred objects, I have a number of different properties for which I have a measurement & associated measurement error. For example, flux density and...
  12. J

    Statistics: independently distributed mean and variance

    Homework Statement Math and verbal SAT scores are each N(500, 10000) 1)If the math and verbal SAT scores were independently distributed, which is not the case, then what would be the distribution of the overall SAT scores? Find its mean and variance. Homework Equations The...
  13. S

    Variance of Geometric Brownian motion?

    I am trying to derive the Probability distribution of Geometric Brownian motion, and I don't know how to find the variance. start with geometric brownian motion dX=\mu X dt + \sigma X dB I use ito's lemma working towards the solution, and I get this. \ln X = (\mu - \frac{\sigma...
  14. M

    Understanding Variance and Kurtosis: A Brief Explanation

    hello again pf! as a really simple question, can someone talk to me about the difference between variance and kurtosis? i know as kurtosis decreases from 3 (normal distribution) our pdf is shorter and fatter, with less weight in the tails. i also know variance tells us how dispersed data is...
  15. E

    MHB Conditional variance calculations (Crypto-currency reward offered)

    I'm reading a journal article that implies the following but I can't see how it is done. I'll give 100 DogeCoin (or equivalent) to whomever can explain this in full. Given that V(A|B) = s V(A) = r*s + w B = A + C and A & C are independent so V(B) = V(A) + V(C) & V(C) = V(B) - V(A) Then how...
  16. T

    Variance of Estimator: Learn How to Calculate

    Hi there. I would like to ask you one question about variance of estimator. Suppose that Y_i=βX_i+ε_i and β estimator is \bar{Y} / \bar{X}. I calculated mean of estimator. I am not sure if it's correct, but i got that its equal to n*β. But how about variance. Any help would be appreciated!
  17. J

    Probability, Correlation, Variance Statistics Homework Help.

    Homework Statement 1. Assume each birth in a hospital on a given day is independent of one another, and each birth, P(boy)=0.48. What is the probability that the 8th baby born is the 5th girl. 2. Two random variables X and Y have joint distribution given by. What is their correlation. 3. Z...
  18. N

    Distribution of sample mean and variance, and variance of sample means

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  19. S

    Calculating Variance of Eq. with random variables

    Homework Statement I am attempting to calculate a heat transfer across a medium with known material properties. I have the equation and all but one variable I have an exact answer for. I require the variance of my answer. Homework Equations I know ALL variables (ie numerical value) except...
  20. U

    MHB Manipulating Taylor Expansion for Sample Mean, Variance, Skewness & Kurtosis

    I have the following expression: $$\frac{1}{p} \ln\left(1+\frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n} \sum_{i=1}^n x_i^2 + \frac{p^3}{3!n} \sum_{i=1}^n x_i^3 + \frac{p^4}{4!n} \sum_{i=1}^n x_i^4 + \cdots \right)$$ Now let $$Y = \frac{p^1}{1!n}\sum_{i=1}^n x_i + \frac{p^2}{2!n}...
  21. E

    Estimate the error variance of Autoregressive model

    Hi everyone, hopefully someone can help For an vector AR(1) model of the form y(t)=Ay(t-1) + et How does one estimate the variance of the white noise input. I was under the impression that one can simply use the residual of the model fit and estimate the variance from this, when I do a...
  22. M

    Probability - expectation and variance from a coin toss

    Homework Statement A coin is flipped repeatedly with probability p of landing on heads each flip. Calculate the average <n> and the variance \sigma^2 = <n^2> - <n>^2 of the attempt n at which heads appears for the first time. Homework Equations \sigma^2 = <n^2> - <n>^2 The...
  23. A

    MHB How do I find the variance and what is the answer to the question?

    Ive been stuck at this question for over 5 hours now trying to figure out what the answer is. Please help!:confused: We have observed 250 cars on a motorway with speed limit of 90 km/h Speed number of cars 75 32 85...
  24. U

    Average and variance of a unit sphere

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  25. R

    Mean, variance of non-parametric estimator

    Homework Statement For the nonparameteric estimator \hat{f}(x)=\frac{1}{2hn}\sum\limits_{i=1}^n I_i(x) of a pdf, (a) Obtain its mean and determine the bias of the estimator (b) Obtain its variance Homework Equations The Attempt at a Solution For (a), I think it goes like this...
  26. M

    Negative variance of an observable quantity

    Quantum mechanics has a well-known procedure for evaluating the expectation value of an observable quantity in a given quantum state. First one must obtain the quantum operator O that is associated with the observable quantity. Then the rule for computing the expectation value is: Apply O to the...
  27. E

    Optimizing Point Estimates: Bias and Variance Analysis for Mean Estimation

    Homework Statement Suppose that: E(X1) = μ, Var(X1) = 7, E(X2) = μ, Var(X2) = 13, E(X3) = μ, and Var(X3) = 20, and consider the point estimates: μˆ1 = X1/3 + X2/3 + X3/3 μˆ2 = X1/4 + X2/3 + X3/5 μˆ3 = X1/6 + X2/3 + X3/4 + 2 (a) Calculate the bias of each point estimate. Is...
  28. Y

    Distribution of Log of Variance

    Homework Statement If Y_1, Y_2, ... are iid with cdf F_Y find a large sample approximation for the distribution of \log(S^2_N), where S^2_N is the sample variance. Homework Equations The Attempt at a Solution The law of large numbers states that for large N S^2_N converges in...
  29. G

    Some probability questions - mean , variance , uniformly chosen points

    Hi there , I have some simple (on 1st look) questions. I just need some help.http://img59.imageshack.us/img59/3018/5hby.png 1st question. I think the expected value should be 150 since the probability of getting a head is 1/2 .. 300x1/2 = 150 so the mean should also be 150. the variance...
  30. U

    Error Propagation - Estimating Variance

    Homework Statement Not exactly a homework question, but rather a section in Statistical Data Analysis: Suppose there is a pdf y(x)[/SUB] that is not completely known, but μi and Vij are known: Homework Equations The Attempt at a Solution I understand how <y(x)> ≈ y(μ), My confusion: Why...
  31. U

    MHB Finding the conditional variance and CDF

    Question: Assume a bivariate GARCH process as follows: \begin{align} r_{mt} &= \sigma_{mt}\epsilon_{mt} \ \ \ \cdots \ \ \ \text{(1)} \\ r_{it}&=\sigma_{it}\rho_{it}\epsilon_{mt}+\sigma_{it}\sqrt{1-\rho_{it}^2}\xi_{it} \ \ \ \cdots \ \ \ \text{(2)} \\ (\epsilon_{mt}, \xi_{it}) & \sim S...
  32. G

    Variance of Y^p: How to Calculate Using Conditional Probabilities

    X ~ standard uniform random variable We toss a coin randomly and define Y := { X if the coin toss is heads ...{ 1 is the coin toss is tails Question wants the Var(Y^p) for any p > 0. My work: Var(Y^p) = E(Y^(2p)) - E(Y^p)^2 I'm not sure how to go about finding E(Y^p) and E(Y^(2p)). I thought...
  33. F

    MHB Mean and variance from a probability distribution function

    f(x)=f(x)={█(2/(√2π) e^(〖-x〗^2/2)@0 otherwise)┤for 0<x<∞ Find the mean and variance of X The hint says, compute E(X) directly and then compute E(X2) by comparing that integral with the integral representing the variance of a variable that is N (0, 1)
  34. D

    How Do You Calculate Standard Deviation for Linear Combinations?

    I have attached the problem of interest. I think I am having trouble calculating the standard deviation for part a. D=A-B-C E(A)=10 E(B)=2 E(C)=2 sigma(A)=0.1 sigma(B)=0.05 sigma(C)=0.1 where sigma=standard deviation so to find SD...
  35. T

    Show the two forms of the sample variance are equivalent

    Homework Statement Showthe two forms of the sample variance are equivalent: \frac{1}{n-1}\sum_{i=1}^\n (Yi-Ybar)2 = \frac{1}{n(n-1)}\sum_{i=1}^\n \sum_{j>i}\n (Yi-Yj)2 The first summation is from i=1 to n, the second is i=1 to n and the third is j>i to n. Sorry, I don't know how to format...
  36. D

    What is the difference between Variance and Covariance?

    Could someone please explain it to me what is the difference between Variance and Covariance?
  37. dexterdev

    To find mean wedivide the sum of no.s with 'n'for variance why 'n-1'

    Hi all, For finding average we take the sum of sequence numbers and divide by the number of elements. Why for variance this changes to number of elements minus 1. -Devanand T
  38. DavideGenoa

    Variance of statistic used in runs test

    Hi, friends! Since this is my first post, I want to present myself as an Italian who is trying to teach himself mathematics and natural sciences, while having a strictly humanities-centered school background, and I am tempted very much to enrol in a university scientific course. I read in the...
  39. S

    Variance captured in coordinate axis.

    Hi all, Note: The text below is the motivation for my question. To jump to the question immediately, please skip to the line that says HI! I have a set of data points, let's call it A, and I ran principal component analysis to get the top 3 principal components to be able to represent the...
  40. A

    Why Is Standard Deviation Defined Using Squared Differences?

    1) For the normal distribution it seems that the integral of the propability density function from \mu-\sigma to \mu+\sigma is independent of \sigma. I guess that gives kind of a nice interpretation of \sigma. But how do you prove this, when the antiderivative of an exponential with a square...
  41. C

    How is the Variance in Particle Number Derived?

    I saw an equation on wikipedia: (http://en.wikipedia.org/wiki/Fermi-Dirac_statistics) Does anybody know how this is derived?
  42. W

    MHB Establish Whether populations have equal variance

    day 1 distance equals 52.175m, std = 0.015m and n = 10 day 2 distance equals 52.193m, std = 0.021m and n = 11 Establish whether the two populations have equal variance at the 0.05 signifcance level. Can someone help me out with how to get to the answer and what the answer actually is?
  43. T

    Proving variance with moment generating functions

    Moment generating functions: How can I show that Var(X)=\frac{d^2}{dt^2}ln M_X(t)\big |_{t=0} Recall: M_X(t)=E(e^{tx})=\int_{-\infty}^{\infty}e^{tx}f(x)dx E(X^n)=\frac{d^n}{dt^n}M_X(t)\big |_{t=0} Var(X)=E(X^2)-[E(X)]^2=E[(X-E(X))^2] ------------ I tried just applying the equation...
  44. G

    Variance of normally distributed RV

    Homework Statement The time it takes for a nurse to look up a patient journal is uniformly distributed between three and seven minutes. One morning there's 96 journal orders for the nurse to take care of. Calculate the probability that she will get them all done during an eight hour work...
  45. A

    MHB Calculate the Expected value, Variance, density and 2nd moment

    I have to solve this exercise. Here are also my solutions but I don't know if they're correct. Let Ω ⊆ R ^ 2 a finite set of points in R ^ 2. Notation v_i = (x_i, y_i). Pr [] is a probability measure on either Ω. Random variables X, Y: Ω-> R project a point on the coordinate. For example Y(v_i)...
  46. E

    Finding mean of Y if Y=(X1+X2+X3)/3 given mean and variance of x's

    Let X1, X2, X3 be three random variables. Suppose all three have mean μ and variance 1. The sample mean is Y = (X1 +X2 +X3)/3. (a) Can you compute the mean of Y? If so, what is it? If not, why not? I have that it is either μ OR that it is not possible to find, since we don't know if they...
  47. Z

    Three-Factorial Analysis of Variance

    Homework Statement I understand how the two factor analysis of variance works but adding a third level is giving me some trouble. The model I am trying to analyze has three factors with 2,3, and 4 levels respectively. Can anyone give me an example of a three-factor analysis of variance...
  48. J

    Instead, you should be using the t distribution with 39 degrees of freedom.

    Homework Statement Measure of Contaminatino has a normal distribution w/ unknown mean and unknown variance. Random sample of n=40 provides sample mean = 28.30 and sample variance = 17.38 1)Find Upper 95% Confidence Interval 2)Test Null Hyp = 25.5 vs Alt Hyp = 28.30... Upper Tailed...
  49. P

    Calculating Variance of Y with X1, X2,...,X15

    The random variable X1, X2,...,X15 are independent and take each values ​​ +1,-1 with probability 1/2. We define Y = sum from j=1 to 15(j*Xj) whats is the variance VAR(Y)=? i will find this with VAR(Y)=E(X^2)-(E(X)^2) but how i can find them?
  50. C

    Variations of the Variance Formula

    The variance is denoted by σ^{2} It is calculated with this equation: σ^{2}=\frac{\sum^{N}_{i=1}(Xi-μ)^{2}}{N} Which makes sense. To calculate the average (deviation from the mean)^2 you need to sum up the (deviations from the mean)^2 and then divided by the number of deviations. The reason...
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