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)}
.

View More On Wikipedia.org
Recent contents Top users
  1. K

    Finding expectation and variance

    So for example, if I have a random variable X, take it to be normally distributed. How do you find the expectation and variance of the random variable e^X in terms of μ and σ? Integrating the entire normal function with the f(x) is it?
  2. B

    The Speed of Light: Variance in Non-Vacuum Mediums

    Does the speed of light vary with different frequencies(or wavelengths) in mediums other than vacuum, or is it constant for all light regardless the frequency?
  3. C

    Exploring Kurtosis: The Role of 4th Central Moment & Variance

    I'm studying on statistics. Then, I saw 'Kurtosis', that represents 'peakness' of the distribution. In the text, the kurtosis is defined as 4-th central moment devided by square of variance. But, I can't understand why the standized 4-th central moment is used. What is the role of the...
  4. T

    A little help understanding standard deviation & variance

    Hello all This is not homework, i work in engineering. I have some data in a table, there are 3 columns and 5 rows. The data relates to how high or low one rail is to the other. The data was collected across 3 days. At each 7m intervals the height of the right hand rail was...
  5. ArcanaNoir

    Bivariate expected value and variance

    Homework Statement I need to know these formulas to answer the homework problems, but I can't squeeze the forumlas out of the gibberish in the book, so I'm asking for varification of the formulas. For a bivariate probablity density function, for example f(x,y)= 2xy when x and y are...
  6. R

    Is the Variance of a 2D Random Walk Simply 2n?

    I've made a 2D walker to compare different RNG's. I'm measuring the succes of each walk as the distance from the origin to the endpoint, using the regular 2-norm. The thing I can't seem to work out is the variance. D_n=\sqrt(x_n^2+y_n^2) Var(D_n)=E[D_n^2]=E[Z_1^2+...+Z_n^2] Since...
  7. J

    Graphs: Expected Number of Triangles and Variance

    Homework Statement Let G be a random graph on n vertices: 1) What is the expected number of triangles in G? 2) What is the variance in the number of triangles? Homework Equations N/A The Attempt at a Solution I think I can do (1) by using indicator variables. In particular, let...
  8. Z

    Probability: The birth problem - mean and variance

    Homework Statement Suppose that a married couple in Canada decide to have babies until they get the first girl baby. It is well-known that in high-latitude countries, the chance of have a girl is slightly higher than the chance of having boy. Suppose that the chance of having a girl in Canada...
  9. H

    Independent random varables with common expectation and variance

    Homework Statement Suppose X1 , X2 , . . . , Xn are independent random variables, with common expectation μ and variance σ^2 . Let Sn = X1 + X2 + · · · + Xn . Find the variance of Sn. The attempt at a solution Expected value: E[S_n] = n E[X_i] = n\mu \hspace{10 cm} (1)...
  10. R

    Mean and Variance of Lognormal Distributions

    Homework Statement Given that the natural log of the growth consumption rate is conditionally normally distributed. I am trying to convert it to a lognormal distribution, but I keep getting a variance that is different from what is in the solution manual. The problem is #3 in the document below...
  11. D

    Variance and Expected Value Problem

    Give an example of a random variable (i.e. give the range of values it takes and its p.m.f.) with the following properties: EX = 4, VAR(X)=4. Now give an example of a random variable with a different p.m.f. than...
  12. K

    Need help with a conditional variance proof.

    need urgent help with a conditional variance proof. I have been given this problem and I'm pretty stumped. I want to prove that Y=g(X) if and only if var(YlX) = 0. so if var(YlX)=0 then E(Y^2lX) - E(YlX)^2 = 0 E(Y^2lX) =E(YlX)^2 so what should I do now? I tried showing that this...
  13. F

    Bayes formula and picking variance from distribution

    This should be rather simple bayesian problem, but I can't figure it out for myself. If i pick numbers from normal distributionS, where the variance of the distribution at each pick v1, is in turn picked out of a normal distribution with variance v2. What is then the distribution of the...
  14. K

    How Do You Calculate the Standard Deviation of Profit Per Lottery Ticket?

    [solved]Expected Value and Variance Hi, I have a problem on Expected Value and Variance, and having spent hours but still couldn't figure out :( One state lottery has 200 prizes of $1 100 prizes of $5 40 prizes of $25...
  15. E

    Calculating Overall Variance & Standard Deviation for 3 Sets of Data

    Hi If I have measured the resonance frequency of three sets of resonators and calculated the mean, variance and standard deviation for each set. How do I add the three variances and standard deviations to get an overall variance and standard deviation? Well, I know that the standard...
  16. B

    NMR Heat Variance: Magnetic & Electromagnetic Fields

    In the Nuclear Magnetic Resonance, do the applied magnetic and electromagnetic fields correspond linearly to the heat generated? If not, how do they vary? Gracias
  17. Rasalhague

    Analysis of Variance: Is Toy Color a Factor?

    Well, just as I thought I'd got the hang of this... Koosis: Statistics: A Self-Teaching Guide, 4th ed., §§ 6.29-43. The "degreeses of freedom" are 3 and 36. This critical value, 4.38, is found by looking up the score for 1% in the table at the back of the book, or in Excel with...
  18. L

    Variance (error bars) with a binomial proportion

    I have a list of chemicals, their assay test results, and a binomial column of whether or not the assay test result was high enough to be considered a threat (anything >2g/ml). Some chemicals were tested more than once, but others were not. It is understood that it is a poor set of data, but I...
  19. W

    Paired samples-equality of variance and 95% CI around difference in variances

    Hi, Can a few of you please review the approach I plan to take for obvious errors? I have 50 subjects and each have a measure taken on the same variable before and after treatment. So, this is standard paired t-test time, but what I am actually interested in is the variance of the treatment...
  20. K

    Mean and Variance of Random Walk

    I'm reading a stat textbook and it says the following: Let a discrete-time random walk be defined by Xt = Xt-1 + et, where the et's are i.i.d. normal(0,σ2). Then for t≧1, (i) E(Xt) = 0 (ii) Var(Xt) = t σ2 However, the textbook doesn't have a lot of justifications for these results and...
  21. T

    Calculate expected value and variance of d, d = sqrt(x^2+y^2)

    I'm bad at stochastics so really glad for any help Homework Statement I have two normally distributed NON INDEPENDENT stochastic variables X~N(muX,sigX^2) and Y~N(muY,sigY^2) A third variable D is defined as D = sqrt(X^2 + Y^2). Since Y and X are stochastic D will also be stochastic...
  22. J

    Time Variance of y(t)=x(2t) System

    Homework Statement show whether the system y(t) = x(2t) is time variant or notHomework Equations a system is time invariant if a time shift in the input signals results in an identical time shift in the output signal, that is if y[n] is the output of a discrete-time, time invariant system...
  23. J

    Checking regular variance around 0, hypergeometric fucntion

    Homework Statement A function g is \alpha-regularly varying around zero if for all \lambda > 0, \lim_{x\to 0} \frac{g(\lambda x)}{g(x)}=\lambda^{\alpha} For real s and \alpha \in (0,1), define f: f(s)=1-\alpha \int_{0}^{\infty} e^{\alpha t}...
  24. R

    Variance of square of random variable

    Homework Statement Lets say I roll 2 fair dice and take the sum of the square of each dice. What formula will be the variance? Homework Equations var(x)=e(x^2)-e(x)^2The Attempt at a Solution For dice A; E(A)=3.5 E(A^2)=91/6 ^ same for dice B. VAR(A^2+B^2)=E(A^4)-E(A^2)^2+E(B^4)-E(B^2)^2 ?
  25. L

    Calculation of sample variance

    I am new in statistic. I come across the sample variance calculation in a book and it explains that denominator is divided by n-1 instead of n is because variance in samples will be likely to be lower than the population variance, so we divide by n-1 to make the variance larger. However, when...
  26. A

    What is the Variance of Tossed Coins?

    Homework Statement Let Y denote the number of heads obtained when three fair coins are tossed.The variance of Y2 is Homework Equations The Attempt at a Solution MY problem is understanding what Y2 is. i have tried to calculate VAR(Y*Y) but my answer is wrong
  27. V

    Statistics proof, identically distributed RVs and variance

    Homework Statement Show that for identically distributed, but not necessarily independent random variables with positive pairwise correlation ρ, the variance of their average is ρσ^2 + (1-ρ)σ^2/B. ρ - pairwise corellation σ^2 - variance of each variable B - number of samples...
  28. S

    Increasing variance of weights in sequential importance sampling

    Hi all, I know about these facts: 1- The variance of importance weights increases in SIR (also know as the degeneracy problem). 2- It's bad (lol), because in practice, there will be a particle with high normalized weight and many particles with insignificant (normalized) weights. But I...
  29. J

    Negative binomail distribution and its variance

    Hi, I am trying to estimate variance for negative Binomial distribution using maximum likelihood estimation and Expected (Fisher's) information to determine its variance. I know what variance is for this distrubution but I cannot derive it. Here is my solution. Any comments and...
  30. L

    Is the sample mean and variance always unbiased?

    I'm wondering if the sample mean \sum{x_i}/n and sample variance \frac{1}{n-1}\sum{(x_i-\bar{x})^2} is always an unbiased estimate of the true expected value and variance of the random variable X, where x_i are iid samples. Or at least asymptotically unbiased. I don't think it is, since the...
  31. M

    Variance analysis and regression

    Homework Statement Assume a one way variance analysis model on the form: Y_{ij} = \mu + \alpha_{i} + e_{ij} where e_{ij} independent with expectation 0 and constant variance z_{ijl} = \left\{ \begin{array}{rcl} 1 & \mbox{for} & 1 \\ 0 & \mbox{else} \end{array}\right show that: a)...
  32. O

    Calculate Variance of Tomato Crop Income: Steps & Answers

    is known that the Tomato crop (in ton) in some farm are Sampled for 10 years. the Standard deviation of the crop was 2 ton. the Income (Y) from the Tomato Depends on the crop (X) according to following connection Y=3X-2 the Variance Income from the Tomato in this Sampled is 4? if i...
  33. L

    Estimator for variance when sampling without replacement

    Does anyone know the formula for an unbiased estimator of the population variance \frac{1}{n}\sum_{i=1}^{n}(x_i - \bar{x})^2 when taking r samples without replacement from a finite population \{x_1, \dots, x_n\} whose mean is \bar{x}? A google search doesn't find anything useful other than the...
  34. T

    Test Scores: Variance & Maximum Mark

    Say we have a sample of test scores, all marked between 0 and 100. Does the sample variance have to be less than or equal to the maximum mark 100 or can it exceed this?
  35. S

    Why Does the Expected Value of Sample Variance Differ From Population Variance?

    It is defined that the population variance is S^{2}= \frac{1}{N-1}\sum^{N}_{1}\left(y_{i} - \bar{y}_{N}\right)^{2} or \sigma^{2}= \frac{1}{N}\sum^{N}_{1}\left(y_{i} - \bar{y}_{N}\right)^{2}. Also that the V\left[\bar{y}_{n}\right] = \frac{N-n}{N}\frac{S^{2}}{n} = \left(\frac{1}{n} -...
  36. A

    Variance of a function in an infinite space

    Homework Statement It is given a function y(t)=ae(t) where e(t) is the "[URL error function [/URL] I am looking for the variance of this function in an infinite space. Since t is time, I assume that this space is defined as [0,+∞). Thus, the usual variance functions does not apply since...
  37. S

    Expected value nd variance of mean estimator

    Homework Statement A sample of size n is drawn from a population having N units by simple random sampling without replacement. A sub-sample of size n_{1} units is drawn from the n units by simple random sampling without replacement. Let \bar{y_{1}} denote the mean based on n_{1} units and...
  38. T

    Why is the conditional variance of Y equal to (1-rou^2)* variance of y?

    The attached equation is from http://en.wikipedia.org/wiki/Multivariate_normal_distribution can anyone show me why the conditional variance is equal to (1-rou^2)* variance of y thanks
  39. E

    Analytic determination of Expectation, variance

    Hi, I want to proof what the distribution will be when I apply a normal distributed x to a linear function y = a*x + b. What will be the mean and the variance of y ? The expectations can be calculated than with this formula ( probably with this formula what i want can be proofed with...
  40. P

    Quantum variance (uncertainty)

    Homework Statement N=a^+a a=\frac{ip+mwx}{\sqrt{2m\hbar w}} \quad a^+=\frac{-ip+mwx}{\sqrt{2m\hbar w}} |z>=e^{\frac{-|z|^2}{2}}\sum^{\infty}_{n=0}\frac{z^n}{\sqrt{n!}}|n> where <n|n>=1 Show that the variance (uncertainty) in N, \Delta N is |z| i.e. calculate (\Delta...
  41. M

    Mean and Variance Estimation: Importance of Normality for Confidence Intervals

    Homework Statement Why is it normality is much more important for making a confidence interval for the mean than for the variance? You do use the estimated variance to make the mean confidence interval. So why is the mean confidence interval more robust against the normality assumtion...
  42. B

    Find the mean and variance of Y^2

    Let Y by the number of heads obtained if a coin is tossed three times. Find the mean and variance of Y^2. For the mean I get, (0+1+4+9)/4=7/2, and for variance I get (0+1+16+81)/4 - (7/2)^2 = 49/4. Is this correct? For the following question, I'm not sure how to begin: Show that if T...
  43. L

    Expected variance of subset of population

    I want to calculate expected variance of a randomly selected subset of a population. The particular problem I am trying to solve is as follows. There is a set of values X = {x1, ... , xn}. Let Y be subset of X with n-1 elements. I think that if Y is selected at random (that is, if is...
  44. mnb96

    Solving Variance Problem: Computing E(\hat{\theta}) and E(\hat{\theta}^2)

    Hello, we are given N independent random variables z_i defined as follows: z_i = \theta + v_i where the r.v. v_i are zero-mean normal distributions v_i \sim N(0,\sigma^2). I want to compute the variance of the estimator \hat{\theta}=\frac{1}{n}\sum_{i=1}^n z_i However I can't...
  45. J

    Calculating Variance for Randomly Drawn Beads in a Necklace

    1. A necklace consists of 5 beads on a string. The beads for making the necklace are drawn at random from a box containing a very large number of beads. 2/3 of the beads are pink and 1/3 are blue. find the mean and variance of the number of unlike pairs of adjacent beads in the necklace. I am...
  46. Z

    How Can the Variance of a Quadratic Form Be Simplified?

    In the Searle's 1971 book Linear Model, page 57, has a formula for the Variance of Quadratic form: var(Y^{T}AY)=2tr(A\SigmaA\Sigma)+4\mu^{T}A\SigmaA\mu The proof of this showed on page 55 was based on MGF. I'm looking for proofs are less complicated. Some thing that is similar to show the...
  47. D

    Proving Variance of a Random Variable with Moment-Generating Functions

    Suppose that Y is a random variable with moment-generating function m(t) and W = aY + b, with a moment-generating function of m(at) * e^(tb). Prove that V(W) = V(Y) * a^2. I have done an absurd amount of work on this problem, and I know its actual solution doesn't have one and a half pages worth...
  48. M

    How Do I Calculate the Variance of a Transformed Random Variable?

    Homework Statement How do I calculate the variance of \frac{1}{\log{X} + 2} where X is a random variable? The Attempt at a Solution Is it: \frac{1}{\log{var(X)}} Homework Equations The Attempt at a Solution
  49. J

    Find the mean and variance of this set of 40 numbers

    A computer can generate random numbers which are either 0 or 2. On a particular occasion, it generates a set of numbers which consists of 23 zeros and 17 twos. Find the mean and variance of this set of 40 numbers. Please help i just don't know where to start. In fact i am thinking of...
  50. D

    Calculating variance from range

    Hello, I have taken 5 samples and found that my average concentration is 5 mg, with a range of .003 mg in either direction. I would like to calculate the variance with only this information. Is this possible? I am used to calculating the variance from the standard deviation, but I don't...
Back
Top