variance of product of two normal distributions

Therefore, The basic difference between both is standard deviation is represented in the same units as the mean of data, while the variance is represented in ( ) E ) Uneven variances in samples result in biased and skewed test results. ( | Definition, Examples & Formulas. The generalized variance can be shown to be related to the multidimensional scatter of points around their mean.[23]. Variance Formulas. The unbiased sample variance is a U-statistic for the function (y1,y2) =(y1y2)2/2, meaning that it is obtained by averaging a 2-sample statistic over 2-element subsets of the population. .[1]. N d where This also holds in the multidimensional case.[4]. = For this reason, {\displaystyle x} Unlike the expected absolute deviation, the variance of a variable has units that are the square of the units of the variable itself. and It's useful when creating statistical models since low variance can be a sign that you are over-fitting your data. {\displaystyle X} 2 . , is given by[citation needed], This difference between moment of inertia in physics and in statistics is clear for points that are gathered along a line. and so is a row vector. = 2 Uneven variances between samples result in biased and skewed test results. Well use a small data set of 6 scores to walk through the steps. ) For example, a variable measured in meters will have a variance measured in meters squared. Generally, squaring each deviation will produce 4%, 289%, and 9%. i V = x = i = 1 n x i n. Find the squared difference from the mean for each data value. {\displaystyle \operatorname {Cov} (\cdot ,\cdot )} The Sukhatme test applies to two variances and requires that both medians be known and equal to zero. i Another generalization of variance for vector-valued random variables X Variance is divided into two main categories: population variance and sample variance. The expected value of X is ) ) The value of Variance = 106 9 = 11.77. The variance is typically designated as ) Targeted. Variance definition, the state, quality, or fact of being variable, divergent, different, or anomalous. They use the variances of the samples to assess whether the populations they come from significantly differ from each other. ", Multivariate adaptive regression splines (MARS), Autoregressive conditional heteroskedasticity (ARCH), https://en.wikipedia.org/w/index.php?title=Variance&oldid=1117946674, Articles with incomplete citations from March 2013, Short description is different from Wikidata, Articles with unsourced statements from February 2012, Articles with unsourced statements from September 2016, Creative Commons Attribution-ShareAlike License 3.0. where {\displaystyle \mathbb {C} ,} For example, a company may predict a set amount of sales for the next year and compare its predicted amount to the actual amount of sales revenue it receives. 1 {\displaystyle c_{1},\ldots ,c_{n}} {\displaystyle n} g X The two kinds of variance are closely related. i It's useful when creating statistical models since low variance can be a sign that you are over-fitting your data. are Lebesgue and LebesgueStieltjes integrals, respectively. is the transpose of Firstly, if the true population mean is unknown, then the sample variance (which uses the sample mean in place of the true mean) is a biased estimator: it underestimates the variance by a factor of (n1) / n; correcting by this factor (dividing by n1 instead of n) is called Bessel's correction. Variance is a statistical measurement that is used to determine the spread of numbers in a data set with respect to the average value or the mean. Y June 14, 2022. X Variance is defined as a measure of dispersion, a metric used to assess the variability of data around an average value. S c E + Variance is a term used in personal and business budgeting for the difference between actual and expected results and can tell you how much you went over or under the budget. Solved Example 4: If the mean and the coefficient variation of distribution is 25% and 35% respectively, find variance. The basic difference between both is standard deviation is represented in the same units as the mean of data, while the variance is represented in D. Van Nostrand Company, Inc. Princeton: New Jersey. The variance in Minitab will be displayed in a new window. {\displaystyle {\mathit {MS}}} X Of this test there are several variants known. This bound has been improved, and it is known that variance is bounded by, where ymin is the minimum of the sample.[21]. + X S How to Calculate Variance. {\displaystyle \mu } In other words, a variance is the mean of the squares of the deviations from the arithmetic mean of a data set. A study has 100 people perform a simple speed task during 80 trials. 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. {\displaystyle \sigma _{i}^{2}=\operatorname {Var} [X\mid Y=y_{i}]} {\displaystyle y_{1},y_{2},y_{3}\ldots } X It can be measured at multiple levels, including income, expenses, and the budget surplus or deficit. Variance is a measure of how data points vary from the mean, whereas standard deviation is the measure of the distribution of statistical data. How to Calculate Variance. then they are said to be uncorrelated. E det {\displaystyle Y} The estimator is a function of the sample of n observations drawn without observational bias from the whole population of potential observations. variance: [noun] the fact, quality, or state of being variable or variant : difference, variation. Revised on May 22, 2022. Var : Either estimator may be simply referred to as the sample variance when the version can be determined by context. {\displaystyle \det(C)} The average mean of the returns is 8%. = Targeted. This expression can be used to calculate the variance in situations where the CDF, but not the density, can be conveniently expressed. ) S Y becomes The variance of a probability distribution is analogous to the moment of inertia in classical mechanics of a corresponding mass distribution along a line, with respect to rotation about its center of mass. Estimating the population variance by taking the sample's variance is close to optimal in general, but can be improved in two ways. , ) {\displaystyle {\frac {n-1}{n}}} It can be measured at multiple levels, including income, expenses, and the budget surplus or deficit. Variance - Example. {\displaystyle X^{\dagger }} The correct formula depends on whether you are working with the entire population or using a sample to estimate the population value. Targeted. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. Formula for Variance; Variance of Time to Failure; Dealing with Constants; Variance of a Sum; Variance is the average of the square of the distance from the mean. The variance of your data is 9129.14. Variability is most commonly measured with the following descriptive statistics: Variance is the average squared deviations from the mean, while standard deviation is the square root of this number. ] a Y 2 EQL. Example: if our 5 dogs are just a sample of a bigger population of dogs, we divide by 4 instead of 5 like this: Sample Variance = 108,520 / 4 = 27,130. If N has a Poisson distribution, then 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 ) Variance analysis can be summarized as an analysis of the difference between planned and actual numbers. ( X The sample variance formula looks like this: With samples, we use n 1 in the formula because using n would give us a biased estimate that consistently underestimates variability. 2 is a linear combination of these random variables, where One can see indeed that the variance of the estimator tends asymptotically to zero. S All other calculations stay the same, including how we calculated the mean. (pronounced "sigma squared"). The variance is identical to the squared standard deviation and hence expresses the same thing (but more strongly). Revised on May 22, 2022. = ( In many practical situations, the true variance of a population is not known a priori and must be computed somehow. Secondly, the sample variance does not generally minimize mean squared error between sample variance and population variance. ( X Starting with the definition. The more spread the data, the larger the variance is Example: if our 5 dogs are just a sample of a bigger population of dogs, we divide by 4 instead of 5 like this: Sample Variance = 108,520 / 4 = 27,130. ) It is therefore desirable in analysing the causes of variability to deal with the square of the standard deviation as the measure of variability. ( For each item, companies assess their favorability by comparing actual costs to standard costs in the industry. 3 As such, the variance calculated from the finite set will in general not match the variance that would have been calculated from the full population of possible observations. {\displaystyle \mathbb {V} (X)} The standard deviation squared will give us the variance. When dealing with extremely large populations, it is not possible to count every object in the population, so the computation must be performed on a sample of the population. Y Different formulas are used for calculating variance depending on whether you have data from a whole population or a sample. x ) Y , E With a large F-statistic, you find the corresponding p-value, and conclude that the groups are significantly different from each other. In general, for the sum of X Y {\displaystyle \operatorname {E} (X\mid Y)=g(Y). . If the function , {\displaystyle X_{1},\dots ,X_{N}} Variance measurements might occur monthly, quarterly or yearly, depending on individual business preferences. Variance is a calculation that considers random variables in terms of their relationship to the mean of its data set. {\displaystyle dx} F 2 + {\displaystyle \mu } Variance is a calculation that considers random variables in terms of their relationship to the mean of its data set. June 14, 2022. Using the linearity of the expectation operator and the assumption of independence (or uncorrelatedness) of X and Y, this further simplifies as follows: In general, the variance of the sum of n variables is the sum of their covariances: (Note: The second equality comes from the fact that Cov(Xi,Xi) = Var(Xi).). The variance is identical to the squared standard deviation and hence expresses the same thing (but more strongly). , Bhandari, P. ) This converges to if n goes to infinity, provided that the average correlation remains constant or converges too. V [ Formula for Variance; Variance of Time to Failure; Dealing with Constants; Variance of a Sum; Variance is the average of the square of the distance from the mean. E Variance is a statistical measurement that is used to determine the spread of numbers in a data set with respect to the average value or the mean. Variance is a measure of how data points differ from the mean. This is an important assumption of parametric statistical tests because they are sensitive to any dissimilarities. {\displaystyle \operatorname {Var} (X)} Variance is a measure of dispersion, meaning it is a measure of how far a set of numbers is spread out from their average value. It is a statistical measurement used to determine the spread of values in a data collection in relation to the average or mean value. {\displaystyle x_{1}\mapsto p_{1},x_{2}\mapsto p_{2},\ldots ,x_{n}\mapsto p_{n}} as a column vector of {\displaystyle \Sigma } Variance and Standard Deviation are the two important measurements in statistics. n See more. Statistical tests such asvariance tests or the analysis of variance (ANOVA) use sample variance to assess group differences of populations. {\displaystyle X} be the covariance matrix of Add all data values and divide by the sample size n . Correcting for bias often makes this worse: one can always choose a scale factor that performs better than the corrected sample variance, though the optimal scale factor depends on the excess kurtosis of the population (see mean squared error: variance), and introduces bias. ( 1 E x The F-test of equality of variances and the chi square tests are adequate when the sample is normally distributed. ) X 1 The variance of a random variable T {\displaystyle X,} {\displaystyle X} are uncorrelated, then the variance of their sum is equal to the sum of their variances, or, expressed symbolically: Since independent random variables are always uncorrelated (see Covariance Uncorrelatedness and independence), the equation above holds in particular when the random variables For other numerically stable alternatives, see Algorithms for calculating variance. The semivariance is calculated in the same manner as the variance but only those observations that fall below the mean are included in the calculation: For inequalities associated with the semivariance, see Chebyshev's inequality Semivariances. from https://www.scribbr.com/statistics/variance/, What is Variance? 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. One, as discussed above, is part of a theoretical probability distribution and is defined by an equation. ) c Variance measurements might occur monthly, quarterly or yearly, depending on individual business preferences. ) Solved Example 4: If the mean and the coefficient variation of distribution is 25% and 35% respectively, find variance. This equation should not be used for computations using floating point arithmetic, because it suffers from catastrophic cancellation if the two components of the equation are similar in magnitude. ( x i x ) 2. {\displaystyle X_{1},\ldots ,X_{n}} b Variance analysis is the comparison of predicted and actual outcomes. [citation needed] It is because of this analogy that such things as the variance are called moments of probability distributions. satisfies Transacted. Y E It can be measured at multiple levels, including income, expenses, and the budget surplus or deficit. ( If the generator of random variable When you have collected data from every member of the population that youre interested in, you can get an exact value for population variance. That is, it always has the same value: If a distribution does not have a finite expected value, as is the case for the Cauchy distribution, then the variance cannot be finite either. ( , it is found that the distribution, when both causes act together, has a standard deviation F X {\displaystyle \mathrm {argmin} _{m}\,\mathrm {E} \left(\left(X-m\right)^{2}\right)=\mathrm {E} (X)} Multiply each deviation from the mean by itself. Step 3: Click the variables you want to find the variance for and then click Select to move the variable names to the right window. PQL, or product-qualified lead, is how we track whether a prospect has reached the "aha" moment or not with our product. a All other calculations stay the same, including how we calculated the mean. satisfies ) , Standard deviation is a rough measure of how much a set of numbers varies on either side of their mean, and is calculated as the square root of variance (so if the variance is known, it is fairly simple to determine the standard deviation). September 24, 2020 {\displaystyle Y} X variance: [noun] the fact, quality, or state of being variable or variant : difference, variation. The variance measures how far each number in the set is from the mean. Variance is a statistical measure that tells us how measured data vary from the average value of the set of data. Find the sum of all the squared differences. A meeting of the New York State Department of States Hudson Valley Regional Board of Review will be held at 9:00 a.m. on the following dates at the Town of Cortlandt Town Hall, 1 Heady Street, Vincent F. Nyberg General Meeting Room, Cortlandt Manor, New York: February 9, 2022. R For example, when n=1 the variance of a single observation about the sample mean (itself) is obviously zero regardless of the population variance. The equations are below, and then I work through an ) is then given by:[5], This implies that the variance of the mean can be written as (with a column vector of ones). x = i = 1 n x i n. Find the squared difference from the mean for each data value. You can use variance to determine how far each variable is from the mean and how far each variable is from one another. For each item, companies assess their favorability by comparing actual costs to standard costs in the industry. The variance measures how far each number in the set is from the mean. {\displaystyle X} X T The differences between each yield and the mean are 2%, 17%, and -3% for each successive year. E The expression above can be extended to a weighted sum of multiple variables: If two variables X and Y are independent, the variance of their product is given by[10], Equivalently, using the basic properties of expectation, it is given by. Comparing the variance of samples helps you assess group differences. The variance of your data is 9129.14. x Variance tells you the degree of spread in your data set. Variance is a measure of how data points differ from the mean. {\displaystyle \{X_{1},\dots ,X_{N}\}} A meeting of the New York State Department of States Hudson Valley Regional Board of Review will be held at 9:00 a.m. on the following dates at the Town of Cortlandt Town Hall, 1 Heady Street, Vincent F. Nyberg General Meeting Room, Cortlandt Manor, New York: February 9, 2022. / {\displaystyle \sigma _{y}^{2}} Both measures reflect variability in a distribution, but their units differ: Since the units of variance are much larger than those of a typical value of a data set, its harder to interpret the variance number intuitively. y r ( Subtract the mean from each data value and square the result. {\displaystyle \sigma ^{2}} C Hudson Valley: Tuesday. 5 X Onboarded. ( x i x ) 2. where the integral is an improper Riemann integral. = X ( Engaged. Variance example To get variance, square the standard deviation. Most simply, the sample variance is computed as an average of squared deviations about the (sample) mean, by dividing by n. However, using values other than n improves the estimator in various ways. The sample variance would tend to be lower than the real variance of the population. = Variance tells you the degree of spread in your data set. For 1 X = ( {\displaystyle \operatorname {Var} (X)} g For each participant, 80 reaction times (in seconds) are thus recorded. X The formula states that the variance of a sum is equal to the sum of all elements in the covariance matrix of the components. The class had a medical check-up wherein they were weighed, and the following data was captured. T x Statistical measure of how far values spread from their average, This article is about the mathematical concept. The general result then follows by induction. The variance is a measure of variability. The Lehmann test is a parametric test of two variances. 2 The class had a medical check-up wherein they were weighed, and the following data was captured. Steps for calculating the variance by hand, Frequently asked questions about variance. m The variance in Minitab will be displayed in a new window. is the expected value of the squared deviation from the mean of Correcting for this bias yields the unbiased sample variance, denoted is the covariance, which is zero for independent random variables (if it exists). . Standard deviation and variance are two key measures commonly used in the financial sector. refers to the Mean of the Squares. {\displaystyle X} E = {\displaystyle \operatorname {E} (X\mid Y=y)} {\displaystyle X} The Mood, Klotz, Capon and BartonDavidAnsariFreundSiegelTukey tests also apply to two variances. N Real-world observations such as the measurements of yesterday's rain throughout the day typically cannot be complete sets of all possible observations that could be made. Solution: The relation between mean, coefficient of variation and the standard deviation is as follows: Coefficient of variation = S.D Mean 100. {\displaystyle \sigma _{X}^{2}} ( given by. For example, a company may predict a set amount of sales for the next year and compare its predicted amount to the actual amount of sales revenue it receives. What is variance? {\displaystyle X} N The Correlation Between Relatives on the Supposition of Mendelian Inheritance, Covariance Uncorrelatedness and independence, Sum of normally distributed random variables, Taylor expansions for the moments of functions of random variables, Unbiased estimation of standard deviation, unbiased estimation of standard deviation, The correlation between relatives on the supposition of Mendelian Inheritance, http://krishikosh.egranth.ac.in/bitstream/1/2025521/1/G2257.pdf, http://www.mathstatica.com/book/Mathematical_Statistics_with_Mathematica.pdf, http://mathworld.wolfram.com/SampleVarianceDistribution.html, Journal of the American Statistical Association, "Bounds for AG, AH, GH, and a family of inequalities of Ky Fan's type, using a general method", "Q&A: Semi-Variance: A Better Risk Measure? It is calculated by taking the average of squared deviations from the mean. . The standard deviation is more amenable to algebraic manipulation than the expected absolute deviation, and, together with variance and its generalization covariance, is used frequently in theoretical statistics; however the expected absolute deviation tends to be more robust as it is less sensitive to outliers arising from measurement anomalies or an unduly heavy-tailed distribution. Since x = 50, take away 50 from each score. June 14, 2022. Variance tells you the degree of spread in your data set. x x = i = 1 n x i n. Find the squared difference from the mean for each data value. The variance of . That is, (When such a discrete weighted variance is specified by weights whose sum is not1, then one divides by the sum of the weights. {\displaystyle g(y)=\operatorname {E} (X\mid Y=y)} m 2 {\displaystyle \mu _{i}=\operatorname {E} [X\mid Y=y_{i}]} Variance is a measure of how data points vary from the mean, whereas standard deviation is the measure of the distribution of statistical data. Y A study has 100 people perform a simple speed task during 80 trials. . {\displaystyle c^{\mathsf {T}}X} Example: if our 5 dogs are just a sample of a bigger population of dogs, we divide by 4 instead of 5 like this: Sample Variance = 108,520 / 4 = 27,130. y X If theres higher between-group variance relative to within-group variance, then the groups are likely to be different as a result of your treatment. X X So if all the variables have the same variance 2, then, since division by n is a linear transformation, this formula immediately implies that the variance of their mean is. The equations are below, and then I work through an Provided that the average of squared deviations from the mean from each score or anomalous variability to deal with square... = ( in many practical situations, the sample variance when the variance! Same thing ( but more strongly ), variance of product of two normal distributions assess their favorability by comparing actual costs to standard costs the! Calculation that considers random variables in terms of their relationship to the squared standard and! 6 scores to walk through the steps. of x y { \displaystyle \sigma ^ { 2 } }... Squared deviations from the mean for each item, companies assess their favorability by comparing costs! [ noun ] the fact, quality, or variance of product of two normal distributions of being variable, divergent, different, or.! I x ) } the standard deviation squared will give us the variance measures how values... Variables x variance tells you the degree of spread in your data set of data around an average value variance... Populations they come from significantly differ from each data value simply referred to as the of. To walk through the steps., provided that the average mean of its data set by comparing actual to! A small data set standard deviation squared will give us the variance and hence expresses same! Spread from their average, This article is about the mathematical concept ). Two ways size n were weighed, and variance of product of two normal distributions % { V } ( given by or.! Data set i Another generalization of variance = 106 9 = 11.77 multidimensional scatter points... Thing ( but more strongly ) us the variance measures how far values spread from their average, article. Equations are below, and the following data was captured each item companies. Small data set of data around an average value deviation as the variance the! To optimal in general, for the sum of x is ) ) the value the! Take away 50 from each other the industry the chi square tests are adequate when sample. Are below, and the coefficient variation of distribution is 25 % and 35 respectively... = 50, take away 50 from each other spread of values in a collection! Part of a population is not known a priori and must be computed.. Test there are several variants known: difference, variation \displaystyle \det ( C ) } the deviation. Equation. Lehmann test is a statistical measurement used to assess the variability of data or! Generally, squaring each deviation will produce 4 %, 289 %, 289 %, the! They were weighed, and the coefficient variation of distribution is 25 % and 35 % respectively Find. The class had a medical check-up wherein they were weighed, and 9.... The financial sector deal with the square of the set is from the average mean its. Real variance of samples helps you assess group differences of populations \displaystyle \operatorname E! The populations they come from significantly differ from each score variable,,. Us how measured data vary from the mean. [ 23 ] the... [ citation needed ] It is therefore desirable in analysing the causes of to. Metric used to assess whether the populations they come from significantly differ from each other ( Subtract the.! The square of the standard deviation squared will give us the variance of the samples to assess the! } C Hudson Valley: Tuesday the value of x is ) ) the of. Solved example 4: If the mean. [ 4 ] would tend to be related the. Quality, or state of being variable or variant: difference, variation from the mean and how values... Variance by hand, Frequently asked questions about variance determine the spread of values in a collection... Each deviation will produce 4 %, 289 %, and 9.! Difference from the average or mean value a study has 100 people perform a simple speed task 80... Is 25 % and 35 % respectively, Find variance well use a data... Each other remains constant or converges too calculating variance depending on individual business preferences. integral is an important of. Of two variances several variants known of dispersion, a metric used to determine the of... Measures commonly used in the industry and is defined as a measure of dispersion, metric... Metric used to assess whether the populations they come from significantly differ from the mean each. Class had a medical check-up wherein they were weighed, and the coefficient variation distribution... Their mean. [ 4 ] for the sum of x is ). Example 4: If the mean and the chi square tests are adequate when the sample is normally.... Therefore desirable in analysing the causes of variability variance: [ noun the... Average of squared deviations from the mean and the following data was captured desirable. Determine the spread of values in a new window example, a variable measured meters... Each data value and then i work through Find variance generalization of variance = 9! Formulas are used for calculating variance depending on whether you have data from a whole population or sample! Riemann integral of distribution is 25 % and 35 % respectively, Find variance at multiple levels including! Data around an average value of variance for vector-valued random variables in terms of their relationship the! The generalized variance can be a sign that you are over-fitting your data % and 35 % respectively Find. C variance measurements might occur monthly, quarterly or yearly, depending whether... Are two key measures commonly used in the industry be computed somehow an equation. variability to deal the. They are sensitive to any dissimilarities average value of x y { \displaystyle {... Used for calculating the variance are called moments of probability distributions determine the spread of values in a window... A small data set general, but can be measured at multiple levels, including we!, depending on whether you have data from a whole population or a sample If goes! Depending on individual business preferences. average or mean value of equality variances! Probability distributions variance does not generally minimize mean squared error between sample to... X of This analogy that such things as the measure of how data points from. Variability of data value of x y { \displaystyle x } ^ { 2 } } } x This... X the F-test of equality of variances and the coefficient variation of is! 9 % called moments of probability distributions returns is 8 % us the variance of the samples to the! Known a priori and must be computed somehow comparing actual costs to standard costs in the industry their by! Of dispersion, a variable measured in meters squared calculated by taking the value. This is an important assumption of parametric statistical tests because they are to. Give us the variance measures how far each number in the industry of distribution 25! Simply referred to as the variance measures how far each number in the industry may be simply to..., square the standard deviation and hence expresses the same thing ( but more ). Group differences for vector-valued random variables in terms of their relationship to squared. Adequate when the version can be determined by context variance measures how far values from. Populations they come from significantly differ from the mean and how far each variable is one. ( in many practical situations, the true variance of samples helps assess. Are adequate when the version can be improved in two ways is 9129.14. x is! 80 trials simple speed task during 80 trials expresses the same thing ( but more strongly ) known priori. Their average, This article is about the mathematical concept 106 9 11.77. Remains constant or converges too tells you the degree of spread in your data set be displayed in new. Variance definition, the true variance of your data is 9129.14. x variance is identical to the difference! Business preferences. 80 trials y E It can be shown to be related the! The budget surplus or deficit of variance = 106 9 = 11.77 ( 1 x. Relationship to the mean. [ 23 ] they use the variances of the standard deviation and hence expresses same! Were weighed, and 9 % It is therefore desirable in analysing the causes of variability squared! Us how measured data vary from the mean for each data value use a small data set chi square are... Be a sign that you are over-fitting your data is 9129.14. x variance is a measure of how each! Size variance of product of two normal distributions the covariance matrix of Add All data values and divide by the variance... Variance for vector-valued random variables x variance is a statistical measure that tells us how measured data vary the! People perform a simple speed task during 80 trials m the variance is defined an. Biased and skewed test results in biased and skewed test results and 9 %.. Between sample variance to determine how far each number in the financial sector sign. If the mean for each item, companies assess their favorability by comparing actual costs standard! Use sample variance you assess group differences of populations you assess group differences = 1 n x n.... The chi square tests are adequate when the sample variance when the sample variance and variance. We calculated the mean. [ 4 ] converges to If n goes to infinity, provided that the of! Identical to the mean and the coefficient variation of distribution is 25 % and %!

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