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MOMENTS: help us characterize distributions. These distributions have different characteristics:

Moments – A Non-Technical Introduction Quantitative Methods: Statistics Dan Hamilton California Lutheran University Revised: Sept. 30, 2013.

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MOMENTS: help us characterize distributions. These distributions have different characteristics:

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  1. Moments – A Non-Technical IntroductionQuantitative Methods: StatisticsDan HamiltonCalifornia Lutheran UniversityRevised: Sept. 30, 2013

  2. EXPECTATIONSExpectations may be taken of a random variable or of powers of a random variable or of its deviations from its mean.These expectations have a special name: “Moments”.Moments are important because they characterize a distribution.

  3. MOMENTSLet X = random variable:First moment E(X) “Mean”Second moment E(X2) Third moment E(X3) 2ndmoment about the mean “Variance” E[(X - μ)2]3rdmoment about the mean “Skewness” E[(X - μ)3] 4thmoment about the mean “Kurtosis” E[(X - μ)4]

  4. MOMENTSFirst moment MeanSecond moment Not used…Third moment Not used…Why not used? They do not help us characterize the distribution.2ndmoment about the meanVariance3rdmoment about the meanSkewness4thmoment about the meanKurtosisAn aside: the Test for Normality: the “test statistic” is a function of these moments.

  5. MOMENTS: help us characterize distributions. These distributions have different characteristics:

  6. Example of skewness: The distribution is “Skewed”. This is measured by Skewness.

  7. Example of Kurtosis: The distribution is relatively “Peaked”. This is measured by Kurtosis.

  8. We distinguish between theoretical vs. sample moments Theoretical : use the expectation operator Theoretical moments are used to mathematically analyze or characterize a distribution. Sample : use sample data Sample moments are calculated using actual data so you better understand the characteristics of the data that you are working with.

  9. Theoretical Moments If Yand X are random variables: E(Y) is the theoretical mean of Y or “first moment” of Y Let μy = E(Y) E(X) is the theoretical mean of X Let μx = E(X)

  10. Theoretical Moments – continued E[(Y -μY)2] is theoretical variance of Y E[(Y -μY)(X -μX)] is theoretical covariance of Y and X E[(Y -μY)(X -μX)]/[σYσX] is theoretical correlation of Y and X

  11. SAMPLE MomentsMean (Σtyt) /T = YbarVariance sy2 = [Σt (yt-Ybar)2]/ (T-1)Covariance [Σt (yt-Ybar)(xt-Xbar)]/(T-2) Correlation [Σt (yt-Ybar)(xt-Xbar)]/[sysx ]

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