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On the Interrelation Between the Sample Mean and the Sample Variance

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  • Ananda Sen

Abstract

The linearity (or lack thereof) of association between sample mean and sample variance is explored in this note with the intent of providing new insights. Of particular interest is a well-known inequality involving the measures of skewness and kurtosis that is derived as a consequence of an identity involving the correlation between sample mean and sample variance. The nature of association between the two is explored further by means of the conditional expectation of the sample variance given the mean. We present several characterization results where the specific relationship of this conditional expectation and sample mean uniquely determines the parent population. The note is presented at a level accessible to graduate or upper-level undergraduate students with several illustrative examples included as teaching aid.

Suggested Citation

  • Ananda Sen, 2012. "On the Interrelation Between the Sample Mean and the Sample Variance," The American Statistician, Taylor & Francis Journals, vol. 66(2), pages 112-117, May.
    • Handle: RePEc:taf:amstat:v:66:y:2012:i:2:p:112-117
    DOI: 10.1080/00031305.2012.695960
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    Cited by:

    1. B. O'Neill, 2014. "Some Useful Moment Results in Sampling Problems," The American Statistician, Taylor & Francis Journals, vol. 68(4), pages 282-296, November.
    2. Brandts, Jordi & El Baroudi, Sabrine & Huber, Stefanie J. & Rott, Christina, 2021. "Gender differences in private and public goal setting," Journal of Economic Behavior & Organization, Elsevier, vol. 192(C), pages 222-247.
    3. Ogasawara, Haruhiko, 2017. "Extensions of Pearson’s inequality between skewness and kurtosis to multivariate cases," Statistics & Probability Letters, Elsevier, vol. 130(C), pages 12-16.
    4. Lukas Hoesch & Adam Lee & Geert Mesters, 2022. "Robust inference for non-Gaussian SVAR models," Economics Working Papers 1847, Department of Economics and Business, Universitat Pompeu Fabra.
    5. Lukas Hoesch & Adam Lee & Geert Mesters, 2022. "Locally Robust Inference for Non-Gaussian SVAR Models," Working Papers 1367, Barcelona School of Economics.

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