A Note on Some Hidden Properties of The VarianceAnd ‎Other Closely Related Measures

  • Authors

    • Mohammad Fraiwan Al-Saleh Department of Statistics, Yarmouk University-Jordan
    • Hashim Abdallah Jarrah Department of Statistics, Yarmouk University-Jordan
    https://doi.org/10.14419/9n9x7b95

    Received date: December 27, 2025

    Accepted date: February 5, 2026

    Published date: February 10, 2026

  • Population Variance; Sample Variance; Mean Absolute Deviation; Coefficient of ‎Variation; Standard Deviation; Stratified Random Sampling‎.

  • Abstract

    Several important hidden properties about population variance and sample variance are ‎highlighted. Very interesting, important queries are raised about the definition of the ‎mean absolute deviation, the sample variance, and standard deviation. In. In addition, the ‎close relationship between variance and the coefficient of variation is emphasized and ‎highlighted. It is noted that these two measures of variation are functions of the same two ‎quantities, the average of the squares and the square of the average of the data. A‏ ‏very ‎important comment is raised about the definition of the mean absolute deviation.

  • References

    1. AL-Saleh, M. F., “On the confusion about (n-1)-divisor of the standard deviation”, Journal of Prob. and Stat. Science, vol. 5, 139-144, 2005.
    2. AL-Saleh, M. F., “A rich learning lesson using the Poisson distribution”, Statistical Methodology, Vol. 4, no. 4, pp. 504-507, 2007. https://doi.org/10.1016/j.stamet.2007.01.005.
    3. Al-Saleh, M. F., Yousif, A. E., “Properties of the standard deviation that are rarely mentioned in classrooms”, Austrian journal of statistics, vol. 3, pp. 193-202, 2009.
    4. Casella, G., Berger, R., “Statistical Inference”, Wadsworth & Brooks/Cole Advanced Book & Software, California, USA, 1990.
    5. Balakrishnan, N., Balasubramanian, K., “Revisiting Sen’s inequalities on order statistics”, Statistics and prob. letters, vol. 78, pp. 616-621, 2007. https://doi.org/10.1016/j.spl.2007.09.023.
    6. Herbert, H. Mezbahur, R., “Revisiting the Digits of PI and Their Randomness”, International Journal of Statistical Science (IJSS), vol. 4, pp. 13-24, 2005.
    7. Al-Saleh, M. F. and Maabreh, Arwa, “A Rich Learning Statistical Lesson in the Cauchy Distribution”, Mutah Journal for Natural, Applied and Health Sciences, vol. 40, no. 1, pp. 13-32, 2025. https://doi.org/10.35682/mjnahs.v20i1.1192.
    8. Al-Saleh, M. F. and Jarrah, H., “Hidden Properties in the Main Sampling Techniques”, International Journal of Advanced Statistics and Prob., vol. 12, no. 1, pp. 12-16, 2025. https://doi.org/10.14419/dp63md84.
    9. Heinbockel J. H., “Introduction to Calculus” Volume II, Old Dominion University, VA, USA, 2012.
    10. Khan, R., “A note on the generating function of a negative hyper-geometric distribution”, Sankhya 56, 309-313, 1994.
    11. Bloch, D., “A note on the estimation of the location parameter of the Cauchy distribution”, Journal of the American Statistical Association, 61(315), 852-855, 1966. https://doi.org/10.1080/01621459.1966.10480912.
    12. Copas J. B., "On the unimodality of the likelihood for the Cauchy distribution,” Biometrika, 62, no.3, 701-704, 1975. DOI: 10.1093/biomet/62.3.701. https://doi.org/10.1093/biomet/62.3.701.
    13. Rothenberg T. J., Fisher F. M., Tilanus C. B., " A note on estimation from a Cauchy sample," Journal of the American Statistical Association, vol. 59, no. 306, pp.460-463, 1964. https://doi.org/10.1080/01621459.1964.10482170.
    14. Azmay N., Rosli R., Maat S., Mahmud M. “Reasoning and Statistical Thinking: A Systematic Literature Review”, International journal of academic research in progressive education and development, Vol. 1 2 , No. 2, 2023, E-ISSN: 2226-6348 © 2023.
    15. Schoen, C. , Christopher, B ., Alexandra B., Tim Jacobbe C., Lanrong, d. “ Improving the Teaching and Learning of Statistics”, ILearning and Instruction, Vol. 95, 2025, 102018. https://doi.org/10.1016/j.learninstruc.2024.102018.

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