for the frequency distribution if 2/3 (mean – mode) = 4 and the coefficient of skewness is 0.8 , find its variance​

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1. [tex]\begin{gathered}\begin{gathered}\bf \:Given-\begin{cases} &\sf{\dfrac{2}{3}(Mean – Mode) = 4 } \\ &\sf{Coefficient \: of \: Skewness = 0.8} \end{cases}\end{gathered}\end{gathered}[/tex]

   [tex]\begin{gathered}\begin{gathered}\bf \:To\:find-\begin{cases} &\sf{Variance} \end{cases}\end{gathered}\end{gathered}[/tex]

   [tex]\large\underline{\sf{Solution-}}[/tex]

   Given that

   [tex]\rm :\longmapsto\:\dfrac{2}{3} (Mean – Mode) = 4[/tex]

   [tex]\bf\implies \:Mean – Mode = 6[/tex]

   and

   [tex]\rm :\longmapsto\:Coefficient \: of \: Skewness \: = 0.8[/tex]

   We know that,

   Coefficient of Skewness is given by

   [tex]\rm :\longmapsto\:Coefficient \: of \: Skewness =\dfrac{Mean – Mode}{S. D.} [/tex]

   [tex]\rm :\longmapsto\:0.8 = \dfrac{6}{S. D.} [/tex]

   [tex]\rm :\longmapsto\:S. D. = \dfrac{6}{0.8} [/tex]

   [tex]\bf\implies \:S. D. = \dfrac{15}{2} [/tex]

   We know,

   [tex]\rm :\longmapsto\:Variance = {(S. D.)}^{2} [/tex]

   [tex]\rm :\longmapsto\:Variance = {\bigg(\dfrac{15}{2} \bigg) }^{2} [/tex]

   [tex]\rm :\longmapsto\:Variance = \dfrac{225}{4} [/tex]

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   Additional Information :-

   • The coefficient of skewness is a measure of asymmetry in the distribution.

   • A positive skew indicates a longer tail to the right, while a negative skew indicates a longer tail to the left.

   • A perfectly symmetric distribution, like the normal distribution, has a skew equal to zero.

   Interpretation :-

   • If skewness is less than −1 or greater than +1, the distribution is highly skewed.

   • If skewness is between −1 and −½ or between +½ and +1, the distribution is moderately skewed.

   • If skewness is between −½ and +½, the distribution is approximately symmetric.

   Another Formula :-

   [tex]\rm :\longmapsto\:Coefficient \: of \: Skewness =\dfrac{3(Mean – Mode)}{S. D.} [/tex]

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