Mean Deviation
Mean Deviation about Mean of a Raw Data
Let \[{{x}_{1}},{{x}_{2}}{{x}_{3}},---,{{x}_{n}}\] be the n observation, then the mean of the data is given by:
\[\overline{X}=\frac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+----+{{x}_{n}}}{n}\]
\[\Rightarrow \,\overline{X}\frac{1}{n}\sum\limits_{k=1}^{n}{{{X}_{k}}\Rightarrow \overline{X},=\frac{1}{n}}\sum\limits_{k=1}^{n}{{{X}_{k}}}\]
Then the deviation of the data from the mean is given by:
\[\left| \left. {{x}_{1}}-\overline{X} \right|,\left| {{x}_{2}},-\left. \overline{X} \right|,\left| \left. {{x}_{3}}-\overline{X} \right|,---,\left| \left. {{x}_{n}}-\overline{X} \right| \right. \right. \right. \right.\]
Now the mean deviation of the data is given by:
Mean Deviation \[=\,\Rightarrow M.D.=\frac{1}{n}\sum\limits_{k=1}^{n}{\left| \left. {{X}_{n}}-\overline{X} \right| \right.}\]
Mean Deviation about Mean of a Grouped Data Let
\[{{x}_{1}},{{x}_{2}},{{x}_{3}},---,{{x}_{n}}\] be the n - observation and \[{{f}_{1}},{{f}_{2}},{{f}_{3}},---{{f}_{n}}\] be the corresponding frequencies of the data. Then the mean of the data is given by:
\[\overline{X}=\frac{{{x}_{1}}{{f}_{1}}+{{x}_{2}}{{f}_{2}}+----+{{x}_{n}}{{f}_{n}}}{{{f}_{1}}+{{f}_{2}}++---+{{f}_{n}}}\,\] Or, \[\overline{X}=\frac{\sum\limits_{k=1}^{n}{{{X}_{k}}}{{f}_{k}}}{\sum\limits_{k=1}^{n}{{{f}_{k}}}}\]
Then the mean deviation is given by \[\Rightarrow \,M.D.=\frac{\sum\limits_{k=1}^{n}{{{f}_{k}}}\left| \left. {{X}_{k}}-\overline{X} \right| \right.}{\sum\limits_{k=1}^{n}{{{f}_{k}}}}\]
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