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Arithmetic Mean of grouped data can be found by $\bar{x}=\frac{\sum f_ix_i}{\sum f_i}$ where $x_i$ is the midpoint of each class.
Aother equation to evaluate Arithmetic Mean of grouped data is using Step Deviation $$ \boxed{\bar{x}=a+\frac{\sum f_id_i}{\sum f_i}} $$ where $a$ is the assumed mean and $d_i=x_i-a$ are the deviations of $x_i$ from the assumed mean, and $N=\sum f_i$
How do I make sense of the second formula which is called the Step Deviation method ?
Is there a proof which shows the above equation actually is the mean ?
1 Answer
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Thanx @StubbornAtom for the hint. $$ x_i=d_i+a\\ \bar{x}=\frac{\sum f_ix_i}{\sum f_i}=\frac{\sum f_i(d_i+a)}{N}=\frac{\sum f_id_i}{N}+a\frac{\sum f_i}{N}=\frac{\sum f_id_i}{N}+a $$
