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Variance and Standard Deviation - Shortcut Method to Find Variance and Standard Deviation

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notes

By using step-deviation method, it is possible to simplify the procedure. 
Let the assumed mean be ‘A’ and the scale be reduced to `1/h` times (h being the width of class-intervals). Let the step-deviations or the new 
values be `y_i`.
i.e. `y_i = (x_i - A) /h  or x_i = A +hy_i`                    ...(1)
we know that \[\bar{x} =\frac{\displaystyle\sum_{i=1}^{n} f_ix_i}{N} \]

Replacing xi from (1) in (2), we get

\[\bar{x} =\frac{\displaystyle\sum_{i=1}^{n} f_i(A + hy_i)}{N} \]
=\[\frac {1}{N}   (\displaystyle\sum_{i=1}^{n}
 f_i A + =   \displaystyle\sum_{i=1}^{n}
hf_iy_i )\] = =\[\frac {1}{N}  (A \displaystyle\sum_{i=1}^{n}
 f_i + h =  \displaystyle\sum_{i=1}^{n}
 f_i y_i) \]

=A .\[\frac{N}{N} + h \frac {\displaystyle\sum_{i=1}^{n} f_iy_i}{N} \] 

(because \[\displaystyle\sum_{i=1}^{n} f_i\] = N )

Thus `bar x = A + h bar y`         ...(3)

Now Variance of the variable x, \[\sigma_x^2 =\frac {1}{N}   \displaystyle\sum_{i=1}^{n}
 f_i (x_i - \bar x)^2 \]

=\[\frac {1}{N}   \displaystyle\sum_{i=1}^{n}
 f_i (A +hy_i - A - h  \bar y)^2 \]  (Using (1) and (2))

=\[\frac {1}{N}   \displaystyle\sum_{i=1}^{n}
 f_i  h^2  (y_i -       \bar y)^2 \] 

= \[\frac {h^2}{N}   \displaystyle\sum_{i=1}^{n}
 f_i (y_i -\bar y)^2 = h^2 × \] variance of the variable yi

i .e.`sigma_x^2 = h^2 sigma_x^2`

or `σ  _x = hσ _y`                        ...(4)
From (3) and  (4), we have 

σx  = \[{\frac{h}{N}}\sqrt{N\displaystyle\sum_{i=1}^{n} f_i y_i ^2 - (\displaystyle\sum_{i=1}^{n} f_i y_i) ^2 }  \]  ...(5)

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