Standard Deviation

Standard deviation is the position square root of the arithmetic mean of the squares of the deviations of the given observation from their arithmetic mean.

Amongst all the methods of finding out dispersion, the standard deviation is regarded as the best. It is free from those defects with which the earlier methods (range, quartile deviation and mean deviation) suffer. Its value is based upon each and every item of the series and it also take into account algebraic signs. Standard deviation is also known as 'Root-Mean-Square Deviation' because it is the square root of the arithmetic mean of the square of the deviations. It is denoted by the Greek letter \(\sigma\) (known as Sigma).

1. While computing mean deviation, algebraic signs are ignored, whereas while calculating standard deviation, algebraic signs are taken into account.
2. The mean deviation can be computed either from mean, median or mode, whereas standard deviation is always computed from arithmetic mean.

Calculation of Standard Deviation

Individual Series

(1) Direct Method or Actual Mean Method:

Let X₁, X₂, X₃, .......................... X_n be the N-variate and \(\overline{X}\) be the arithmetic mean.

Define: x = X - \(\overline{X}\) = deviation taken from actual mean.

Then, standard deviation is given by the following formula:

\begin{align*} \sigma = \sqrt {\frac {\sum{x^2}}N}\\ \end{align*}

(2) Short-cut Method of Assumed Mean Method:

Let X₁, X₂, X₃, ..........................X_n be the N-variate values and A be the assumed mean.

Define: d = X - A = deviation taken from assumed mean.

Then, standard deviation is defined by the following formula:

\begin{align*} \sigma &= \sqrt {\frac {\sum{d^2}}N - (\frac {\sum {d}}N)^2}\\ \end{align*}

This method is suitable when the actual mean is in fraction (or decimal).

Discrete Series

Let X₁, X₂, X₃, ........................ X_n be the variable values and f₁, f₂, f₃, ....................... f_n is their frequencies.

Let, \(\overline{X}\) and A be the actual mean and assumed mean respectively. Then the standard deviation is defined by the following formulae:

(1) Direct Method or Actual Mean Method:

If x = X - \(\overline{X}\), then \(\sigma\) = \(\sqrt {\frac {\sum {fx^2}}N}\)

(2) Short cut Method or Assumed Mean Method:

If d = X - A, then \(\sigma\) = \(\sqrt {\frac {\sum{fd^2}}N - (\frac {\sum {fd}}N)^2}\)

(3) Step-Deviation Method:

If d' = \(\frac {X - A}h\), where h is common factor, then \(\sigma\) = \(\sqrt {\frac {\sum {fd'^2}}N - (\frac {\sum {fd'}}N)^2}\)× h

Continuous Series

Let X₁, X₂, X₃, ............................. X_n be the mid-values of the classes and f₁, f₂, f₃, ....................... f_n be their frequencies.

Let \(\overline{X}\) and A be the actual mean and assumed mean. Then, the standard deviation is defined by the following formulae:

(1) Direct Method:

If x = X - \(\overline{X}\), then \(\sigma\) = \(\sqrt {\frac {\sum {fx^2}}N}\)

(2) Short-cut Method:

If d = X - A, then \(\sigma\) = \(\sqrt {\frac {\sum {fd^2}}N - ({\frac {\sum {fd}}{N}})^2}\)

(3) Step Deviation Method:

If d' = \(\frac {X - A}h\), h = class size or common factor, then \(\sigma\) = \(\sqrt {\frac {\sum {fd'^2}}N - (\frac {\sum {fd'}}N)^2}\)× h

Coefficient of standard deviation and coefficient of variation

Standard deviation is the absolute measure of dispersion based on the standard deviation is known as the coefficient of standard deviation. Thus,

Coefficient of S.D. = \(\frac {S.D.}{Mean}\)

Also, the coefficient of standard deviation multiplied by 100 is known as the coefficient of variation (C.V.). Thus,

Coefficient of Variation (C.V.) = \(\frac {S.D.}{Mean}\)× 100

Merits and Demerits of Standard Deviation

Merits

1. The value of standard deviation is based on each and every item of the data.
2. It is free from those defects with which other methods like range, quartile deviation, mean deviation suffer.
3. It is less affected by fluctuations of sampling.

Demerits

1. As compared to the measure, it is somewhat more difficult to understand and compute.
2. Its value is unduly affected by extreme observations.