Standard Deviation, Variance

Subject: Mathematics

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This note helps you get the knowledge, merits, demerits and the important formulae related to standard deviation and variance.
Standard Deviation, Variance

Merits and Demerits of M.D.

Merits

  • It is easy to understand and calculate.
  • It is based on all observations.
  • As the deviations are taken from the central values, so the comparison of two distributions about their formation can easily be made.

Demerits

  • The greatest drawback of mean deviation is that the algebraic signs are ignored.
  • It is not capable of algebraic treatments.
  • It cannot be computed in case of open end classes.
  • It does not give satisfactory result when a deviation is taken from mode as the mode is ill-defined.

Standard Deviation

Standard deviation (S.D.) is defined as the positive square root of the mean of the square of the deviations taken from the arithmetic mean. It is denoted by \(\sigma\).

If x be the variate values and \(\overline{x}\) their arithmetic mean then the S.D denoted by

\(\sigma_x\) or simply \(\sigma\) is given by:

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$$\sigma\;=\;\sqrt{\frac{\Sigma(x-\overline{x})^2}{n}}=\sqrt{\frac{\Sigma\;x^2}{n}-\bigg(\frac{x}{n}\bigg)^2}$$

$$Also,\;\sigma=\sqrt{\frac{\Sigma\;f(x-\overline{x})^2}{N}}=\sqrt{\frac{\Sigma\;fx^2}{N}-\bigg(\frac{fx}{N}\bigg)^2}$$

Where n = no. of observation, N = total frequency

The first formula is for individual series and the second one is for discrete as well as continuous series, x is the mid value of each class for continuous series.

When deviations are taken from the assumed mean, then the formula for S.D. take the following forms

$$\sigma=\sqrt{\frac{\Sigma\;d^2}{n}-\bigg(\frac{d}{n}\bigg)^2}$$

$$Also,\;\sigma=\sqrt{\frac{\Sigma\;fd^2}{N}-\bigg(\frac{fd}{N}\bigg)^2}$$

$$And,\;\sigma=h\times\;\sqrt{\frac{\Sigma\;fd’^2}{N}-\bigg(\frac{fd’}{N}\bigg)^2}$$

$$where\;d=x-a,\;d’=\frac{x-a}{h},a=assumed\;mean,\;h=common\;fator.\;$$

Merits and Demerits of Standard Deviation

Merits

  • It is rigidly defined.
  • It is based on all observations.
  • It is least affected by the fluctuation of sampling.
  • It is suitable for further mathematical treatment.

Demerits

  • It is difficult to compute.
  • It gives greater weight to the extreme values and less to those which are near to the mean.
  • The standard deviation cannot be calculated for open end classes.

Variance and Mean Square deviation

Variance is the square of the standard deviation. Hence standard deviation and variance are equivalent in measuring variation. But due to difficulty in the computation of standard deviation, a variance is more preferred. Variance plays an important role in advance work of data analysis and is one of the most useful measurements in statistics. The square of the standard deviation is known as the variance. It is denote by \(\sigma\^2\;\mu_2\). Variance is calculated by the following formula:

$$\sigma^2\;=\;\frac{\Sigma(x-\overline{x})^2}{n}=\frac{\Sigma\;x^2}{n}-\bigg(\frac{x}{n}\bigg)^2$$

The mean square deviation denoted by s2 is defined by

$$s^2=\frac{\Sigma\;f(x-a)^2}{N}$$

Where a is an arbitrary number.

So in a mean square deviation, the deviations are taken from the arbitrary number ‘a’/

The positive square root of the mean square deviation is known as the root mean square deviation. It is denoted by s. Thus,

$$s=\sqrt{\frac{\Sigma\;f(x-a)^2}{N}}$$

Combined standard deviation

If n­1 ­and n­2 ­ be the sizes, \(\overline{x}_1\;and\overline{x}_2\;the\;arithmetic\;means\;\sigma_1\;and\;\sigma_2\) the respective standard deviations of two component series, then their combined standard deviation denoted by \(\sigma_{12}\) is given by;

$$\sigma_{12}=\sqrt{\frac{n_1\sigma_1^2+ n_2\sigma_2^2+ n_1d_1^2+ n_2d_2^2}{n_1+n_2}}

$$where\;d_1\;=\overline{x}_1-\overline{x}12\;\;\; d_2\;=\overline{x}_2-\overline{x}12$$

$$and\;\overline{x}_{12}=combined\;mean=\frac{n_1\overline{x}_1+n_2\overline{x}_2}{n_1+n_2}$$

Difference between Mean deviation and Standard deviation:

Though mean deviation and standard deviation are the measures of dispersion including all the items, they differ in the following cases;

  • Deviations are taken from the mean median or both in the case of mean deviation whereas the deviations are taken only from mean in case of standard deviation.
  • Whether the deviation is positive or negative, only the positive value is considered in the case of mean deviation but no such problem occurs in standard deviation.
  • Standard deviation follows different properties but mean deviation does not

Properties of standard deviation

The standard deviation satisfies the following properties:

  • In discrete distribution, the standard deviation is not less than the mean deviation from the name.
  • Standard deviation is the least possible of not mean square deviation
  • Standard deviation is independent of the change of origin but not of scale.

Taken reference from

( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )

Things to remember
  • Standard deviation (S.D.) is defined as the positive square root of the mean of the square of the deviations taken from the arithmetic mean. It is denoted by \(\sigma\).
  • $$\sigma\;=\;\sqrt{\frac{\Sigma(x-\overline{x})^2}{n}}=\sqrt{\frac{\Sigma\;x^2}{n}-\bigg(\frac{x}{n}\bigg)^2}$$

  •  The square of the standard deviation is known as the variance. It is denote by \(\sigma\^2\;\mu_2\). Variance is calculated by the following formula:

    $$\sigma^2\;=\;\frac{\Sigma(x-\overline{x})^2}{n}=\frac{\Sigma\;x^2}{n}-\bigg(\frac{x}{n}\bigg)^2$$

  • If n­­and n­2 ­ be the sizes, \(\overline{x}_1\;and\overline{x}_2\;the\;arithmetic\;means\;\sigma_1\;and\;\sigma_2\) the respective standard deviations of two component series, then their combined standard deviation denoted by \(\sigma_{12}\) is given by;

    $$\sigma_{12}=\sqrt{\frac{n_1\sigma_1^2+ n_2\sigma_2^2+ n_1d_1^2+ n_2d_2^2}{n_1+n_2}}

    $$where\;d_1\;=\overline{x}_1-\overline{x}12\;\;\; d_2\;=\overline{x}_2-\overline{x}12$$

  • It includes every relationship which established among the people.
  • There can be more than one community in a society. Community smaller than society.
  • It is a network of social relationships which cannot see or touched.
  • common interests and common objectives are not necessary for society.

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