Subject: Mathematics
Merits and Demerits of M.D.
Merits
Demerits
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:
$$\sigma\;=\;\sqrt{\frac{\Sigma(x-\overline{x})^2}{n}}=\sqrt{\frac{\Sigma\;x^2}{n}-\bigg(\frac{x}{n}\bigg)^2}$$
$$Also,\;\sigma=\sqrt{\frac{\Sigma\;f(x-\overline{x})^2}{N}}=\sqrt{\frac{\Sigma\;fx^2}{N}-\bigg(\frac{fx}{N}\bigg)^2}$$
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
Demerits
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 s^{2} 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 n_{1 }and n_{2 } 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}$$
Difference between Mean deviation and Standard deviation:
Though mean deviation and standard deviation are the measures of dispersion including all the items, they differ in the following cases;
Properties of standard deviation
The standard deviation satisfies the following properties:
Taken reference from
( Basic mathematics Grade XII and A foundation of Mathematics Volume II and Wikipedia.com )
$$\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 n_{1 }and n_{2 } 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$$
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