Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Relation between Mean, Median and Mode

(i) For a symmetrical distribution

Mean = Mode = Median

(ii)  For a moderately asymmetrical distributions

Mode =  3 Median –  2 Mean

Measure of Dsipersion:

(i) Range -:

The range is the difference between two extreme observations

i.e. Range =  x_\text{man}-x_\text{min}

Where,  x_\text{man} = Greatest observation of distribution

 x_\text{min} = Lowest observation of distribution

(ii)  Mean Deviation (M.D) -:

For a set of  n- observations  (x_1,x_2,...,x_n) the mean deviation about an average  A (mean/median/mode) is given by

 \text{M.D}=\dfrac{1}{n}\sum_{i=1}^{n}\ \left| x_i-A \right|

For a frequency distribution  \dfrac{x_i}{f_i},\ i=1,2,3,,,.

 \text{M.D}=\dfrac{1}{n}\sum_{i=1}^{n}f_i\ \left| x_i-A \right|

(iii) Standard Deviation (S.D) -:

Standard deviation is defined as the positive square root of the arithmetic mean of the squares of the deviations of the given values from their arithmetic mean.

\text{S.D.}=\sigma=\sqrt{\dfrac{1}{n}\sum_{i=1}^{n}\left( x_i-\overline{x} \right)^2}

 For frequency distribution  \dfrac{x_i}{f_i}, i=1,2,3,...

 \sigma^2=\sqrt{\dfrac{1}{N}\sum_{i=1}^{n} f_i\left( x_i-\overline{x} \right)^2}

 \to Standard deviation is regarded as the best measure of dispersion because it satisfies almost all the properties of an ideal measure of dispersion.

(iv) Variance -:  (\sigma^2)

Square of the standard deviation is called as variance  (\sigma^2) .

 \sigma^2=\dfrac{1}{N}\sum_{i=1}^{n} f_i\cdot x^2-\overline{x}\ ^2

Root Mean Square Deviation (S):

 S=\sqrt{\dfrac{1}{N}\sum_{i=1}^{n}f_i\left( x_i-A \right)^2}

Mean Square Deviation -:  (S^2)

 S^2={\dfrac{1}{N}\sum_{i=1}^{n}f_i\left( x_i-A \right)^2}

Analysis of Frequency Distribution -:

The coefficient of variation is defined as

 \text{C.V}=\dfrac{\sigma}{\overline{x}}\times 100, \overline{x}\neq 0

Where  \sigma and  \overline{x} are the standard deviation and mean of the data, respectively.

Shortcut Method to Find Variance -:

 \sigma^2=\dfrac{h^2}{N^2}\left[ N\sum_{i=1}^{n} f_i\ y_i\ ^2-\left( \sum_{i=1}^{n}\ f_iy_i \right)^2\  \right]

Where  y_i=\dfrac{x_i-A}{n}

Shortcut Method to Find Standard Deviation -:

 \sigma=\dfrac{h}{N}\sqrt{N\sum_{i=1}^{n}f_iy_i\ ^2 -\left( \sum_{i=1}^{n}\ f_iy_i \right)^2}

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