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

Quantiles

Quantiles or partition values are the value of the variets which divides the distribution into a number of equal parts. Some important partition value are quartiles, deciles and percentiles.

(i) Quartiles denoted by  Q_1,\ Q_2,\ Q_3 divides the distribution into  4 equal parts.

  Q_i=l+\dfrac{h}{f}\left( \dfrac{iN}{4}-c \right),\ i=1,2,3

(ii) Deciles denoted by  D_1,D_2,...,D_9 divides the distribution into  10 equal parts.

 D_i=l+\dfrac{h}{f}\left( \dfrac{iN}{10}-c \right),\ i=1,2,3,...,9

 (iii) Percentiles denoted by  P_1,P_2,...,P_{99} divides the distribution into  100 equal parts.

 P_i=l+\dfrac{h}{f}\left( \dfrac{iN}{100}-c \right),\ i=1,2,3,...,99

Geometric Mean:

Geometric mean of a set of  n observations  (x_1,x_2,...,x_n) is the  n^{th} root of their product.

G.M =  (x_1, x_2,...x_n)^\frac{1}{n}

 \Rightarrow \log G=\dfrac{1}{n}\left\{ \log x_1+\log x_2+...+\log x_n \right\}

 \Rightarrow \log G=\dfrac{1}{n}\sum_{i=1}^{n}\log x_i

Merits -:

It is defined based on all observations suitable for further mathematical treatment not affected much by sampling fluctuations.

Demerits -:

Not very much easy to calculate and understand by the non-mathematical persons. If one observation is zero, then the geometric mean is zero.

It gives comparatively more weight to small items.

Uses -:

A geometric mean is used to study the population growth, rate of component interest, construction of index numbers, study of cell divisions, etc.

Harmonic Mean -:

Harmonic mean of a set of  n observations  (x_1,x_2,...,x_n) is the reciprocal of the arithmetic mean or simple mean of the reciprocal of the observations.

 H=\dfrac{1}{\dfrac{1}{n}\sum_{i=1}^{n}\dfrac{i}{x_i}}

Merits -:

Rigidly defined, based on all observations, suitable for further mathematical treatment; not affected much by sampling fluctuations.

Demerits -:

 \to It’s not easy to understand or calculate. If one observation is zero, then the harmonic mean is infinite.

 \to It gives more important to small items.

Mode -:

Mode is the value corresponding to maximum frequency.

 x: 1\ 2\ 3\ 4\ 5\ 6\

 f: 4\ 9\ 16\ 25\ 22\ 15\

 \Rightarrow  mode is  4 .

The mode is determined by the method of grouping.

In the case of continuous frequency distribution, the class interval corresponding to the maximum frequency to the maximum frequency is called the ‘modal class’, and the mode is obtained by the following formula.

 \begin{aligned}  \text{Mode}&= l+\dfrac{h(f_1-f_0)}{(f_i-f_0)-(f_2-f_1)}\\ &=l+\dfrac{h(f_1-f_0)}{2f_i-f_2-f_0} \end{aligned}

Merits

(i) Readily comprehensible and easy to calculate

(ii) Not affected by extreme values

(iii) Useful for open-end class

Demerits

(i) Mode is ill-defined, i.e. for some distribution mode can’t be clearly defined as there may exist more than one mode.

(ii) It is not based on all observations, not suitable for further mathematical treatment.

(iii)  Affected much by sampling fluctuations.

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