Statistics - Notes, Concept and All Important Formula

STATISTICS

MEASURES OF CENTRAL TENDENCY:

An average value or a central value of a distribution is the value of variable which is representative of the entire distribution, this representative value are called the measures of central tendency. Generally there are following five measures of central tendency:

(a) Mathematical average

(i) Arithmetic mean.           (ii) Geometric mean.           (iii) Harmonic mean

(b) Positional average

(i) Median.       (ii) Mode

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All Chapter Notes, Concept and Important Formula

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1. ARITHMETIC MEAN/MEAN:

(i) For ungrouped dist.: If $x _{1}, x _{2}, \ldots \ldots x _{ n }$ are $n$ values of variate $x$ then their mean $\bar{x}$ is defined as

$\overline{ x }=\dfrac{ x _{1}+ x _{2}+\ldots . .+ x _{ n }}{ n }=\frac{\displaystyle \sum_{ i =1}^{ n } x _{ i }}{ n }$

$\Rightarrow\displaystyle \Sigma x _{ i }= n \overline{ x }$

(ii) For ungrouped and grouped freq. dist.: If $x_{1}, x_{2}, \ldots . x_{n}$ are values of variate with corresponding frequencies $f _{1}, f _{2}, \ldots f _{ n }$ then their mean is given by

$\overline{ x }=\frac{ f _{1} x _{1}+ f _{2} x _{2}+\ldots+ f _{ n } x _{ n }}{ f _{1}+ f _{2}+\ldots+ f _{ n }}=\frac{\displaystyle \sum_{ i =1}^{ n } f x _{ i }}{ N }, \text { where } N =\displaystyle \sum_{ i =1}^{ n } f _{ i }$

(iii) By short cut method :

Let $d _{ i }= x _{ i }- a$

$\therefore \overline{ x }= a +\frac{\displaystyle \Sigma f _{i}d _{ i }}{ N }$, where a is assumed mean.

(iv) By step deviation method:

Let $\quad u_{i}=\frac{d_{i}}{h}=\frac{x_{i}-a}{h}$

$\therefore \quad \overline{ x }= a +\left(\frac{\Sigma f _{ i } i _{ i }}{ N }\right) h$

(v) Weighted mean : If $w_{1}, w_{2}, \ldots \ldots w_{n}$ are the weights assigned to the values $x _{1}, x _{2}, \ldots \ldots x _{ n }$ respectively then their weighted mean is defined as

$\text { Weighted mean }=\dfrac{w_{1} x_{1}+w_{2} x_{2}+\ldots . .+w_{n} x_{n}}{w_{1}+\ldots . .+w_{n}}$$=\frac{\displaystyle \sum_{i=1}^{n} w_{i} x_{i}}{\displaystyle \sum_{i=1}^{n} w_{i}}$

(vi) Combined mean : If $\overline{ x }_{1}$ and $\overline{ x }_{2}$ be the means of two groups having $n _{1}$ and $n _{2}$ terms respectively then the mean (combined mean) of their composite group is given by combined mean

 $=\dfrac{ n _{1} \overline{ x }_{1}+ n _{2} \overline{ x }_{2}}{ n _{1}+ n _{2}}$

If there are more than two groups then,

$\text { combined mean }=\dfrac{ n _{1} \overline{ x }_{1}+ n _{1} \overline{ x }_{2}+ n _{3} \overline{ x }_{3}+\ldots}{ n _{1}+ n _{2}+ n _{3}+\ldots}$

(vii) Properties of Mean:

• Sum of deviations of variate about their mean is always zero i.e. $\Sigma\left(x_{i}-\bar{x}\right)=0, \Sigma f_{i}\left(x_{i}-\bar{x}\right)=0$
• Sum of square of deviations of variate about their mean is minimum i.e. $\Sigma\left( x _{ i }-\overline{ x }\right)^{2}$ is minimum
• If $\bar{x}$ is the mean of variate $x_{i}$ then mean of $\left(x_{i}+\lambda\right)$ is $\bar{x}+\lambda$, mean of $\left(\lambda x_{1}\right)=\lambda \bar{x}$, mean of $\left(a x_{i}+b\right)$ is $a \bar{x}+b$ (where $\lambda, a, b$ are constant)

• Mean is independent of change of assumed mean i.e. it is not effected by any change in assumed mean.

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2. MEDIAN :

The median of a series is the value of middle term of the series when the values are written in ascending order. Therefore median, divides an arranged series into two equal parts.

Formulae of median:

(i) For ungrouped distribution : Let $n$ be the number of variate in a series then

$\scriptsize{Median=\left[\begin{array}{l} \left.\left(\frac{ n +1}{2}\right)^{ th } \text { term, (when } n \text { is odd }\right) \\ \text { Mean of }\left(\frac{ n }{2}\right)^{\text {th }} \text { and }\left(\frac{ n }{2}+1\right)^{ th } \text { terms, (when } n \text { is even) }\end{array}\right.}$

(ii) For ungrouped freq. dist. : First we prepare the cumulative frequency (c.f.) column and Find value of $N$ then

$\scriptsize{Median =\left[\begin{array}{l} \left.\left(\frac{ N +1}{2}\right)^{ th } \text { term, (when } N \text { is odd }\right) \\ \text { Mean of }\left(\frac{ N }{2}\right)^{\text {th }} \text { and }\left(\frac{ N }{2}+1\right)^{\text {th }} \text { terms, (when } N \text { is even) }\end{array}\right.}$

(iii) For grouped freq. dist : Prepare c.f. column and find value of $\frac{ N }{2}$ then find the class which contain value of c.f. is equal or just greater to $N / 2$, this is median class

$\therefore \text { Median }=\ell+\frac{\left(\frac{ N }{2}- F \right)}{ f } \times h$

where $\mathbf{\ell}$ - lower limit of median class

f - freq. of median class

F - c . f. of the class preceding median class

h - Class interval of median class

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3. MODE :

In a frequency distribution the mode is the value of that variate which have the maximum frequency.

Method for determining mode :

(i) For ungrouped dist.: The value of that variate which is repeated maximum number of times.

(ii) For ungrouped freq. dist. : The value of that variate which have maximum frequency.

(iii) For grouped freq. dist.: First we find the class which have maximum frequency, this is model calss

$\therefore \text { Mode }=\ell+\frac{ f _{0}- f _{1}}{2 f _{0}- f _{1}- f _{2}} \times h$

where $\mathbf \ell$ - lower limit of model class

$\mathbf f _{0}-$ freq. of the model class

$\mathbf f _{1}-$ freq. of the class preceding model class

$\mathbf f _{2}-$ freq. of the class succeeding model class 

$\mathbf h$ - class interval of model class

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4. RELATION BETWEEN MEAN, MEDIAN AND MODE :

In a moderately asymmetric distribution following is the relation between mean, median and mode of a distribution. It is known as impirical formula.

Mode $=3$ Median - 2 Mean

Note: (i) Median always lies between mean and mode

(ii) For a symmetric distribution the mean, median and mode coincide.

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5. MEASURES OF DISPERSION :

The dispersion of a statistical distribution is the measure of deviation of its values about the their average (central) value. Generally the following measures of dispersion are commonly used.

(i) Range

(ii) Mean deviation

(iii) Variance and standard deviation

(i) Range : The difference between the greatest and least values of variate of a distribution, are called the range of that distribution. If the distribution is grouped distribution, then its range is the difference between upper limit of the maximum class and lower limit of the minimum class.

Also, coefficient of range $=\dfrac{\text { difference of extreme values }}{\text { sum of extreme values }}$

(ii) Mean deviation (M.D.) : The mean deviation of a distribution is, the mean of absolute value of deviations of variate from their statistical average (Mean, Median, Mode).

If $A$ is any statistical average of a distribution then mean deviation about $A$ is defined as

Mean deviation $=\dfrac{\displaystyle \sum_{ i =1}^{ n }\left| x _{ i }- A \right|}{ n } \quad$ (for ungrouped dist.)

Mean deviation $=\dfrac{\displaystyle \sum_{ i =1}^{ n } f _{ i }\left| x _{ i }- A \right|}{ N }$ (for freq. dist.)

Note :- It is minimum when taken about the median

Coefficient of Mean deviation $=\dfrac{\text { Mean deviation }}{ A }$

(where A is the central tendencyabout which Mean deviation is taken)

(iii) Variance and standard deviation : The variance of a distribution is, the mean of squares of deviation of variate from their mean. It is denoted by $\sigma^{2}$ or var(x).

The positive square root of the variance are called the standard deviation. It is denoted by $\sigma$ or S.D.

Hence standard deviation $=+\sqrt{\text { variance }}$

Formulae for variance :

(i) for ungrouped dist. :

$\begin{array}{l}\sigma_{x}^{2}=\frac{\Sigma\left(x_{i}-\bar{x}\right)^{2}}{n} \\\sigma_{x}^{2}=\frac{\Sigma x_{i}^{2}}{n}-\bar{x}^{2}=\frac{\Sigma x_{i}^{2}}{n}-\left(\frac{\Sigma x_{i}}{n}\right)^{2} \\\sigma_{d}^{2}=\frac{\Sigma d_{i}^{2}}{n}-\left(\frac{\Sigma d_{i}}{n}\right)^{2}, \text { where } d_{i}=x_{i}-a\end{array}$

(ii) For freq. dist.:

$\sigma_{ x }^{2}=\frac{\sum f _{ i }\left( x _{ i }-\overline{ x }\right)^{2}}{ N }$

$\begin{array}{l}\sigma_{x}^{2}=\frac{\Sigma f_{1} x_{i}^{2}}{N}-(\bar{x})^{2}=\frac{\Sigma f_{i} x_{i}^{2}}{N}-\left(\frac{\Sigma f_{i} x_{i}}{N}\right)^{2} \\\sigma_{d}^{2}=\frac{\Sigma f_{i} d_{i}^{2}}{N}-\left(\frac{\Sigma f_{i} d_{i}}{N}\right)^{2} \\\sigma_{u}^{2}=h^{2}\left[\frac{\Sigma f_{i} u_{i}^{2}}{N}-\left(\frac{\Sigma f_{i} u_{i}}{N}\right)^{2}\right] \quad \text { where } u_{i}=\frac{d_{i}}{h}\end{array}$

(iii) Coefficient of S.D. $=\frac{\sigma}{\overline{ x }}$

Coefficient of variation $=\frac{\sigma}{\overline{ x }} \times 100 \quad$ (in percentage)

Note :- $\sigma^{2}=\sigma_{ x }^{2}=\sigma_{ d }^{2}= h ^{2} \sigma_{ u }^{2}$

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6. MATHEMATICAL PROPERTIES OF VARIANCE :

$\operatorname{Var} .\left( x _{ i }+\lambda\right)=\operatorname{Var}\left( x _{ i }\right)$

$\operatorname{Var} \cdot\left(\lambda x _{ i }\right)=\lambda^{2} \cdot \operatorname{Var}\left( x _{ i }\right)$

$\operatorname{Var}\left(a x_{i}+b\right)=a^{2} \cdot \operatorname{Var}\left(x_{i}\right)$

where $\lambda, a , b$, are constant

If means of two series containing $n _{1}, n _{2}$ terms are $\overline{ x }_{1}, \overline{ x }_{2}$ and their variance's are $\sigma_{1}^{2}, \sigma_{2}^{2}$ respectively and their combined mean is $\bar{x}$ then the variance $\sigma^{2}$ of their combined series is given by following formula

$\sigma^{2}=\dfrac{ n _{1}\left(\sigma_{1}^{2}+ d _{1}^{2}\right)+ n _{2}\left(\sigma_{2}^{2}+ d _{2}^{2}\right)}{\left( n _{1}+ n _{2}\right)}$ $\text{where }d _{ i }=\overline{ x }_{1}-\overline{ x }, d _{2}=\overline{ x }_{2}-\overline{ x }$

i.e. $\sigma^{2}=\dfrac{ n _{1} \sigma_{1}^{2}+ n _{2} \sigma_{2}^{2}}{ n _{1}+ n _{2}}+\dfrac{ n _{1} n _{2}}{\left( n _{1}+ n _{2}\right)^{2}}\left(\overline{ x }_{1}-\overline{ x }_{2}\right)^{2}$

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