# Positional Measures and Mean

## Positional Measures and Mean

These are the values of the variable which divide the total frequency into a number of equal parts example median divides the total frequency into two equal parts. Some of these are as follows

(i) Quartiles: Those values of the variable which divides total distribution into four equal parts are known as quartile. thus, there will be three quartiles dividing the total distribution into four equal parts. Three quartiles are Q₁ (Lower quartiles), Q₂ (median), Q₃ (upper quartile).

 Individual (or) Discrete Series Continuous Series Formula Qr $$\frac{r(n+1)}{2}$$th term $$\frac{m}{4}$$th term $${{Q}_{r}}=l+\frac{h}{f}\left( \frac{m}{4}-c \right)$$th term

(ii) Deciles: Those values of variables which divides the distribution the total distribution into ten equal parts, are known as deciles.

(iii) Percentiles: Those values of variables which divides the total distribution into hundred equal parts, are known as Percentiles.

Mean: The sum of all the observation is divided by the number of observations is called Mean and it is denoted by x

Mean of Individual Data: If x₁, x₂, …, xn are n observations, then mean by

(a) Direct Method:

$$\bar{x}=\frac{{{x}_{1}}+{{x}_{2}}+…+{{x}_{n}}}{n}$$.

(b) Shortcut Method:

$$\bar{x}=A+\frac{1}{n}\sum\limits_{e=1}^{n}{{{d}_{e}}}$$,

Where A is assumed mean and dₑ = xₑ – A

Weighted Mean: Corresponding weight of x₁, x₂, …, xn are w₁, w₂, …, wrespectively mean, then weighted mean $$=\frac{{{w}_{1}}{{x}_{1}}+{{w}_{2}}{{x}_{2}}+…+{{w}_{n}}{{x}_{n}}}{{{w}_{1}}+{{w}_{2}}+….+{{w}_{n}}}$$.

Combined Mean: If two sets of observation are given, then the combined mean of both the sets can be calculated

$${{\bar{x}}_{12}}=\frac{{{n}_{1}}{{{\bar{x}}}_{1}}+{{n}_{2}}{{{\bar{x}}}_{2}}}{{{n}_{1}}+{{n}_{2}}}$$,

Where,

$${{\bar{x}}_{1}}$$= mean of first set of observations

n₁ = No. of observations in first set.

$${{\bar{x}}_{2}}$$= mean second set of observations

n₂ = number of second set of observations.

Properties of Mean:

(i) The sum of the squares of the deviations of a set of values is minimum when taken about mean.

(ii) Mean is affected by the change or shifting of origin and scale.