STATISTICS

STATISTICS:Stats meaning immediately and stics means into

MEAN FOR UNGROUPED DATA

Mean=SumofallobservationsNumberofobservations$$Mean=\frac{Sumofallobservations}{Numberofobservations}$$

Mean(X¯¯¯¯)=∑xin$$Mean\left(\overline{X}\right)=\frac{\sum x_i}{n}$$

MEAN FOR A GROUPED DATA

 There are three methods for this

1) DIRECT METHOD

Mean(X¯¯¯¯)=∑fixi∑fior∑fixin$$Mean\left(\overline{X}\right)=\frac{\sum f_i{x}_i}{\sum f_i}or\frac{\sum f_i{x}_i}{n}$$

Where:xi=Midvalues,∑fi=sumofallfrequencies$$Where:x_i=Midvalues,\sum f_i=sumofallfrequencies$$

2) ASSUMED MEAN METHOD

Mean(X¯¯¯¯)=a+∑fidi∑fiora+∑fidin$$Mean\left(\overline{X}\right)=a+\frac{\sum f_i{d}_i}{\sum f_i}ora+\frac{\sum f_i{d}_i}{n}$$

Where:a=AssumedMean,di=xi−a$$Where:a=AssumedMean,d_i=x_i-a$$

3) STEP DEVIATION METHOD

Mean(X¯¯¯¯)=a+h×∑fiui∑fiora+h×∑fiuin$$Mean\left(\overline{X}\right)=a+h\times \frac{\sum f_i{u}_i}{\sum f_i}ora+h\times \frac{\sum f_i{u}_i}{n}$$

Where:h=heightoftheinterval,ui=xi−ah

MEDIAN

FOR UNGROUPED DATA

ALGORITHM:-  a) First write the given observation in ascending order.

b)Find the number of terms let it is n.

IfnisoddthenMedian=(n+12)th$$IfnisoddthenMedian={\left(\frac{n+1}{2}\right)}^{th}$$

Ifniseventhen,Median=12[(n2)thterm+(n2+1)th]

MEDIAN FOR GROUPED DATA  

ALGORITHM:-     1) Find the cumulative frequency of the observations.

2)Find∑fi=N,thenfindN2$$2)Find\sum f_i=N,thenfind\frac{N}{2}$$

3) Then find the Median class and then Median.

MedianClass=Classcorrespondingtothec.f≥N2iscalledMedianClass$$MedianClass=Classcorrespondingtothec.f\ge \frac{N}{2}iscalledMedianClass$$

Median=l+(N2−c.ff)×h$$Median=l+\left(\frac{\frac{N}{2}-c.f}{f}\right)\times h$$

Wherel=LowerlimitofMedianClass$$Wherel=LowerlimitofMedianClass$$

c.f=CumulativefrequencyoftheclassprecedingtheMedianClass$$c.f=CumulativefrequencyoftheclassprecedingtheMedianClass$$

f=FrequencyoftheMedianClass$$f=FrequencyoftheMedianClass$$

h=HeightoftheMedianClass$$h=HeightoftheMedianClass$$

STATISTICS

MODE

 (FOR UNGROUPED DATA)

Most frequent observation is called mode, in other words, the observation whose frequency is maximum is called mode.

MODE :- (FOR GROUPED DATA)

First of all find the modal class

MODAL CLASS:- Class corresponding to the highest frequency is called the modal class. 

MODE=l+(f1−f02f1−f0−f2)×h$$MODE=l+\left(\frac{f_1-f_0}{2f_1-f_{0-f_2}}\right)\times h$$

                     Where:-

l=Lowerlimitofthemodalclass$$l=Lowerlimitofthemodalclass$$

f1=Frequencyofthemodalclass$$f_1=Frequencyofthemodalclass$$

f0=Frequencyoftheclassprecedingthemodalclass$$$$f0=Frequencyoftheclassprecedingthemodalclass$$\mathrm{<span\; style="color:\; \#757575;"><br></span>}$$$$$$

EMPIRICAL FORMULA SHOWING THE RELATION BETWEEN MEAN, MODE MEDIAN
Mode = 3 Median - 2 Mean

MEAN DEVIATION ABOUT MEAN

FOR UNGROUPED DATA

MeanDeviationAbout(X¯¯¯¯)=∑|xi−x¯¯¯|n$$MeanDeviationAbout(\overline{X})=\frac{\sum \left|x_i-\overline{x}\right|}{n}$$

FOR GROUPED DATA

MeanDeviationAbout(X¯¯¯¯)=∑fi|xi−x¯¯¯|n$$MeanDeviationAbout(\overline{X})=\frac{\sum f_i\left|x_i-\overline{x}\right|}{n}$$

MEAN DEVIATION ABOUT MEDIAN

FOR UNGROUPED DATA

M.D.(M)=∑|xi−M|N,WhereM=Median$$M.D.(M)=\frac{\sum \left|x_i-M\right|}{N},WhereM=Median$$

FOR GROUPED FREQUENCY

M.D.(M)=∑fi|xi−M|N,WhereN=Sumofallfrequencies

VARIANCE

VARIANCE:- Mean of the squares of the deviations from the mean is called the variance.

FOR UNGROUPED DATA

Variance(σ2)=∑(xi−x¯¯¯)2N,WhereNisthesumofallfrequencies$$Variance({\sigma }^2)=\frac{\sum {\left(x_i-\overline{x}\right)}^2}{N},WhereNisthesumofallfrequencies$$

VARIANCE FOR GROUPED DATA

Variance(σ2)=∑fi(xi−x¯¯¯)2N,WhereN=∑fi$$Variance({\sigma }^2)=\frac{\sum f_i{\left(x_i-\overline{x}\right)}^2}{N},WhereN=\sum f_i$$

SHORT-CUT METHOD OF FINDING THE VARIANCE

Variance(σ2)=h2N[∑fiy2i−(∑fiyi)2]$$Variance({\sigma }^2)=\frac{h^2}{N}\left[\sum f_i{y}_i^2-{\left(\sum f_i{y}_i\right)}^2\right]$$

Variance(σ2)=h2N[∑y2i−(∑yi)2],iffiisnotgiven$$Variance({\sigma }^2)=\frac{h^2}{N}\left[\sum y_i^2-{\left(\sum y_i\right)}^2\right],if{f}_{i}isnotgiven$$

COEFFICIENT OF VARIATION

C.V=StandardDeviationMean×100orC.V=σx¯¯¯×100$$C.V=\frac{StandardDeviation}{Mean}\times 100orC.V=\frac{\sigma }{\overline{x}}\times 100$$

If C.V is more then observations are more variable, unstable and less consistence and vice-versa.

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f2=Frequencyoftheclasssucceedingthemodalclass