BRM-Statistics in Research

8/9/2019 BRM-Statistics in Research

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GOOD MORNING

ALL OF YOU





The role of statistics in r esearch is toFunction as a tool in designing r esearch, analyzing its

data & drawing conclusions ther efrom.

` Classification & tabulation achieve this objective to some extent but a r esearcher have to go step further &develop certain indices or measur es to

summarize & analyze collected & classified data«

` After this we adopt the process of generalization fromsmall groups to population.



DESCRIPTIVE STATISTICS

&

INFERENTIAL STATISTICS

Descriptive statistics is concerned with development of certain indices from raw data

Inf erential statistics is mainly concerned with twomajor problems:-` Estimation of population parameters..` Testing of statistical hypothesis..



Measures

Of

Central tendency

Measures of

dispersion

Measures of

Asymmetry/skewness

Measures of

relationship

Other measures-indexnumbers, time

series..etc..



Also known as Statistical average, it tells us the point about which items have tendency to cluster.It is consider ed as the most r epr esentative figur e

for entir e mass of data.

Mean Median Mode

Geometric

mean

Harmonic

mean



Mean is also known as Arithmetic average which may be defined as the value which we get by dividing the total of the values of various given items in a series by the total no. of items .

CALCULATION OF MEAN

INDIVIDUAL DISTRIBUTION

ME AN or X= Xi/n = X1+X2+X3+««.+Xn/n

FREQUENCY DISTRIBUTION

X =f iXi/f i = f1X1+ f2 X2+««««««..fnXnf1+f2+f3«.+fn



WEIGHTED MEAN

X = WiXi / wi

ADVANTAGES OF MEAN` It is simple to understand & easy to calculate.` It is based on all obser vations` It is capable of further algebraic tr eatment.` It is rigidly defined.

DISADVANTAGES` It is highly aff ected by extr eme values` It can¶t be calculated in open-ended series.` It can¶t be ascertained graphically.` It can¶t be determined in situation when one value is

missing.



MEDIAN

It is defined as the middle value of the series whenarranged in ascending or descending orders. It is the value which divides the series into two equal parts. Itis thus a positional average & is used mostly in

context of qualitative phenomena.

CALCULATION OF MEDIAN

M = size of n + 1 th item2



ADVANTAGES OF MEDIAN` It is easy to understand.` It is easy to locate or compute.` It is not aff ected by extr eme values.` It is most appropriate average in case of open-ended classes.` It is the most suitable average in dealing with qualitative facts

such as beauty, honesty etc.

DISADVANTAGES` It r equir es arranging of data in ascending or descending order.` It is not based on all obser vations.` It is not capable of further algebraic tr eatment.` It is difficult to calculate if the no of items is very small or

large.



It is the most commonly or fr equently occurring value inthe series. Or it is the size of the item which has the maximum fr equency.

CALCULATION OF MODE1. Inspection method2. Grouping method

Calculation in continuous seriesZ = L1 + f1-f0 * i

2f1-f0-f2



ADVANTAGES OF MODE` It is easy to understand & simple to calculate.` It is not aff ected by extr eme values.` It can be located graphically.` It can be easily calculated in case of open ended classes.` All fr equencies ar e not needed for its calculation.

DISADVANTAGES` It is not suited to algebraic tr eatment.` Ther e can be bimodal fr equency series.` It is not based on all items of series.` It is not rigidly defined.



It is defined as the nth root of the product of the values of n items in a given series .

Symbolically, we can put it thus:

Geometric mean= ¥X1 * X2 * X3«.* Xn



It is defined as the r eciprocal of the average of the r eciprocal of the values of items of a series.

Symbolically, we can expr ess it as under:

H.M = Rec. r ec Xin

=Rec. r ec X1 + r ec X2 + r ec X3«+Xn

n



Dispersion is the measur e of the variations of the items. Itdefined as scatterness or spr eadness of the individualitems in a given series.

RANGE MEAN

DEVIATION

STANDARD

DEVIATION

COEFFICIENT OF STANDARD DEVIATION &

COEFFICIENT OF VARIATION



RANGE is the biggest possible measur e of dispersion &is defined as the diff er ence between the values of the extr eme items of a series.

RANGE= highest value of an - lowest value of Item in a series) the item in series

MEAN DEVIATIONIt is the average of diff er ence of the value of items fromsome average of the series.



CALCULATION OF MEAN DEVIATION

MEAN DEVIATION FROM MEAN (x ) = Xi ± Xn

MEAN DEVIATION FROM MEDIAN (m) = Xi ± M

n

COEFFICIENT OF MEAN DEVIATIONWhen mean deviation is divided by the average used in finding

out the mean deviation itself.

for eg. ± Mean deviationMean or median



It is defined as the squar e root of the average of squar esof deviations, when such deviations for the values of individual items in a series ar e obtained from the arithmetic average. It is denoted by symbol .

CALCULATIONStandard deviation () = (X ± X)²

n



COEFFICIENT OF STANDARD DEVIATION

When we divide standard deviation by arithmetic average.Thus the r elative measur e of standard deviation is called

coefficient of S.D.

COEFFICIENT OF S.D =

X

VARIANCEVariance is the squar e of the standard deviation.

Variance = (S.D.)² = ²



COEFFICIENT OF VARIATION= S.D. * 100

X

ADVANTAGES OF STANDARD DEVIATION` It is rigidly defined.`

It is based on all obser vations.` It is capable of being tr eated mathematically.` It is not very much aff ected by the fluctuations of sampling.

DISA

DVA

NTAG

ES

` It is difficult to understand & difficult to calculate.` It gives mor e importance to extr eme obser vations.



Skewness is degr ee of asymmetry or departur e from symmetry of a distribution....It means lack of symmetry in a fr equencydistribution..the significance of skewness lies in the fact that we can study the formation of series & can have an idea about

shape of cur ve, whether normal or otherwise, when items of series ar e plotted on graph.

The fr equency distribution is skewed when

X =M

= Z

It is either positively skewed when X > M > Z or it isnegatively skewed when X < M < Z...



KARL PEARSON¶S METHOD

Skewness = X ± Z

When mode is ill-defined , we calculate skewness as:

skewness = 3( X ± M )

COEFFICIENT OF SKEWNESS

= X ± Z OR 3( X ± M )



Under this we study :- Bivariate population- A. Corr elation ±it is studied through1. Spearman¶s rank corr elation.

2. Karl Pearson's coefficient of corr elation. A. Regr ession analysis Multivariate population- A. Corr elation ± it can be studied through a) Coefficient of multiple corr elation.b) Coefficient of partial corr elation A. Multiple r egr ession



KARL PEARSON¶S COEFFICIENT OFCORRELATION

r = xy¥ x² * y²

x = X ± X , y = Y - Y

SIMPLE REGRESSION ANALYSIS

Y = a + bX

Y = Na + bXX Y = aX + bX²



Multiple correlation

R. = ¥r² + r² - 2r .r .r

1- r² Partial correlation

r . = r - r . r

¥1-r² ¥1- r²

Multiple regression

Y = a + bX + bX

Y = na + bX + bX

XY=aX + bX² + bXX

XY=aX + bXX + bX²



INDEX NUMBERS

An index number is a statistical measur e designed to show changes ina variable or group of r elated variables with r espect to time, geographic location or other characteristics.

` They ar e helpful in making pr edictions.

` Index numbers also measur es changes taking place ineconomic & social phenomena, in business & provide for

comparison of these changes.

Thus index numbers can be def ined as a specialized type of averages which measures the relative changes in diff variablesat diff point of time.



` Index numbers measur e only r elative changes.` It measur es cost of living of diff class of people.

TIME SERIES ANALYSIS

In context of economic & business r esearches, we may obtainoften data r elating to some time period concerning some phenomenon..Such data is labeled as TIME SERIES.

Such series ar e usually the eff ects of one or mor e of the followingfactors:

1. Secular trend or long term trend2. Short time oscillations.a) Cyclical f luctuations(C)b) Seasonal f luctuations(S)c) Irregular f luctuations( I )



For analyzing time series we usually have two models:- A). Multiplicative model

Y = T * C * S * I

B). ADDITIVE MODELY = T + S + C + I

Methods of time series analysis

Fr ee hand

cur ve method

Semi-average method

Method of moving

averages Method of least

squar es



IMPORTANCE OF TIME SERIES ANALYSIS

` For Understanding the dynamic conditions.` Evaluating success or f ailure of business policies by study

of past trends` Formulating f uture policies on basis of past trends

Thus , analysis of time series is important in context of longterm short term f orecasting is theref ore considered apower f ul tool in hand of researchers.



THANK

YOU

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BRM-Statistics in Research