PROBABILITY NORMAL DISTRIBUTION

• PROBABILITY DISTRIBUTION
  NORMAL
  RESEARCH METHODOLOGY
  
  
• NORMAL DISTRIBUTION

1. WIDELY USED CONTINUOUS PROBABILITY DISTRIBUTION
2. MAINLY USED TO STUDY THE BEHAVIOR OF CONTINUOUS RANDOM VARIABLES  LIKE HEIGHT,WEIGHT ETC
3. IT WAS FIRST DISCOVERED BY AN ENGLISH MATHEMATICIAN ABRAHAM DE MOIVRE IN 1733
4. LATER DISCOVERED AND APPLIED BY LAPLACE AND KARL GAUSS
5. ALSO KNOWN AS GAUSSIAN DISTRIBUTION AFTER THE NAME OF KARL GAUSS
6. NORMAL DISTRIBUTION AS LIMITING FORM OF BINOMIAL DISTRIBUTION
7. UNDER CERTAIN CONDITION
8.  n ,THE NUMBER OF TRIALS IN INFINITELY IN LARGE ,I.E n----∞ NEITHER p NOR q IS VERY SMALL

• GRAPH OF NORMAL DISTRIBUTION
• THE GRAPH OF THE NORMAL DISTRIBUTION IS CALLED NORMAL CURVE. THE SHAPE OF THE NORMAL CURVE DEPENDS ON THE VALUE OF MEAN AND STANDARD DEVIATION. THERE WILL BE DIFFERENT SHAPES OF NORMAL CURVE FOR DIFFERENT VALUES OF MEAN AND STANDARD DEVIATION
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• ASSUMPTION OF NORMAL DISTRIBUTION

1. INDEPENDENT CAUSES:- THE FORCES AFFECTING THE EVENT MUST BE INDEPENDENT OF ONE ANOTHER
2. CONDITION OF SYMMETRY:-THE OPERATION OF CAUSAL FORCES MUST BE SUCH THAT THE DEVIATIONS FROM MEAN ON EITHER SIDE IS EQUAL IN NUMBER AND SIZE
3. MULTIPLE CAUSATION: THE CAUSAL FORCES MUST BE NUMEROUS AND OF APPROXIMATELY EQUAL WEIGHT OR IMPORTANCE
4. PROPERTIES OF NORMAL DISTRIBUTION CURVE
5. PERFECTLY SYMMETRICAL AND BELL SHAPED
6. UNI MODAL DISTRIBUTION : IT HAS ONLY ONE MODE
7. EQUALITY OF MEAN,MEDIAN AND MODE
8. ASYMPTOTIC TO THE BASE LINE:- IT IS ASYMPTOTIC TO THE BASE LINE ON EITHER SIDE WHICH MEANS IT HAS TENDENCY TO TOUCH THE BASE LINE BUT IT NEVER TOUCHES IT.
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• PROPERTIES

1. THE NORMAL CURVE EXTENDS TO INFINITY ON EITHER SIDE
2. THE TOTAL AREA UNDER THE NORMAL CURVE IS 1
3. THE ORDINATES OF THE NORMAL CURVE AT THE MEAN IS MAXIMUM
4. THE MEAN ORDINATES DIVIDE THE WHOLE AREA UNDER THE CURVE INTO TWO EQUAL PARTS 50% ON THE RIGHT SIDE AND 50% ON THE LEFT SIDE
5. EQUIDISTANT OF QUARTILES:- IN NORMAL DISTRIBUTION Q1 AND Q3 ARE EQUIDISTANT FROM THE MEDIAN

Q₃-M=M-Q₁

                         

6. QUARTILE DEVIATION

• Q.D= 2/3 STANDARD DEVIATION

7. MEAN DEVIATION:

• M.D= 4/5 S.D

8. POINT OF INFLEXION:-

• HAS TWO POINTS OF INFLEXION THE POINT WHERE THE CURVE CHANGES ITS CURVATURE AT MEAN -1 AND MEAN + 1

9. DISTRIBUTION OF CONTINUOUS PROBABILITY DISTRIBUTION

a)    MEAN =X‾ OR ц OR m

b)    S,D=

c)    MOMENT COEFFICIENT OF SKEWNESS=√β1 =0

d)    MOMENT OF COEFFICIENT OF KURTOSIS=β2 =3

e)    VARIANCE IS SQUARE OF STANDARD DEVIATION =²

10.  MAIN PARAMETERS: TWO PARAMETER :

a)    MEAN

b)    STANDARD DEVIATION

11.  AREA PROPERTY: THE TOTAL AREA UNDER THE NORMAL CURVE IS ONE

a)    AREA UNDER NORMAL CURVE BETWEEN X‾-1  AND X‾+1 =.6826AREA UNDER NORMAL CURVE BETWEEN X‾-1  AND X‾+1 =.6826

b)    AREA UNDER NORMAL CURVE BETWEEN X‾-2  AND X‾+2 =.9545

c)    AREA UNDER NORMAL CURVE BETWEEN X‾-3  AND X‾+3 =.9973

12. IMPORTANCE OF NORMAL DISTRIBUTION

1. THE NORMAL DISTRIBUTION IS WIDELY USED IN THE STUDY OF NATURAL PHENOMENON LIKE LENGTH OF LEAVES OF THE TREES,BIRTH RATES AND DEATH RATES ETC
2. NORMAL DISTRIBUTION IS THE BASIS OF SAMPLING THEORY, WITH THE HELP OF NORMAL DISTRIBUTION,ONE CAN TEST WHETHER THE SAMPLES DRAWN FROM THE UNIVERSE REPRESENT THE UNIVERSE SATISFACTORY OR NOT
3. IT HELPS IN DETERMINING THE TOLERANCE OR SPECIFICATION LIMITS WITHIN WHICH THE QUALITY OF THE PRODUCT LIES. THE VARIATIONS IN THE QUALITY OF A PRODUCT ARE ACCEPTABLE WITH IN THESE TOLERANCE LIMITS
4. ALSO WIDELY USED IN CASE OF LARGE SAMPLES
5. SERVE AS A GOOD APPROXIMATION TO MANY THEORETICAL DISTRIBUTION SUCH AS BIONOMINAL,POISSON ETC

• MEASURMENT OF AREA UNDER NORMAL CURVE

1. THE GIVEN VALUES OF THE NORMAL VARIATE IS TRANSFORMED INTO STANDARD UNITS BY SUBSTITUTION
2. THE FORMULA Z=(X- X‾)/ 
3. SUPPOSE MEAN = 40
4. X=45,AND STANDARD DEVIATION IS 5
5. Z=45-40/5=5/5=1 THUS THE Z TRANSFORMATION WILL BE 1 WE HAVE TO SEE THE TABLE FOR ANY PARTICULAR VALUE OF Z