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Statistics formulae

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  • 1. STATISTICS FORMULAEMEASURES OF CENTRAL TENDENCY: 1) Range = Highest Value – Lowest Value 2) No. Of Class Intervals = 1+ 3.2 log N (Struge’s Rule) 3) Class Width = Range / No. Of Class Intervals. OR Class Intervals = Range / Class Width 4) Arithmetic Mean = ∑ ‫/ ݔ‬n, OR ∑݂‫ / ݔ‬N ొ ି ୔.େ.୊ L+ቈ ቉×H ୬ାଵ మ ଶ ୊ 5) Median = or Where: L = Lower limit of median class, PCF = Previous Cumulative Frequency, F= Frequency of the Median Class, & H= Width of the CI ‫+ܮ‬ቂ ቃ×ℎ ௙ଵି௙଴ ଶሺ௙ଵሻି௙௢ି௙ଶ 6) Mode = Where: Where: L = Lower limit of modal class, f1 = Highest Frequency in the Distribution, f0 = Preceding Frequency to the Highest Frequency in the Distribution, f2 = Succeeding Frequency to the Highest Frequency in the Distribution, & H= Width of the CI ௡ ୒ భ ౜ ∑ ∑ 7) Harmonic Mean = or ೣ ౮ ∑ ௅௢௚ ௫ ∑ ௙௅௢௚ ௫ 8) Geometric Mean = ‫݃݋݈݅ݐ݊ܣ‬ ‫݃݋݈݅ݐ݊ܣ‬ ௡ ே orPraveenKumar Keskar 1
  • 2. ಿ ௥ቀ ర ቁି௉.஼.ி ‫ݎ‬ቀ ቁ ‫ݎ‬ቀ ቁ ݈+ቈ ቉×ℎ ௡ାଵ ேାଵ ସ ସ ி 9) Quartiles = or or ݊+1 ܰ+1 ‫ݎ‬ቀ ቁ or ‫ݎ‬ቀ ቁ 10 10 10) Deciles = or ಿ ௥ቀ ቁି௉.஼.ி ݈+ቈ భబ ቉×ℎ ி ‫ ݎ‬ቀ ଵ଴଴ ቁ ‫ ݎ‬ቀ ଵ଴଴ ቁ ௡ାଵ ேାଵ 11) Percentile = or or ಿ ௥ቀ ቁି௉.஼.ி ݈+ቈ భబబ ி ቉×ℎMEASURES OF DISPERSION: 12) Range = Highest Value – Lowest Value ୌ୧୥୦ୣୱ୲ ୚ୟ୪୳ୣ – ୐୭୵ୣୱ୲ ୚ୟ୪୳ୣ ୌ୧୥୦ୣୱ୲ ୚ୟ୪୳ୣା ୐୭୵ୣୱ୲ ୚ୟ୪୳ୣ Co-efficient of Range = ୕ଷିொଵ ଶ 13) Quartile Deviation = ୕ଷିொଵ ொଷାொଵ Co-efficient of Q.D = ഥ ∑ |௫ି௫| ഥ ∑ ௙|௫ି௫| ௡ ே 14) Mean Deviation = or ெ௘௔௡ ஽௘௩௜௔௧௜௢௡ ௙௥௢௠ ௠௘௔௡ ெ௘௔௡ Co-efficient of MD =PraveenKumar Keskar 2
  • 3. Standard Deviation = ට ට ∑ሺ௫ି௫̅ ሻమ ∑ ௙ሺ௫ି௫̅ ሻమ ௡ ே 15) or OR ∑x ଶ ∑ fx ଶ ට ∑ x2 −ቀ nቁ ට ∑ fx2 −ቀ ቁ n ୒ ୒ or × 100 ஢ ത ୶ Co-efficient of Variation =PROBABILITY: 16) Additional Theorem = PሺA ⋃ Bሻ = PሺAሻ + PሺBሻ − PሺA ⋂ Bሻ P ቆBൗAቇ = ୔ሺ୅ ⋂ ୆ሻ ୔ሺ୅ሻ 17) Conditional Theorem = ୔ሺ୅ሻ×୔ሺ୅ Baye’s Theorem = P ൬Aൗ ൰ = ൗ୉ ሻ Eଵ భ 18) ୅ൗ ሻ ∑ ୔ሺ୉భ ሻ×୔ሺ ୉ భ 19) Binomial Distribution: Probability Mass Function (P.M.F) = nେ౮ × ሺpሻ୶ × ሺqሻ୬ି୶ Where, ‫ݔ‬ = 0, 1, 2 _ _ _ n, p+q = 1 20) Poison Distribution ሺ௘ ష೘ ሻ×ሺ௠ሻೣ ௫! P.M.F = Where, x = 0, 1, 2, 3 _ _ _ _ _ _ _ _ ∞PraveenKumar Keskar 3
  • 4. ௫ି௠ ఙ 21) Normal Distribution = Where m = mean & ߪ = Standard DeviationCORRELATION: 22) Karl Pearson’s Coefficient of Correlation: ௡ஊ௫௬ିሺஊ௫ሻሺஊ୷ሻ ඥ௡ஊ௫ మష ሺஊ௫ሻమ ඥ௡ஊ୷మష ሺஊ୷ሻమ r= 23) Spearman’s Rank Correlation: ଺ሺஊୈమ ሻ r = 1- [௡ሺ௡ ିଵሻ] మ where D = R1 – R2 If any rank is repeated then; ଺ሺஊୈమ ା େ.୊.ሻ r=1– [ ௡ሺ௡మ ିଵሻ ] ௠ሺ௠మ ିଵሻ ଵଶ C.F. = Where, m = No. of times the rank is repeatedREGRESSION: 24) If X is dependent on Y= ത) (‫ = ) ̅ݔ- ݔ‬bxy (‫ݕ- ݕ‬ ఙ௫ ఙ௬ bxy = r 25) If Y is dependent on X =PraveenKumar Keskar 4
  • 5. ത) (y - ‫ = ݕ‬byx (‫) ̅ݔ- ݔ‬ ఙ௬ ఙ௫ byx = r ௡ஊ୶୷ିሺஊ୶ሻሺஊ୷ሻ ௡ሺஊ୶మ ሻ ି ሺஊ୶ሻమ byx = ௡ஊ୶୷ିሺஊ୶ሻሺஊ୷ሻ ௡ሺஊ୷మ ሻି ሺஊ୷ሻమ bxy = r = ඥܾ௫௬ × ܾ௬௫TIME SERIES 26) Method of Moving Averages 27) Method of Least Squares a) Linear Equation: y = a + bx ∑௬ ∑ ௫௬ ௡ ∑ ௫మ Where, a = ; b= b) Quadratic Equation: y = a + bx + cx2 Normal Equations are: Σy = na + bΣx + cΣx2 Σxy = aΣx + bΣx2 + cΣx3 Σx2y = aΣx2 + bΣx3 + cΣx4INDEX NUMBERS 28) Unweighted Simple Average: a) Unweighted Arithmetic Mean Index Numbers = ஊ௉ ௡ P01 =PraveenKumar Keskar 5
  • 6. b) Unweighted Geometric Mean Index Number = ஊ୪୭୥୔ ௡ P01 = Antilog ( ) 29) Unweighted Simple Aggregative Index Numbers = ஊ୔భ ஊ୔బ P01 = × 100 30) Weighted Average Relative Index Numbers: a) Weighted Arithmetic Mean Index Number = ஊ୛୔ ஊ୛ P01 = b) Weighted Geometric Mean Index Number = P01 = Antilog (ஊ୛୪୭୥୔) ஊ୛ 31) Weighted Aggregative Index Number = a) Laspeyre’s Price Index Number = ஊ୔భ ୕బ ஊ୔బ ୕బ P01 = × 100 b) Paasche’s Price Index Number = ஊ୔భ ୕భ ஊ୔బ ୕భ P01 = × 100 c) Marshall Edgeworth’s Price Index Number = ஊ୔భ ୕బ ା ஊ୔భ ୕భ ஊ୔బ ୕బ ା ஊ୔బ ୕భ P01 = × 100 d) Fisher’s Price Index Number = P01 = ඥP଴ଵ ሺLaspeyre′sሻ × P଴ଵ ሺPaasche′sሻ = ඨΣP1 Q0 × ΣP1 Q1 ΣP Q ΣP Q 0 0 0 1PraveenKumar Keskar 6