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Business statestics formullas
- 1. BUSINESS STATISTICS PRESENTED BY :- ABHISHEK SHARMA MBA 1ST 17PBA003
- 2. CENTRAL TENDANCY MEAN MEDIAN ( M ) MODE ( Z)
- 3. AIRTHMATIC MEAN Formula Individual Discrete Continuous DIRECT 𝛴𝑋 𝑁 𝛴𝑓𝑋 𝑁 𝛴𝑓𝑚 𝑁 SHORT CUT A+ 𝛴𝑑 𝑁 A+ 𝛴𝑓𝑑 𝑁 A+ 𝛴𝑓𝑑 𝑁 STEP DEVIATION A+ 𝛴𝑓𝑑′ 𝑁 ∗ 𝑖 A+ 𝛴𝑓𝑑 𝑁 ∗ i (d=X-X͞ )
- 4. WEIGHTED MEAN • * WEIGHTED MEAN - ∑𝑾𝑿 ∑𝑾 • HERE, ∑𝑾 IS SUM OF WEIGHT • ∑𝑾𝑿 IS SUM OF WX = W *X
- 5. COMBINED MEAN • X123 = HERE, N1 IS NO. OF VALUE of 1ST SERIES , N2 IS NO. OF VALUE OF 2ND SERIES
- 6. MEDIAN ( M ) Individual & Discrete M= Size of N+1 th item 2 (when in points add = L+U 2 N = NO. OF SERIES OR SUM OR FREFUANCY Continuous M = Size of N th item 2 N/2 lies in particular C.I. M= l1 𝑁 2 −𝑐𝑓 𝑓 ∗ 𝑖
- 7. MODE ( Z ) • (1. INDIVIDUAL SERIES = MOST RECCURING NO. • (2. DISCRETE SERIES = BIGGEST NO. IN F , IT STRAIGHT VALUE IN ‘X’ (3. CONTINUE SERIES = f1 =biggest no. in ‘F’ , f0 = previous no. from f1 and f2 = next no. from f1. Z=𝐿1 + 𝑓1−𝑓0 2𝑓1−𝑓0−𝑓2 *I OR Z=3M-2X͞ ( AIRHTMATIC MEAN )
- 8. DISPERSION RANGE QUARTILE DEVIATION MEAN DEVIATION STANDERED DEVIATION
- 9. RANGE (R) =Largest value-Smallest value (L-S), Coefficient CR= 𝐿−𝑆 𝐿+𝑆
- 10. QUARTILE DEVIATION • Individual & Discrete Continuous Q1= Size of 𝑵+𝟏 𝟒 th item Q3= Size of 𝟑 𝑵+𝟏 𝟒 th item QD= 𝑸𝟑−𝑸𝟏 𝟐 , Q1= Size of 𝑵 𝟒 th item Q3= Size of 3 𝑵 𝟒 th item QD=𝑳 + 𝑵 𝟒 −𝒄𝒇 𝒇 ∗ 𝒊 Coefficient of Q.D. = 𝑸𝟑−𝑸𝟏 𝑸𝟑+𝑸𝟏
- 11. MEAN DEVIATION • Individual Discrete & CONTINOUS COFFICENT MD= 𝜮│𝒅/ 𝑵 MD= 𝜮𝒇│𝒅│ 𝑵 𝑴𝑫 𝑴𝑬𝑫𝑰𝑨𝐍 𝑶𝑹 𝑴𝑬𝑨𝑵
- 12. Coefficient of SD = ϭ 𝑋 And Coefficient of variation = ϭ 𝑋 ∗ 100 Combined SD = 𝑵 𝟏ϭ 𝟏+𝑵 𝟐ϭ 𝟐+𝑵 𝟏 𝒅 𝟏+𝑵₂𝒅₂ 𝑵 𝟏+𝑵₂ STANDERED DEVIATION ( ) DIRECT SHORT CUT STEP DEVIATION Individual =SD= 𝜮𝐱 𝟐 𝑵 OR 𝜮(𝑿−𝑿 𝑵 𝜮𝒅𝒙 𝟐 𝑵 − ( 𝜮𝒅𝒙 𝑵 )² = 𝜮𝒅𝒙 𝟐 𝑵 − ( 𝜮𝒅𝒙 𝑵 )² ∗ 𝑪 Discrete & continuous S.D. = 𝜮𝒇𝐱 𝟐 𝑵 𝜮𝒇𝒅𝒙² 𝑵 − ( 𝜮𝒇𝒅𝒙 𝑵 )² 𝜮𝒅𝒙′ 𝟐 𝑵 − ( 𝜮𝒅𝒙′ 𝑵 )² ∗ 𝑪 dx’= 𝒅𝒙 𝑪
- 13. SKEWNESS Karl pearson Bowleey Kelly SKP = 𝐌𝐞𝐚𝐧−𝐌𝐨𝐝𝐞 𝑺𝑫 SKB = 𝑸𝟑+𝑸𝟏−𝟐 𝑴𝑬𝑫𝑰𝑨𝑵 𝑸𝟑−𝑸𝟏 SKk = 𝐝𝟗+𝐝𝟏−𝟐𝐌𝐞𝐝𝐢𝐚𝐧 𝐝𝟗−𝐝𝟏
- 14. INDEX NUMBER • Simple Methods Formula Price Quantity Simple aggregative 𝜮𝒑𝟏 𝜮𝒑𝟎 ∗ 𝟏𝟎𝟎 𝜮𝒒𝟏 𝜮𝒒𝟎 ∗ 𝟏𝟎𝟎 SimpleAverage Price Relative 𝚺 𝒑𝟏 𝒑𝟎 ∗ 𝟏𝟎𝟎 𝑵 𝚺 𝒒𝟏 𝒒𝟎 ∗ 𝟏𝟎𝟎 𝑵
- 15. INDEX NUMBER • Weighted aggregative Methods • Time Reversal Test : Po1 X P1 = 1 • Factor Reversal Test P01 X Q01 = ΣP1q1/ΣP0q0 Formula Price Quantity Laspeyre’s 𝚺𝐩𝟏𝐪𝟎 𝚺𝐩𝟎𝐪𝟎 ∗ 𝟏𝟎𝟎 𝚺𝐪𝟏𝐩𝟎 𝚺𝐪𝟎𝐩𝟎 ∗ 𝟏𝟎𝟎 Paaschee’s 𝚺𝐩𝟏𝐪𝟏 𝚺𝐩𝟎𝐪𝟏 ∗ 𝟏𝟎𝟎 𝚺𝐪𝟏𝐩𝟏 𝚺𝐪𝟎𝐩𝟏 ∗ 𝟏𝟎𝟎 Fisher’s 𝚺𝐩𝟏𝐪𝟎 𝚺𝐩𝟎𝐪𝟎 ∗ 𝚺𝐩𝟏𝐪𝟏 𝚺𝐩𝟎𝐪𝟏 ∗ 𝟏𝟎𝟎 𝚺𝐪𝟏𝐩𝟎 𝚺𝐪𝟎𝐩𝟎 ∗ 𝚺𝐪𝟏𝐩𝟏 𝚺𝐪𝟎𝐩𝟏 ∗ 𝟏𝟎𝟎 Dorbish & Bowley’s 𝚺𝐩𝟏𝐪𝟎 𝚺𝐩𝟎𝐪𝟎 + 𝚺𝐩𝟏𝐪𝟏 𝚺𝐩𝟎𝐪𝟏 ∗ 𝟏𝟎𝟎 𝟐 𝚺𝐪𝟏𝐩𝟎 𝚺𝐪𝟎𝐩𝟎 + 𝚺𝐪𝟏𝐩𝟏 𝚺𝐪𝟎𝐩𝟏 ∗ 𝟏𝟎𝟎 𝟐 Marshall edgeworth’s 𝚺𝐩𝟏𝐪𝟎 + 𝚺𝐩𝟏𝐪𝟏 𝐩𝟎𝐪𝟎 + 𝐩𝟎𝐪𝟏 ∗ 𝟏𝟎𝟎 𝚺𝐪𝟏𝐪𝟎 + 𝚺𝐪𝟏𝐩𝟏 𝚺𝐪𝟎𝐩𝟎 + 𝚺𝐪𝟎𝐩𝟏 ∗ 𝟏𝟎𝟎
- 16. CO-RELATION • KARL PEARSON METHODS :- 2. SPEARMEN METHODS :- (3. CON CURRENT DEVIATION :- rc = + + 𝟐𝒄 −𝒏 𝒏 (Actual mean) Assumed mean) Actual data r= 𝚺𝐱𝐲 𝚺𝐱 𝟐.𝚺𝐲 𝟐 r= 𝐍.𝚺𝐝𝐱𝐝𝐲−𝚺𝐝𝐱.𝚺𝐝𝐲 𝐍𝚺𝐝𝐱²− 𝚺𝐝𝐱 𝟐. 𝐍.𝚺𝐝𝐲²−(𝚺𝐝𝐲)² r= 𝐍.𝚺𝐗𝐘−𝚺𝐗.𝚺𝐘 𝐍.𝚺𝐗²− 𝚺𝐗 𝟐. 𝚺𝐘²−(𝚺𝐘)² WHEN RANK ARE GIEVAN WHEN RANK ARE NOT GIVEN WHENRANK ARE EQUAL R= 1 - 𝟔𝜮𝐃𝟐 𝐍𝟑 −𝐍 R= 1 - 𝟔𝜮𝐃𝟐 𝐍𝟑 −𝐍 R= 1 - 𝟔(𝜮𝐃𝟐+ 𝟏 𝟏𝟐 𝒎𝟑−𝒎+ 𝟏 𝟏𝟐 𝒎𝟑−𝒎+ 𝟏 𝟏𝟐 𝒎𝟑−𝟑) 𝐍𝟑 −𝐍
- 17. REGRESSION • (1. ALGEBRIC METHOD :- ( 2. NON – ALGEBRIC METHOD – ( I. ACTUAL METHOD :- co- efficient = bxy x byx (3. ASSUMED METHOD :- co- efficient = bxy x byx 1.Y on X Y=a+ bX ΣY=Na+bΣx ΣXY=aΣX+bΣX² 2.X on Y X=a+ bY ΣX=Na+bΣY ΣXY=aΣY+bΣY² 1.Y on X Y-Y̅=byx(X-X̅) byx= 𝚺𝐱𝐲 𝚺𝐱² 2.X on Y X-X̅=bxy(Y-Y̅ ) bxy= 𝚺𝐱𝐲 𝚺𝐲² 1.Y on X Y-Y̅=byx(X-X̅) , byx= 𝐍.𝚺𝐝𝐱𝐝𝐲−𝚺𝐝𝐱.𝚺𝐝𝐲 𝐍.𝚺𝐝𝐱²−(𝚺𝐝𝐱)² 2. X on Y X-X̅=bxy(Y-Y̅) , bxy= 𝐍.𝚺𝐝𝐱𝐝𝐲−𝚺𝐝𝐲.𝚺𝐝𝐱 𝐍.𝚺𝐝𝐲²−(𝚺𝐝𝐲)²
- 18. TIME SERIES • Least square method - Yс = a +bX • if x =0 then a = ∑𝒚 𝑵 b = ∑𝒙𝒚 ∑𝐱²