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STATISTICS FORMULA
Abhishek Puri
17PBA002
Baddi University of emerging
sciences and technology
Measure of central tedendancy
Median
Mean
Mode
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 )
Mean
 COMBINED MEAN
𝑵 𝟏X͞ 𝟏+𝑵 𝟐X͞ 𝟐
𝑵𝟏+𝑵𝟐
,
 WEIGHTED MEAN
∑𝑾𝑿
∑𝑾
Median
Individual
N+1/2 (when in
points add L+U/2)
Continuous
M= L+
𝑁
2
−𝑐𝑓
𝑓
∗ 𝑖
MODE
Z=𝐿1 +
𝑓1−𝑓0
2𝑓1−𝑓0−𝑓2
*I
 OR
Z=3M-2X͞
Measure of dispersion
Quartile
deviation
Mean deviation
Standard deviation
QUARTILE DEVIATION
Individual Continuous
Q1=
𝑁+1
4
Q3=
3 𝑁+1
4
QD=
𝑄3−𝑄1
2
,
QD=𝐿 +
𝑁
4
−𝑐𝑓
𝑓
∗ 𝑖
Coeff. =
𝑄3−𝑄1
𝑄3+𝑄1
STANDARD DEVIATION
DIRECT SHORT CUT STEP
DEVIATION
Individual
=SD=
𝚺𝐗 𝟐
𝐍
OR
𝚺(𝐗−𝐗͞
𝐍
𝚺𝐝𝐱 𝟐
𝐍
− (
𝚺𝐝𝐱
𝐍
)²
=
𝚺𝐝𝐱 𝟐
𝐍
− (
𝚺𝐝𝐱
𝐍
)² ∗
𝐂
Discrete &
continuous
𝚺𝐟𝐝𝐱²
𝐍
− (
𝚺𝐟𝐝𝐱
𝐍
)²
𝚺𝐝𝐱′ 𝟐
𝐍
− (
𝚺𝐝𝐱′
𝐍
)² ∗
𝐂
dx’=
𝐝𝐱
𝐂
STANDARD DEVIATION
 Coefficient of SD=
ϭ
𝑋
AND
 Coefficient of variation or C.V =
ϭ
𝑋
∗ 100
 Combined SD=
𝑵 𝟏ϭ 𝟏+𝑵 𝟐ϭ 𝟐+𝑵 𝟏 𝒅 𝟏+𝑵₂𝒅₂
𝑵 𝟏+𝑵₂
MEAN DEVIATION
Individual Discrete &
CONTINOUS
COFFICENT
MD=
𝛴│𝑑
𝑁
MD=
𝛴𝑓│𝑑│
𝑁
𝑀𝐷
𝑀𝐸𝐷𝐼𝐴N 𝑂𝑅 𝑀𝐸𝐴𝑁
SKEWNESS
Karl pearson Bowleey Kelly
𝐦𝐞𝐚𝐧 − 𝐦𝐨𝐝𝐞
𝑺𝑫
𝑸𝟑 + 𝑸𝟏 − 𝟐 𝑴𝑬𝑫𝑰𝑨𝑵
𝑸𝟑 − 𝑸𝟏
𝒅𝟗 + 𝒅𝟏 − 𝟐𝒎𝒆𝒅
𝒅𝟗 − 𝒅𝟏
Weighted aggregative Methods
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
INDEX NUMBERS
 Time Reversal Test : P01 X P10 = 1
 Factor Reversal Test P01 XQ01 = ΣP1q1/ΣP0q0
CORELATION
(Actual
mean)
Assumed mean) Actual data
r=
𝛴𝑥𝑦
𝛴𝑥2.𝛴𝑦2
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 ranks are not given
Rank are equal
R = 1 −
6[∑𝐷2+
1
12
(𝑚3−𝑚)+
1
12
(𝑚3−𝑚)
𝑛3−𝑛
REGRESSION
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
Equation using coefficient (Using Mean)
1.Y on X
Y-
X̅)
byx=
𝑁.𝛴𝑥𝑦−𝛴𝑥.𝛴𝑦
𝑁.𝛴𝑥²−(𝛴𝑥)²
2.X on Y
X-X̅=bxy(Y-
bxy=
𝑁.𝛴𝑥𝑦−𝛴𝑦.𝛴𝑥
𝑁.𝛴𝑦²−(𝛴𝑦)²
Assumed mean
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
Time series
Least square method
 YC = a +bx
 if x =0 then
 a =
∑y
𝑁
 b =
∑xy
∑x²

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Abhishek[1]

  • 1. STATISTICS FORMULA Abhishek Puri 17PBA002 Baddi University of emerging sciences and technology
  • 2. Measure of central tedendancy Median Mean Mode
  • 3. 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. Mean  COMBINED MEAN 𝑵 𝟏X͞ 𝟏+𝑵 𝟐X͞ 𝟐 𝑵𝟏+𝑵𝟐 ,  WEIGHTED MEAN ∑𝑾𝑿 ∑𝑾
  • 5. Median Individual N+1/2 (when in points add L+U/2) Continuous M= L+ 𝑁 2 −𝑐𝑓 𝑓 ∗ 𝑖
  • 7. Measure of dispersion Quartile deviation Mean deviation Standard deviation
  • 8. QUARTILE DEVIATION Individual Continuous Q1= 𝑁+1 4 Q3= 3 𝑁+1 4 QD= 𝑄3−𝑄1 2 , QD=𝐿 + 𝑁 4 −𝑐𝑓 𝑓 ∗ 𝑖 Coeff. = 𝑄3−𝑄1 𝑄3+𝑄1
  • 9. STANDARD DEVIATION DIRECT SHORT CUT STEP DEVIATION Individual =SD= 𝚺𝐗 𝟐 𝐍 OR 𝚺(𝐗−𝐗͞ 𝐍 𝚺𝐝𝐱 𝟐 𝐍 − ( 𝚺𝐝𝐱 𝐍 )² = 𝚺𝐝𝐱 𝟐 𝐍 − ( 𝚺𝐝𝐱 𝐍 )² ∗ 𝐂 Discrete & continuous 𝚺𝐟𝐝𝐱² 𝐍 − ( 𝚺𝐟𝐝𝐱 𝐍 )² 𝚺𝐝𝐱′ 𝟐 𝐍 − ( 𝚺𝐝𝐱′ 𝐍 )² ∗ 𝐂 dx’= 𝐝𝐱 𝐂
  • 10. STANDARD DEVIATION  Coefficient of SD= ϭ 𝑋 AND  Coefficient of variation or C.V = ϭ 𝑋 ∗ 100  Combined SD= 𝑵 𝟏ϭ 𝟏+𝑵 𝟐ϭ 𝟐+𝑵 𝟏 𝒅 𝟏+𝑵₂𝒅₂ 𝑵 𝟏+𝑵₂
  • 11. MEAN DEVIATION Individual Discrete & CONTINOUS COFFICENT MD= 𝛴│𝑑 𝑁 MD= 𝛴𝑓│𝑑│ 𝑁 𝑀𝐷 𝑀𝐸𝐷𝐼𝐴N 𝑂𝑅 𝑀𝐸𝐴𝑁
  • 12. SKEWNESS Karl pearson Bowleey Kelly 𝐦𝐞𝐚𝐧 − 𝐦𝐨𝐝𝐞 𝑺𝑫 𝑸𝟑 + 𝑸𝟏 − 𝟐 𝑴𝑬𝑫𝑰𝑨𝑵 𝑸𝟑 − 𝑸𝟏 𝒅𝟗 + 𝒅𝟏 − 𝟐𝒎𝒆𝒅 𝒅𝟗 − 𝒅𝟏
  • 13. Weighted aggregative Methods 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
  • 14. INDEX NUMBERS  Time Reversal Test : P01 X P10 = 1  Factor Reversal Test P01 XQ01 = ΣP1q1/ΣP0q0
  • 15. CORELATION (Actual mean) Assumed mean) Actual data r= 𝛴𝑥𝑦 𝛴𝑥2.𝛴𝑦2 r= 𝑁.𝛴𝑑𝑥𝑑𝑦−𝛴𝑑𝑥.𝛴𝑑𝑦 𝑁𝛴𝑑𝑥²− 𝛴𝑑𝑥 2. 𝑁.𝛴𝑑𝑦²−(𝛴𝑑𝑦)² r= 𝑁.𝛴𝑋𝑌−𝛴𝑋.𝛴𝑌 𝑁.𝛴𝑋²− 𝛴𝑋 2. 𝛴𝑌²−(𝛴𝑌)²
  • 16. When ranks are not given
  • 17. Rank are equal R = 1 − 6[∑𝐷2+ 1 12 (𝑚3−𝑚)+ 1 12 (𝑚3−𝑚) 𝑛3−𝑛
  • 18. REGRESSION 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
  • 19. Equation using coefficient (Using Mean) 1.Y on X Y- X̅) byx= 𝑁.𝛴𝑥𝑦−𝛴𝑥.𝛴𝑦 𝑁.𝛴𝑥²−(𝛴𝑥)² 2.X on Y X-X̅=bxy(Y- bxy= 𝑁.𝛴𝑥𝑦−𝛴𝑦.𝛴𝑥 𝑁.𝛴𝑦²−(𝛴𝑦)²
  • 20. Assumed mean 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
  • 21. Time series Least square method  YC = a +bx  if x =0 then  a = ∑y 𝑁  b = ∑xy ∑x²