math forulae for competitive exams, CA-CPT, bank PO, etc ratio, proportion, permutation, combination, derivatives, ap, gp...

Author:
mukeshnt

270 downloads 722 Views 902KB Size

Numerator = Antecedent , Denominator = Consequent Order is important Types of Ratios: a) Original a : b Inverse = b : a b) Original a : b Duplicate = a2 : b 2 c) Original a : b Triplicate = a3 : b 3 d) Original a : b

Sub Duplicate

=

e) Original a : b

Sub Duplicate

=

f)

a c e , , b d f

a : b 3

a :3b

a b a = b

Compound ratio =

(any 3 or more ratios) g) Continued ratio

c d c d

e f e f

= a : b : c

PROPORTION: 1.

If 4 Quantities a, b, c, d are in proportion: a b

c d

or

Result: 1) ad = bc

a : b = c : d. (Product of extremes = Product of means)

2) By k – method :

a b

c = k d

a = bk c = dk. 2.

If 3 Quantities a, b, c are in proportion

a b

b c

or

a : b : c :

Result: 1) b2 = ac i.e b is geometric mean of a & c (or mean proportional) a = 1st proportional b = Mean proportional (or Geometric mean) c = 3rd proportional. 2) By k – method :

a b

b = k. c

b = ck a = ck2. 3.

If 4 Quantities a, b, c, d are in continued proportion: a b c , , b c d

or

a : b : c : d

Result: 1) b2 = ac ; c2 = bd ; ad = bc 2) By k – method :

a b

b c

c = dk b = dk2 c = dk3

c = k. d

[Increase the power of k only]

PROPERTIES OF PROPORTION: 1. 2. 3. 4. 5.

a c b d a c Alternendo : b d a c Componendo : b d a c Dividendo : b d

Invertendo

:

a

d c b d c

a

b

b

d d

c

b a b

Componendo – Dividendo : Note :

b a a c b

a a

c d

d d b b

c c

d d

Addition in N r Subtraction in D r

To simplify a ratio that is in the form of componendo – dividend, apply componendo – dividendo on it. (1st term in Nr & 2nd term in Dr)

6.

Addendo

:

a b

c d

e f

Each ratio =

a b

c d

e f

7.

Subtrahendo

:

a b

c d

e f

Each ratio =

a b

c d

e f

INDICES: ap = m i.e. a x a x a ………… p times = m a = base p = power or index or exponent. m = value (or answer) of ap LAWS OF INDICES: 1.

am x an = am + n

2.

am

3.

(am )n = am x n

4.

(a x b)m = am x bm

a b

5.

an = am - n

m

am bm

same base in multiplication Result : Power add up. same base in division Result : Power subtract. (Large single base Result : Power multiply. different base in multiplication Result : Power get distributed.. a b Use : a b

Different powers Different powers - Small) 2 Different powers

different bases in division

Single power (Split) single power.

Result : Power get distributed.. Use 6.

a

= 1

7.

a- m =

:

a b

a b

(Split)

Any base power zero Result : Answer = 1. 1

am

Single base raised to negative power. Result : Only the base gets reciprocated (power does not get reciprocated) Power changes in sign only.

8.

n

am

= am/n = (am)1/n = (a1/n)m

m = actual power n = root part. (radical) a = base (radicand)

NOTE: 1.

In case of cyclic powers : Usual Answer = L.

2.

If x = p1/3 + If x = p1/3 –

1 p

1/ 3

p

1/ 3

1

x3 – 3x = p +

1 p

x3 + 3x = p –

1 p

Question

Answer

LOGARITHMS: If ap = m then loga m = p.

& vice versa

In logb a = c

a = Subject (to which log is applied) b = base c = logarithmic value (or answer)

Usual base = 10

(a.k.a. common base) Take base = 10, if no base is given.

Natural base = e

(e = 2.71828) (Used in limits, derivatives & integration)

REMEMBER: 1. 2. 3. 4. 5.

a = 1 loga 1 = 0 1 a = a loga a 1 = 1 Base of log cannot be ‘0’ Base of Log cannot be negative. log10 10 = 1, log10 100 = 2, log10 1000 = alogam = m

[log 1 to any base = 0] [log a to same base a = 1] [loga 0 = – ]

3 and so on.

LAWS OF LOGARITHMS: 1.

Product Law: NOTE :

2.

logm (a x b) = logma + logmb

1) log (a + b) log a + logb 2) (log a) log b) log a + logb

Quotientt Law: NOTE :

1) log (a - b) 2)

3.

Power Law: NOTE :

4.

Change of base:

a b

Logm

log a log b

= logm a - logm b

loga - logb

log a - logb

log(

logm an = n. logma

1) (log a)n

n. loga log p a

i) logm a = 1 ii) logm a = log p m log a m

(logm a = loga m = 1)

a b

EQUATIONS: I]

II]

SIMPLE LINEAR EQUATION : General form :

Ax + B = 0

x = variable A, B = constants (coefficients) Max power of x = 1. SIMULTANEOUS LINEAR EQUATIONS IN TWO VARIABLES : General form : A1, x + B1, y + C1 = 0 A2, x + B2, y + C2 = 0 x , y = Variables. Methods of solving: 1) Substitution : Express x in terms of y & substitute in other equation. 2) Elimination : Eliminate any one variable & find value of other variable. Replace this in any equation to get the value of 1st variable. (eliminated) (Remember : DASS) 3) Cross Multiplication : B1 C1 A1 B1 B2 C2 A2 B2 x=

B1 C 2 A1 B2

B2 C 1 A2 B1

y =

C1 A2 A1 B2

C 2 A1 A2 B1

4) In case of MCQ’s : Substitute the options to satisfy the equations.

III] QUADRATIC EQUATIONS : (Q.E.) *

General form : Ax2 + Bx + C = 0

x = variable A, B, C = Constants A 0. Max power = 2. No. of answers = 2. (solutions/roots)

If A = 1 Reduced form. If B = 0 or C = 0 Incomplete Q.E. When B = 0 Use a2 – b2 = (a + b) (a – b) to factorise Roots : Same value, different signs. When C = 0 Take x common. One Root = 0. If A = C *

Roots : Reciprocals of each other one root =

Methods of Solving : 1) Factorisation : Involves

p q other root = q p

Splitting of middle terms ax2 + bx + c = (x – ) (x – β) Taking x common Difference of squares i.e. a2 – b2 = (a + b) (a – b) 2) Formula Method : For a Q.E. Ax2 + Bx + C = 0 x= 2

B

B2

4 AC

2A

B – 4AC = Discriminants (∆)

*

Nature of Roots : B2 – 4AC (∆)

B2 – 4AC = 1. Roots : 0REAL & EQUAL 2. Each Root = –

B2 – 4AC < 0 (Negative) 1. Root: NOT REAL (Imaginary or complex)

B 2A

2. One root = a + bi

3. QE is a perfect square.

Other root = a – bi (i =

1)

B2 – 4AC > 0 (Positive) Is also a perfect square

Is not a perfect square

1. Roots : REAL, UNEQUAL & RATIONAL

B2 – 4AC = 0 B2 – 4AC < 0

REAL & EQUAL NOT REAL

1. Roots : REAL, UNEQUAL IRRATIONAL 2. One root = a + b Other root = a – b

(IMAGINARI – CONJUGATE)

RATIONAL (If perfect square) 2

B – 4AC > 0

REAL, UNEQUAL IRRATIONAL (conjugates) (If not a perfect square)

2

B – 4AC

*

0

REAL

Relation between Roots & Coefficients: Q.E. Ax2 + Bx + C 2 roots : Sum of Roots : Product of Roots :

&

= – =

&

B A

C A

*

Formation of Quadratic Equation : 2 roots : & 2 Q.E. x – ( + ) x + = 0 2 x – (Sum of roots) x + Product of roots = 0

*

Symmetric functions of Roots: 1) 2 + 2 = ( + )2 – 2 2) ( – )2 = ( + )2 – 4 3) 3 + 3 = ( + )3 – 3 4) 3 – 3 = ( – )3 + 3

( (

+ –

) )

IV] CUBIC EQUATIONS : General form : Ax3 + Bx2 + Cx + D = 0

x = variable Max power = 3 No. of solution = 3

Method of Solving : 1) Synthetic Division 2) In case of MCQ’S : Use options. Note: Test of Divisibility 1) If Sum of all coefficients = 0 (x – 1) is a factor (i.e. x = 1 is a root) 2) If Sum of coefficients = sum of coefficients (x + 1) is a factor) of odd powers of x of even powers of x (i.e. x = –1 is a root.)

V]

STRAIGHT LINES :

*

SLOPE of line (m) : Inclination of line w.r.t. + ve X axis. m

=

y2 x2

y1 x1

=

tan

=

–

=

m

A B

Diff . of Ordinates Diff . of abscissa

= Angle between line & X-axis If equation of line Ax + By + C = 0 is given. If equation of line is in the form of Y = m X + c. or Y = a + b X (Slope = b)

For 2 PARALLEL LINES: (having slopes m1 & m2) * Slopes are EQUAL *

m1 = m2

m1 =

y – y1 = m(x – x1)

TWO – POINT FORM : y x

3

1 m2

Lines are

Equations differ in constants, coefficient & sign. One line : Ax + By + C = 0 Perpendicular line : Bx – Ay + K = 0

FORMATION OF EQUATION OF LINE : 1. SLOPE – POINT FORM :

2

Lines are

Equations differ in constants only. One line : Ax + By + C = 0 Parallel line : Ax + By + K = 0

For 2 PERPENDICULAR LINES: (having slopes m1 & m2) * Slopes are NEGATIVE RECIPROCALS

*

2 points on line A (x1, y1) ; B (x2, y2)

y1 x1

y2 x2

y1 x1

DOUBLE INTERCEPT FORM : x a

y b

1

Requirement : Slope = m Point = (x1, y1) Requirement : 1st point = (x1, y1) 2nd Point = (x2, y2) (RHS = Slope)

Requirement : X intercept = a Y intercept = b

r

4.

SLOPE – INTERCEPT FORM :

Requirement : Slope = m Y – intercept = c Other form : y = a + bx. (a. k. a. DISPLAY EQUATION)

y = mx + c

5.

GENERAL FORM : Ax + By + C = 0 Slope = –

A B

X – intercept = – Y – intercept = –

1. 2.

C A C A

Other Important Notes : 3 points A, B, C are COLLINER 3 lines are concurrent Pt. of concurrency : Condition for concurrency :

3.

r

Ax 1

By 1 A

x2

y2

1

x3

y3

1

0

2

B

C 2

2

B2

Distance between 2 parallel line Ax + By + C = 0 & Ax + By + K = 0 C

=

A

K

2

B2

Distance formula : A(x1 y1) AB =

5.

1

C A

4.

y1

Distance of a line Ax + By + C = 0 from line Origin (O, O) =

r

x1

Distance of a point (x1, y1) from line Ax + By + C = 0 =

r

Slope AB = Slope BC = Slope AC 3 lines intersect at 1 point only pt. of intersection of 3 lines.

(x 2

x 1)

2

& B(x2 y2) y 1)2

(y 2

Section formula : Internal division. A P (x1 y1)

(x, y) m

Px =

mx 2 m

nx 1 n

B (x2 y2) n

Py =

my 2 m

ny 1 n

6.

Midpoint formula : A (x1 y1) Px =

x1

P

B

(x, y)

x2

(x2 y2) Py =

2

y1

y2 2

INEQUALITIES : Max availability At most Min requirement At least

SIMPLE INTEREST: 1.

SI =

Pnr 100

2.

A P SI A r

P + SI = P + Pin = P(I + in) Principal (in Rs.) Simple Interest (in Rs.) Amount (in Rs.) rate of interest (in % p.a.)

= = = = =

i =

Pin

rate % 100

rate of interest (in decimal)

n = Period or Time (in years) If time in months, divide by 12 In days , divide by 365.

1. 2. 3. 4.

5.

1.

2.

COMPOUND INTEREST: A = P(1 + i)n CI = A – P = P(1 + i)n – P = P[(1 + i)n – 1] CI for nth year = Amount in n years – Amount in (n – 1) years For compounding more than once in a year Mode of compounding Divide Rate Multiply Time Half yearly 2 2 Quarterly 4 4 Monthly 12 12 Effective Rate of Interest : (To be calculated if compounding done more than once in year) E = [(1 + i)n – 1] 100% ANNUITY : Immediate Annuity or Annuity Regular or Annuity Certain. (Ordinary Annuity) Payments are made/received at the END of reach period. Annuity Due : Payments are made / received at the START of each period. Formulae

Ordinary Annuity 1) FV =

C [ (1 i ) 2 i

Annuity Due 1]

1) FV =

C [ (1 i ) n i

1 ] (1

i)

2) PV = FV PV C n r

= = = = =

C 1 i

1 (1 i )

r 100

PERMUTATION AND COMBINATION Factorial Notation : n ! = Product of 1st n natural nos. = 1 x 2 x 3 x 4 x .........x n

NOTE :

Remember : 0! = 1 1! = 1 2! = 2

n ! = n(n – 1) (n – 2) . . . . . . . x 3 x 2 x 1. n ! = n(n – 1)! = n(n – 1) (n – 2)! = n(n – 1) (n – 2) (n – 3)! & so on. 3! = 6 4! = 24 5! = 120

6! = 720 7! = 5040 8! = 40320

2.

Fundamental Principal : 1st job = p 2nd job = q Addition Rule : (OR) (p + q) ways Multiplication Rule : (AND) (p x q) ways.

3. *

PERMUTATION (ARRANGEMENT) – Order important n = No. of places available. r = No. of objects to be arranged n!

n

Pr = No. of arrangements

* *

C 1 i

Future Value Present Value (LOAN) Annuity or Periodic Payment or Instalment. Period or No. of instalments. rate of interest (in %)

i = rate of interest (in decimal)

1.

2) PV =

n

(n

r )!

(n > r)

If No. of places = no. of objects (arrangements amongst themselves) n Then No. of arrangements = Pr = n! No. of places available = n No. of objects to be arranged = r Condition : 1 Particular place is never occupied. No. of arrangements = n-1Pr Condition : 1 particular place is always occupied. No. of arrangements = r x n - 1Pr - 1 [ nPr = n-1Pr + r. n - 1Pr – 1] Condition : Balls in boxes. [Each place can take in all r objects] No. of arrangements = nr Condition :

Permutation with Repetitions. Total no. of objects ( = places) = n No. of alike objects = p of 1st kind = q of 2nd kind = r of 3rd kind. & rest are different.

1 (1 i ) n

(1 + i)

No. of arrangements =

n! p !q ! r !

No. of arrangements of (3p) things in 3 groups =

No. of arrangements of (2p) things in 2 groups = *

(3 p ) ! (P ! ) 3 (2 p) ! (P ! ) 2

Circular permutations : No. of objects ( = places) = n. Condition : To be arranged in a circle. [eg. Circular table] No. of arrangements = (n – 1)! Condition : Does not have same neighbour (necklace) No. of arrangements =

*

1 (n – 1)! 2

COMBINATIONS (SELECTIONS) – Order not important. No. of objects available = n No. of objects to be selected = r No. of selections

=

n

Cr

=

n! r ! (n r ) !

(n > r)

Remember : 1)

n

2) 3) 4) 5) 6)

n

n

Cr =

Pr r!

Cr = nCn - r *** n C0 = nCn = 1 n C1 = n If nCx = nCy then x = y n Cr + nCr - 1 = n + 1Cr

or

x + y = n. (Pascals Law)

*

Total no. of ways of dealing with n things = 2n No. of ways in which all ‘n’ things are rejected = 1 No. of ways in which one or more things are selected = 2 n – 1 Note : nC0 + nC1 + nC2 + . . . . . . . + nCn = 2n n C1 + nC2 + . . . . . . . + nCn = 2n – 1

*

No. of points in a plane = n

Condition : No. 3 points are collinear. No. of Straight lines = nC2 No. of triangles = nC3 Condition : P points are collinear No. of Straight lines = nC2 – pC2 + 1 No. of triangles = nC3 – pC3 * *

(take it or leave it)

(= No. of handshakes)

(p

3)

Maximum no. of diagonals that can be drawn in an n – sided polygon = nC2 – n. [No. of lines – No. of sides] n = No. of parallel line in 1st set . (Sleeping lines) m = No. of parallel lines in 2nd set. (Standing lines) No. of parallelograms = nC2 x

m

C2

SEQUENCE AND SERIES : AP/GP. I]

ARITHMETIC PROGRESSION (AP) : Sequence in which the terms (numbers) increase/decrease by a constant difference. AP : a, a + d, a + 2d, a + 3d, . . . . . . . tn = a + ( n – 1) d. a = 1st term ER n [ 2a (n 1) d ] 2 n [a t n ] 2 n [1st term last term ] 2

Sn = = =

d = common difference ER n = no. of terms (position) EN tn = nth term (any term) ER Sn = Sum of n terms ER

For convenience : No. of terms 3 4 5 *

Terms a – d, a, a + d a – 3d, a – d, a + d, a + 3d a – 2d, a – d, a, a + d, a + 2d

If a, b, c are 3 terms in AP

b =

a

c 2

(A. M. between 2 nos.is half their sum)

Remember : 1) Sum of 1st n natural nos : 2) Sum of squares of 1st n natural nos:

n (n 1) 2

3) Sum of cubes of 1st n natural nos:

13 + 23 + 33 + . . . . . . . + n3 =

4) Sum of 1st n odd natural nos: 5) Sum of 1st n even natural nos:

1 + 3 + 5 + . . . . + (2n – 1) = n2 2 + 4 + 6 + . . . . + (2n) = n(n + 1)

TRIVIA : 1) n tn = m tm 2) tp = q & tq = p 3) Sm = Sn 4) II]

n (n 1) 2 n (n 1) (2n 1) 12 + 22 + 32 + . . . . . . . + n2 = 6

1+2+3+....... +n =

tm + n = 0 tr = p + q – r Sm + n = 0

m2

Sm Sn

n

d = 2a &

2

tm tn

2m 2n

1 1

GEOMETRIC PROGRESSION (GP) Sequence in which the terms increase/decrease by a constant ratio. GP : a, ar, ar2, ar3, . . . . . . . . . tn = arn – 1 a = 1st term ER rn r

Sn = a = a S

=

1 1

1 rn 1 r

a 1

r

if r > 1 if r < 1 (only if r < 1)

r = common ratio ER n = no. of terms EN. (Position) tn = nth term ER (Any term) Sn = Sum of n terms ER. S = Sum of infinite terms.

2

*

For Convenience : No. of Terms

Terms a , a, ar r a a , , ar , ar 3 3 r r a a , , a, ar , ar 2 2 r r

3 4 5 *

Common Ratio

If a, b, c are 3 terms in GP

r r2 r

b2 = ac

b =

ac . (b = G. M. of a & c)

TRIVIA 1) a + aa + aaa + aaaa + ……….. =

a 10 (10 n 9 9

2) 0.a + 0.aa + 0.aaa + 0.aaaa + . . . . . . . = 3) 0.a + 0.0a + 0.00a + . . . . . . . = Best term fro AP : 1, 2, 3. GP : 1, 2, 4

a 9

or

1)

a n 9

n 1 (1 9

0.1n )

( 0.1) n

1

2, 4, 8.

SETS RELATIONS & FUNCTIONS : I]

SETS : Notations : 1. - Belongs to 2. - Does not belong to 3. - Subset 4. - Proper Subset. 5. or { } - Empty set or Null Set. 6. - Union 7. - Intersection. Basic Operations of sets: 1. Union : A B = {x / A or x B or x Both A & E (Common as well as uncommon) 2. Intersection : A B = {x / x A and x B} (common only) c 3. Complement : A or A = {x / x U, x A} (not contained in A) Properties : 1) Union a) A B = B A b) A A = U c) A = A d) A U = U e) If A B then A

B = B

2) a) b) c) d) e)

Formulae : For 2 sets A & B: 1) Addition Theorem : n(A B) = n(A) + n(B) – n(A = n(A) + n(B) 2) n(A B) + n(A B) = n(S). 3) n(Only A) = n(A – B) = n(A = n(A) – n(A B).

Intersection A B = B A. A A = A = A U = A If A B then A

B = A

B) (if A B)

B =

i.e. A & B are disjoint)

4)

n(Only B) = = n(A B) = n(A B) =

5)

n(B – A) = n(A n(B) – n( A B) n(A B) n(A B)

B) De Morgan’s Law.

For 3 sets A, B, C. 1) Additional Theorem : n(A B C) = n(A) + n(B) + n(C) – n(A RELATION : 1) Reflexive 2) Symmetric 3) Transitive 4) Equivalence

B) – n(B

C) – n(C

A) + n(A

: x Rx : If x Ry then y Rx. : If x Ry and y Rz, then xRz. : All if above

LIMITS AND CONTINUITY 1)

0 f (x ) or is of the form , then 0 g( x ) lim f ' ( x ) lim f ' ' ( x ) .......... .. x a g' ( x ) x a g' ' ( x )

L’ HOSPITAL RULE : If lim x

f (x) a g( x )

IMPORTANT FORMULAE : 1) 2) 3) 4) 5) 6) 7) 8) *

lim x lim x

i) ii)

k.

k. f (x )

a

lim

xn a x

lim

ax

x x

0

x

0

lim x

1

1

0

1

a

f (x )

1

log e a

The coefficient of x in Nr must be repeated in Dr

1

x

(1 x )

x

na n

log (1 x ) 0 x

lim x

an a

ex

lim

k.

x

lim

x

1 e

Limit at Infinity:: lim x

Also *

k

a

x

1

1 x

x

lim x

lim

1 x lim

lim

1 x

1 f (x )

2

x

1 x

3

.......... .

lim x

0

x

e

(Also see ⑦)

CONTINUITY : A function f(x) is said to be continuous at x = a if f(a) exists. lim x a

f ( x ) exists.

1 xn

0.

B

C)

iii) iv)

lim x

f ( x ) exists.

a

f(a) =

lim x a

lim x a

f (x )

f ( x ).

EQUATIONS: Key phrase : Rate of change / Gradient / Slope. If y = f(x) is a function involving the variable x, then dy dx

f (x) =

lim h

f (x o

h) h

dy dx

= f (x) is its derivative.

f (x )

Standard Formulae : y = f(x) A L G E B R A I C

c x cx xn

dy dx

f (x)

0 1 c nxn - 1 1

P O W E R

x

1 x

2 x 1

x2 1

1 x

2x

x

x

a ex logx xx

Exponential Logarithmic

x

a loga ex 1/x x x (1 + logx)

Let u & v be two functions involving the variable x. 1. 2.

3. 4. 5. 6.

d ( ) dx d ( . ) dx

d dx dv dx d v dx

d dx d dx dv dx

d dx v2 d du (c ) c dx dx d d f [ g ( x ) ] f ' [ g ( x ) ]. g( x ) dx dx d 2y d dy 2 dx dx dx

(Additional / Subtraction Rule (Multiplication Rule)

(Division Rule) [constant x function] [Chain Rule] [ 2nd Order Derivative or f" (x)]

APPLICATIONS & TYPES: 1.

Slope (or Gradient) of Tangent to a curve: y or f(x) = function representing a curve. dy or f (x) = function representing the slope of tangent to the curve. dx dy or f (a) = Slope of tangent to the curve at any point x = a. on the curve. dx x a

2.

Maxima & Minima : A function f(x) is said to have a maxima at x = a if i) f (x) = 0 at x = a & ii) f"(x) < 0 at x = a. A function f(x) is said to have a minima at x = a if i) f (x) = 0 at x = a & ii) f"(x) > 0 at x = a.

3.

Logarithmic Differentiation : Recognise : xx or [f(x)]g(x) or (function)function. Method : Take log on both sides & then differentiate. Note

: Also applicable if

f ( x ) . g( x ) f ( x ). g( x ) or r (x ) r ( x ). s( x )

i.e. Many functions in multiplication & division. Why Log? : - Log simplified complex multiplication, division, powers. 4.

Implicit Functions : Recognise : x & y scattered throughout the equation.

i.e. f(x,y) = 0

Method : i] See if a single ‘y’ can be isolated from the function. If so, then isolate and then differentiate. ii] If y cannot be isolated, then differentiate the function. w.r.t. x. This gives a new equation involving Isolate 5.

dy dx

dy on LHS. dx

Parametric Functions : Recognise : 2 different functions involving a 3rd variable (t or m) i.e. x = f(t) y = g(t) or x = f(m) y = g(m) or x = f(θ) y = g(θ). Method : Differentiate the functions separately w.r.t. the variable present. i.e. Note

dy dx

dy / dt . dx / dt

: Also applicable if differentiate f(x) w.r.t g(x). Take u = f(x) & v = g(x) & diff w.r.t. x.

CORRELATION: I]

Karl Pearson’s coefficient of correlation OR Product moment correlation coefficient. r

=

Cov ( x , y ) x . y

(x

=

x ) (y

(x

x)

n (x

= (x

2

n

x

x) 2

u2

n

y)

(y

y)

xy (

x)

2

n y)

2

n

=

(y

x ) (y

n

=

y) / n

x 2

n

uv ( u)2

u n

2

y y2

y )2

(

v v2

( v )2

Results : 1) – 1

r 1 2) If r = 1 r = –1 r = 0 3) If r > r <

II]

Perfect Positive correlation. Perfect Negative correlation No correlation.

0 0

Positive correlation Negative correlation Strong – ve Weak – ve Weak + ve

–1 0.5 Spearman’s Rank correlation coefficient : i) For Non-Repeated Ranks : R = 1 –

0

Strong + ve

0.5

1

6 d2 n(n 2 - 1)

ii) For Repeated Ranks : d2

R = 1 – 6 d n m1 m2 III]

= = = =

1 {( m13 12

m1 ) n(n 2

(m 2 3

m 2) )

........}

1)

R1 – R2 = Difference in Ranks. no. of pairs of obsvns. no. of obsvns forming 1st group having repeated ranks. no. of obsvns forming 2nd group having repeated ranks.

Concurrent Deviations coefficient : Rc =

(2c

m) m

c = No. of concurrent deviations. (No. of ‘+’ signs) m = No. of deviations. (= n – 1) (No. of + & – signs in all) n = No. of pairs of obsvns. Other Important Formulae 1) Cov (x, y) =

(x

x ) (y n

y)

2) Coefficient of Determination or Explained Variations.

= r2 x 100%

= (1 – r2) x 100%

3) Coefficient of Non-Determination or Unexplained variance 4) Effect of shift of origin / scale. * Not affected by shift of origin. * Effect of change of scale : rxy =

b.d b d

. ruv

Where b, d = slopes. [In short, r changes in sign only depending on sign of b & d.] 5) Steps for finding correct R when diff is wrong.

2

2

n(n 2

Step 2 : Correct d = wrong d – (wrong d) Step 3 : Correct R

= 1 –

2

1)

+ (correct d)2

d2

6 Correct n(n 2

d2

6

d2 using R = 1 –

Step 1 : Calculate Wrong

1)

REGRESSION: Regression Equation of * Y on X : y y b yx ( x

x)

y = ? x = Given

* X on Y : x

y)

x = ? y = Given

x

b yx ( y

Regression Coefficients : (Slopes of regression lines) 1) byx =

Cov ( x , y ) x (x

=

x ) (y

n

=

n

n

n

= r .

x 2

uv u

2

y

( x)

u ( u)

Cov ( x , y ) y2

(x

=

x) / n

xy x

y) / n 2

(x

=

2) byx =

2

2

v 2

y x

x ) (y (y

=

n

=

n

n

n

= r .

xy y

y) / n x

2

uv v

y) / n 2

2

y

( y)

u ( v)

2

v 2

x y

Properties : 1) If equation is Ax + By + C + 0 byx = – bxy = –

A B B A

(Slope)

Used for Identifying the equations.

2) 3)

byx . bxy < 1 r = byx . b xy

4) 5)

Point of Intersection of 2 regression lines (Solve the 2 equations simultaneously). Effected of shift of origin / change of scale.

(All 3 carry same sign)

(x, y )

byx = bxy =

q x bvu p p x buv q

1. Classical Definition : 2. Statistical Definition : 3. Modern Definition :

u = v = n( A) n(S) lim P(A) = N

x

a p

y

c q

P(A) =

i) P(A) 0 ii) P(S) = 1

FA N

for all A

S

Our partners will collect data and use cookies for ad personalization and measurement. Learn how we and our ad partner Google, collect and use data. Agree & close