RATIO: 1. Numerator = Antecedent , Denominator = Consequent 2. Order is important 3. Types of Ratios:

a) Original a : b Inverse = b : a

b) Original a : b Duplicate = a2 : b2

c) Original a : b Triplicate = a3 : b3

d) Original a : b Sub Duplicate = ba :

e) Original a : b Sub Duplicate = 33 : ba

f) f

e

d

c

b

a,, Compound ratio =

f

e

d

c

b

a

(any 3 or more ratios) = fdb

eca

g) Continued ratio = a : b : c

PROPORTION: 1. If 4 Quantities a, b, c, d are in proportion:

d

c

b

a or a : b = c : d.

Result: 1) ad = bc (Product of extremes = Product of means)

2) By k – method : d

c

b

a= k

a = bk c = dk.

2. If 3 Quantities a, b, c are in proportion c

b

b

a 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 : c

b

b

a= k.

b = ck a = ck2.

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

d

c

c

b

b

a,, or a : b : c : d

Result: 1) b2 = ac ; c2 = bd ; ad = bc

2) By k – method : d

c

c

b

b

a= k.

c = dk b = dk2 c = dk3 [Increase the power of k only]

PROPERTIES OF PROPORTION:

1. Invertendo : d

c

b

a

c

d

a

b

2. Alternendo : d

c

b

a

d

b

c

a

3. Componendo : d

c

b

a

d

dc

b

ba

4. Dividendo : d

c

b

a

d

dc

b

ba

5. Componendo – Dividendo : d

c

b

a

dc

dc

ba

ba

r

r

DinnSubtractio

NinAddition

Note : 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 : f

e

d

c

b

a Each ratio =

fdb

eca

7. Subtrahendo : f

e

d

c

b

a Each ratio =

fdb

eca

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 same base in multiplication Different powers Result : Power add up.

2. am an = am - n same base in division Different powers Result : Power subtract. (Large - Small)

3. (am )n = am x n single base 2 Different powers Result : Power multiply.

4. (a x b)m = am x bm different base in multiplication Single power Result : Power get distributed..

Use : baba (Split)

5. m

mm

b

a

b

a different bases in division single power.

Result : Power get distributed..

Use : b

a

b

a (Split)

6. a = 1 Any base power zero Result : Answer = 1.

7. a- m = ma

1 Single base raised to negative power.

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

8. n ma = am/n m = actual power

= (am)1/n n = root part. (radical) = (a1/n)m a = base (radicand)

NOTE: 1. In case of cyclic powers : Usual Answer = L.

2. If x = p1/3 + 3/1

1

p x3 – 3x = p +

p

1

If x = p1/3 – 3/1

1

p x3 + 3x = p –

p

1

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. a = 1 loga 1 = 0 [log 1 to any base = 0]

2. a1 = a loga a 1 = 1 [log a to same base a = 1] 3. Base of log cannot be ‘0’ [loga 0 = – ] Base of Log cannot be negative. 4. log10 10 = 1, log10 100 = 2, log10 1000 = 3 and so on. 5. alog

am = m

LAWS OF LOGARITHMS: 1. Product Law:

NOTE : 1) log (a + b) log a + logb

2) (log a) log b) log a + logb 2. Quotientt Law:

NOTE : 1) log (a - b) loga - logb

2) b

a

log

log log a - logb log(

b

a

3. Power Law:

NOTE : 1) (log a)n n. loga 4. Change of base:

(logm a = loga m = 1)

logm (a x b) = logma + logmb

Logm b

a = logm a - logm b

logm an = n. logma

i) logm a = m

a

p

p

log

log

ii) logm a = malog

1

EQUATIONS: I] SIMPLE LINEAR EQUATION : General form : Ax + B = 0 x = variable A, B = constants (coefficients) Max power of x = 1. II] 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 = 1221

1221

BABA

CBCB y =

1221

1221

BABA

ACAC

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 = q

p other root =

p

q

* Methods of Solving :

1) Factorisation : Involves 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 = A

ACBB

2

42

B2 – 4AC = Discriminants (∆)

* Nature of Roots :

B2 – 4AC (∆) 1. Roots : REAL & EQUAL 1. Root: NOT REAL (Imaginary or complex)

2. Each Root = – A

B

2 2. One root = a + bi

3. QE is a perfect square. Other root = a – bi (i = 1 )

Is also a perfect square Is not a perfect square

1. Roots : REAL, UNEQUAL & RATIONAL 1. Roots : REAL, UNEQUAL IRRATIONAL

2. One root = a + b

Other root = a – b

B2 – 4AC = 0 REAL & EQUAL

B2 – 4AC < 0 NOT REAL (IMAGINARI – CONJUGATE) RATIONAL (If perfect square)

B2 – 4AC > 0 REAL, UNEQUAL IRRATIONAL (conjugates) (If not a perfect square)

B2 – 4AC 0 REAL * Relation between Roots & Coefficients:

Q.E. Ax2 + Bx + C 2 roots : &

Sum of Roots : & = – A

B

Product of Roots : = A

C

* Formation of Quadratic Equation :

2 roots : &

Q.E. x2 – ( + ) x + = 0 x2 – (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 ( – )

B2 – 4AC =

0

B2 – 4AC < 0 (Negative)

B2 – 4AC > 0 (Positive)

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 = abscissaofDiff

OrdinatesofDiff

xx

yy

.

.

12

12 2 points on line A (x1, y1) ; B (x2, y2)

= tan = Angle between line & X-axis

= – B

A If equation of line Ax + By + C = 0 is given.

= m 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 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 Lines are r

* 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 : Requirement : Slope = m Point = (x1, y1) 2 TWO – POINT FORM : Requirement : 1st point = (x1, y1) 2nd Point = (x2, y2) (RHS = Slope) 3 DOUBLE INTERCEPT FORM : Requirement : X intercept = a Y intercept = b

m1 = m2

m1 = 2

1

m

y – y1 = m(x – x1)

12

12

1

1

xx

yy

xx

yy

1b

y

a

x

4. SLOPE – INTERCEPT FORM : Requirement : Slope = m Y – intercept = c Other form : y = a + bx. (a. k. a. DISPLAY EQUATION) 5. GENERAL FORM :

Ax + By + C = 0

Slope = – B

A

X – intercept = – A

C

Y – intercept = – A

C

Other Important Notes :

1. 3 points A, B, C are COLLINER Slope AB = Slope BC = Slope AC

2. 3 lines are concurrent 3 lines intersect at 1 point only Pt. of concurrency : pt. of intersection of 3 lines.

Condition for concurrency : 0

1

1

1

33

22

11

yx

yx

yx

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

= 22

11

BA

CByAx

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

= 22 BA

C

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

= 22 BA

KC

4. Distance formula : A(x1 y1) & B(x2 y2)

AB = 212

212 )()( yyxx

5. Section formula : Internal division. A P B

(x1 y1) (x, y) (x2 y2)

m n

Px = nm

nxmx 12 Py = nm

nymy 12

y = mx + c

6. Midpoint formula : A P B

(x1 y1) (x, y) (x2 y2)

Px = 2

21 xx Py =

2

21 yy

INEQUALITIES :

Max availability At most

Min requirement At least

SIMPLE INTEREST:

1. SI = PinPnr

100

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

i = 100

%raterate of interest (in decimal)

n = Period or Time (in years) If time in months, divide by 12 In days , divide by 365. COMPOUND INTEREST: 1. A = P(1 + i)n 2. CI = A – P = P(1 + i)n – P = P[(1 + i)n – 1] 3. CI for nth year = Amount in n years – Amount in (n – 1) years

4. 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 5. Effective Rate of Interest : (To be calculated if compounding done more than once in year)

E = [(1 + i)n – 1] 100% ANNUITY : 1. Immediate Annuity or Annuity Regular or Annuity Certain. (Ordinary Annuity)

Payments are made/received at the END of reach period. 2. Annuity Due :

Payments are made / received at the START of each period. Formulae

Ordinary Annuity Annuity Due

1) FV = ]1)1([ 2ii

C 1) FV = )1(]1)1([ ii

i

C n

2) PV = nii

C

)1(

11 2) PV =

nii

C

)1(

11 (1 + i)

FV = Future Value

PV = Present Value (LOAN)

C = Annuity or Periodic Payment or Instalment. n = Period or No. of instalments. r = rate of interest (in %)

i = rate of interest (in decimal) 100

r

PERMUTATION AND COMBINATION 1. Factorial Notation : n ! = Product of 1st n natural nos. = 1 x 2 x 3 x 4 x . . . . . . . . . x n n ! = n(n – 1) (n – 2) . . . . . . . x 3 x 2 x 1. NOTE : n ! = n(n – 1)! = n(n – 1) (n – 2)! = n(n – 1) (n – 2) (n – 3)! & so on. Remember : 0! = 1 3! = 6 6! = 720 1! = 1 4! = 24 7! = 5040 2! = 2 5! = 120 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

nPr = No. of arrangements )!(

!

rn

n (n > r)

* If No. of places = no. of objects (arrangements amongst themselves) Then No. of arrangements = nPr = 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.

No. of arrangements = !!!

!

rqp

n

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

!)3(

P

p

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

!)2(

P

p

* 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 = 2

1(n – 1)!

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

No. of selections = nCr = !)(!

!

rnr

n (n > r)

Remember :

1) nCr = !r

P rn

2) nCr = nCn - r *** 3) nC0 = nCn = 1 4) nC1 = n 5) If nCx = nCy then x = y or x + y = n. 6) nCr + nCr - 1 = n + 1Cr (Pascals Law) * Total no. of ways of dealing with n things = 2n (take it or leave it) No. of ways in which all ‘n’ things are rejected = 1 No. of ways in which one or more things are selected = 2n – 1 Note : nC0 + nC1 + nC2 + . . . . . . . + nCn = 2n nC1 + nC2 + . . . . . . . + nCn = 2n – 1 * No. of points in a plane = n

Condition : No. 3 points are collinear.

No. of Straight lines = nC2 (= No. of handshakes)

No. of triangles = nC3

Condition : P points are collinear ( p 3)

No. of Straight lines = nC2 – pC2 + 1

No. of triangles = nC3 – pC3 * 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 mC2

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

Sn = ])1(2[2

dnan

d = common difference ER

= ][2

ntan

n = no. of terms (position) EN

= ]1[2

termlasttermstn

tn = nth term (any term) ER

Sn = Sum of n terms ER For convenience : No. of terms Terms 3 a – d, a, a + d 4 a – 3d, a – d, a + d, a + 3d 5 a – 2d, a – d, a, a + d, a + 2d

* If a, b, c are 3 terms in AP b = 2

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

Remember :

1) Sum of 1st n natural nos : 1 + 2 + 3 + . . . . . . . + n = 2

)1(nn

2) Sum of squares of 1st n natural nos: 12 + 22 + 32 + . . . . . . . + n2 = 6

)12()1( nnn

3) Sum of cubes of 1st n natural nos: 13 + 23 + 33 + . . . . . . . + n3 = 2

2

)1(nn

4) Sum of 1st n odd natural nos: 1 + 3 + 5 + . . . . + (2n – 1) = n2 5) Sum of 1st n even natural nos: 2 + 4 + 6 + . . . . + (2n) = n(n + 1) TRIVIA :

1) n tn = m tm tm + n = 0

2) tp = q & tq = p tr = p + q – r

3) Sm = Sn Sm + n = 0

4) 2

2

n

m

S

S

n

m d = 2a & 12

12

n

m

tn

tm

II] 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

Sn = a 1

1

r

r n

if r > 1 r = common ratio ER

= a r

r n

1

1 if r < 1 n = no. of terms EN. (Position)

S = r

a

1 (only if r < 1) tn = nth term ER (Any term)

Sn = Sum of n terms ER.

S = Sum of infinite terms.

* For Convenience : No. of Terms Terms Common Ratio

3 r

a , a, ar r

4 3

3,,, arar

r

a

r

a r2

5 2

2,,,, arara

r

a

r

a r

* If a, b, c are 3 terms in GP b2 = ac b = ac . (b = G. M. of a & c)

TRIVIA

1) a + aa + aaa + aaaa + ……….. = na n )110(

9

10

9

2) 0.a + 0.aa + 0.aaa + 0.aaaa + . . . . . . . = )1.01(9

1

9

nna

3) 0.a + 0.0a + 0.00a + . . . . . . . = na)1.0(1

9

Best term fro AP : 1, 2, 3. GP : 1, 2, 4 or 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)

3. Complement : A or Ac = {x / x U, x A} (not contained in A) Properties : 1) Union 2) Intersection

a) A B = B A a) A B = B A.

b) A A = U b) A A =

c) A = A c) A =

d) A U = U d) A U = A

e) If A B then A B = B e) If A B then A B = A Formulae : For 2 sets A & B: 1) Addition Theorem :

n(A B) = n(A) + n(B) – n(A B)

= n(A) + n(B) (if A B = i.e. A & B are disjoint)

2) n(A B) + n(A B) = n(S).

3) n(Only A) = n(A – B) = n(A B )

= n(A) – n(A B).

4) n(Only B) = n(B – A) = n(A B)

= n(B) – n( A B)

5) n(A B ) = n(A B) De Morgan’s Law.

n(A B ) = n(A B) For 3 sets A, B, C. 1) Additional Theorem :

n(A B C) = n(A) + n(B) + n(C) – n(A B) – n(B C) – n(C A) + n(A B C)

RELATION : 1) Reflexive : x Rx 2) Symmetric : If x Ry then y Rx. 3) Transitive : If x Ry and y Rz, then xRz. 4) Equivalence : All if above

LIMITS AND CONTINUITY

1) L’ HOSPITAL RULE : If )(

)(

xg

xf is of the form or

0

0, then

............)(''

)(''lim

)('

)('lim

)(

)(lim

xg

xf

axxg

xf

axxg

xf

ax

IMPORTANT FORMULAE :

1) .lim

kkax

2) )(lim

.)(.lim

xfax

kxfkax

3) 1lim nnn

naax

ax

ax

4) ax

a

xe

x

log1

0

lim

5) 11

0

lim

x

e

x

x

The coefficient of x in Nr must be repeated in Dr

6) 1)1(log

0

lim

x

x

x

7) exx

x1

)1(0

lim

8) Limit at Infinity::

* .01lim

...........1lim1lim1lim

32 nxxxxxxxx

Also 0)(

1lim

xfx

* exx

x1

1lim

(Also see ⑦)

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

ii) )(lim

xfax

exists.

iii) )(lim

xfax

exists.

iv) f(a) = ).(lim

)(lim

xfax

xfax

EQUATIONS: Key phrase : Rate of change / Gradient / Slope.

If y = f(x) is a function involving the variable x, then dx

dy = f (x) is its derivative.

dx

dy f (x) =

h

xfhxf

oh

)()(lim

Standard Formulae :

A L G E

B P R O A W

I E C R Exponential

Logarithmic

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

1. dx

d

dx

d

dx

d)( (Additional / Subtraction Rule

2. dx

d

dx

dv

dx

d).( (Multiplication Rule)

3. 2v

dx

dv

dx

dv

dx

d (Division Rule)

4. dx

ducc

dx

d)( [constant x function]

5. )(].)(['])([ xgdx

dxgfxgf

dx

d [Chain Rule]

6. dx

dy

dx

d

dx

yd2

2

[ 2nd Order Derivative or f" (x)]

y = f(x) dx

dyf (x)

c 0

x 1

cx c

xn nxn - 1

x x2

1

x

1

2

1

x

x

1

xx2

1

ax ax loga

ex ex

logx 1/x

xx xx (1 + logx)

APPLICATIONS & TYPES: 1. Slope (or Gradient) of Tangent to a curve: y or f(x) = function representing a curve.

dx

dy or f (x) = function representing the slope of tangent to the curve.

ax

dx

dyor f (a) = Slope of tangent to the curve at any point x = a. on the curve.

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 )().(

)().(

)(

)(.)(

xsxr

xgxfor

xr

xgxf

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 dx

dy

Isolate dx

dy on LHS.

5. 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. ./

/

dtdx

dtdy

dx

dy

Note : 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 = yx

yxCov

.

),(

=

n

yy

n

xx

nyyxx

22)()(

/)()(

= 22

)()(

)()(

yyxx

yyxx

= 2222 )()( yynxxn

yxxyn

= 2222 )()( vvnuun

vuuvn

Results :

1) – 1 r 1

2) If r = 1 Perfect Positive correlation.

r = – 1 Perfect Negative correlation

r = 0 No correlation.

3) If r > 0 Positive correlation

r < 0 Negative correlation Strong – ve Weak – ve Weak + ve Strong + ve

– 1 0.5 0 0.5 1

II] Spearman’s Rank correlation coefficient : i) For Non-Repeated Ranks :

R = 1 – 1)-(

62

2

nn

d

ii) For Repeated Ranks :

R = 1 – 6 )1(

........})(){(12

1

2

)23

213

12

nn

mmmmd

d = R1 – R2 = Difference in Ranks. n = no. of pairs of obsvns. m1 = no. of obsvns forming 1st group having repeated ranks. m2 = no. of obsvns forming 2nd group having repeated ranks. III] Concurrent Deviations coefficient :

Rc = m

mc )2(

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) = n

yyxx )()(

2) Coefficient of Determination or Explained Variations. = r2 x 100%

3) Coefficient of Non-Determination or Unexplained variance = (1 – r2) x 100% 4) Effect of shift of origin / scale. * Not affected by shift of origin.

* Effect of change of scale : rxy = db

db . . 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.

Step 1 : Calculate Wrong d2 using R = 1 – )1(

62

2

nn

d

Step 2 : Correct d2 = wrong d2 – (wrong d)2 + (correct d)2

Step 3 : Correct R = 1 – )1(

62

2

nn

dCorrect

REGRESSION: Regression Equation of

* Y on X : )( xxbyy yx y = ? x = Given

* X on Y : )( yybxx yx x = ? y = Given

Regression Coefficients : (Slopes of regression lines)

1) byx = 2

),(

x

yxCov 2) byx =

2

),(

y

yxCov

= nxx

nyyxx

/)(

/)()(2

= nyy

nyyxx

/)(

/)()(2

= 22 )( xxn

yxxyn =

22 )( yyn

yxxyn

= 22 )( uun

vuuvn =

22 )( vvn

vuuvn

= r . x

y = r .

y

x

Properties :

1) If equation is Ax + By + C + 0

byx = – B

A (Slope) Used for Identifying the equations.

bxy = – A

B

2) byx . bxy < 1

3) r = xyyx bb . (All 3 carry same sign)

4) ),( yx Point of Intersection of 2 regression lines (Solve the 2 equations simultaneously).

5) Effected of shift of origin / change of scale.

byx = bvuxp

q u =

p

ax

bxy = buvxq

p v =

q

cy

1. Classical Definition : P(A) = )(

)(

Sn

An

2. Statistical Definition : P(A) = N

F

N

Alim

3. Modern Definition : i) P(A) 0 for all A S ii) P(S) = 1