Download - Maths Formulas
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