Statistics for Management and Economics, Eighth Edition

Formulas

Numerical Descriptive techniques

Population mean

= N

xN

ii

=1

Sample mean

n

xx

n

ii

=

=1

Range

Largest observation - Smallest observation

Population variance

2 =

N

)x(N

ii

=

1

2

Sample variance

2s =

1

)(1

2

=

n

xxn

ii

Population standard deviation

= 2

Sample standard deviation

s = 2s

Population covariance

N)y)(x(N

iyixi

xy

=

=1

Sample covariance

11

=

=

n

)yy)(xx(s

n

iii

xy

Population coefficient of correlation

yx

xy

=

Sample coefficient of correlation

yx

xy

sss

r =

Coefficient of determination

R2 = r2

Slope coefficient

21x

xy

s

sb =

y-intercept

xbyb 10 =

Probability

Conditional probability

P(A|B) = P(A and B)/P(B)

Complement rule

P( CA ) = 1 P(A)

Multiplication rule

P(A and B) = P(A|B)P(B)

Addition rule

P(A or B) = P(A) + P(B) - P(A and B)

Bayes Law Formula

)A|B(P)A(P...)A|B(P)A(P)A|B(P)A(P)A|B(P)A(P

)B|A(Pkk2211

iii

+++=

Random Variables and Discrete Probability Distributions

Expected value (mean)

E(X) = =xall

)x(xP

Variance

V(x) = =xall

)x(P)x( 22

Standard deviation

2=

Covariance

COV(X, Y) = xy = )y,x(P)y)(x( yx

Coefficient of Correlation

yx

)Y,X(COV

=

Laws of expected value

1. E(c) = c

2. E(X + c) = E(X) + c

3. E(cX) = cE(X)

Laws of variance

1.V(c) = 0

2. V(X + c) = V(X)

3. V(cX) = 2c V(X)

Laws of expected value and variance of the sum of two variables

1. E(X + Y) = E(X) + E(Y)

2. V(X + Y) = V(X) + V(Y) + 2COV(X, Y)

Laws of expected value and variance for the sum of more than two variables

1. ==

=

k

ii

k

ii XEXE

11

)()(

2. ==

=

k

ii

k

ii XVXV

11

)()( if the variables are independent

Mean and variance of a portfolio of two stocks

E(Rp) = w1E(R1) + w2E(R2)

V(Rp) = 21w V(R1) + 22w V(R2) + 2 1w 2w COV(R1, R2)

= 21w21 +

22w

22 + 2 1w 2w 1 2

Mean and variance of a portfolio of k stocks

E(Rp) = =

k

iii REw

1

)(

V(Rp) = = +==

+k

i

k

ijjiji

k

iii RRCOVwww

1 11

22 ),(2

Binomial probability

P(X = x) = )!xn(!x!n

xnx )p(p 1

np=

)p(np = 12

)p(np = 1

Poisson probability

P(X = x) = !x

e x

Continuous Probability Distributions

Standard normal random variable

=

XZ

Exponential distribution

== /1

xe)xX(P =>

xe1)xX(P = 30

1= nrz S

Time Series Analysis and Forecasting

Exponential smoothing

1)1( += ttt SwwyS

Statistical Process Control

Centerline and control limits for x chart using S

Centerline = x

Lower control limit = n

Sx 3

Upper control limit = n

Sx 3+

Centerline and control limits for the p chart

Centerline = p

Lower control limit = n

)p(pp 13

Upper control limit = n

)p(pp + 13

Decision Analysis

Expected Value of perfect Information

EVPI = EPPI - EMV*

Expected Value of Sample Information

EVSI = EMV' - EMV*

CovarianceCOV(X, Y) = xy =Coefficient of Correlation

Laws of expected valueLaws of varianceLaws of expected value and variance of the sum of two variables

Laws of expected value and variance for the sum of more than two variablesMean and variance of a portfolio of two stocksMean and variance of a portfolio of k stocksSum of squares for errorStandard error of estimateCoefficient of determinationPrediction intervalConfidence interval estimator of the expected value of yStandard Error of EstimateCoefficient of DeterminationAdjusted Coefficient of Determination