Symbols
π=¿
x́=¿
μ=¿
λ=¿
σ 2=¿
σ =¿
x=¿
P(0¿ z )=¿
a=¿
b=¿
N=¿
n=¿
μ x́ =¿
s=¿
df =¿
P(0¿ t)=¿
Chapter 3
Population mean
∑x
μ=
N
Sample mean, raw data
∑x
x́= Weighted mean
n
w , x 1+ ⋯+ wn xn
x́ w =
w1 +…+ wn
Geometric mean
GM =√n ( x1 ) ( x 2 ) … ⋅ ( x n )
Geometric mean rate of increase
Value at theend of period
GM =
√
n
Value at the start of period
−1.0
Population variance
2 ∑ ( x−μ )2
σ = Population standard deviation
N
2
∑ ( x−μ )
σ=
√ N
Sample variance
∑ ( x−x́ )2
s2=
n−1
Sample standard deviation
2
∑ ( x− x́ )
s=
√ n
Sample mean, grouped data
∑ fM
x́=
n
Sample standard deviation, grouped data
2
∑ f ( M − x́ )
s=
√ n
Chapter 4
Location of a percentile
P
LP =( n+1 )
100
Pearson’s coefficient of skewness
3 ( x́−Mⅇⅆⅈan )
sk=
s
Chapter 5
Bayes’ Theorem
P ( A 1 ) P ( B| A 1 )
P ( A 1|B )=
P ( A 1 ) P ( B| A 1 ) + P ( A2 ) P(B∨ A 2)
Number of permutations
n!
n P r=
( n−r ) !
Number of combinations
n!
n Cr =
r ! ( n−r ) !
Chapter 6
Mean of a probability distribution
μ=∑ ( xP ( x ) )
Variance of a probability distribution
σ 2=Σ ( ( x −μ )2 P ( x ))
BINOMIAL
Binomial probability distribution
P ( x ) =n C X π x ( 1−π )n−x
Mean of a binomial distribution
μ=nπ
Variance of a binomial distribution
σ 2=nπ ( 1−π )
HYPERGEOMETRIC
Hypergeometric probability distribution
( N −S C ¿ ¿ n−x)
P ( x ) =( S C x ) ¿
N Cn
POISSON
Poisson probability distribution
μ x ⅇ−μ
P ( x)=
x!
Mean of a Poisson distribution
μ=nπ
Chapter 7
UNIFORM DISTRIBUTION
Mean of a uniform distribution
a+ b
μ=
2
Standard deviation of a uniform distribution
( b−a )2
σ=
√ 12
Uniform probability distribution
1
P ( x) = if a ≤ x ≤ b and 0 elsewhere
b−a
Area
1
Area= ( b−a )
b−a
NORMAL DISTRIBUTION
Normal probability distribution
2
( x− μ)
−( ) Standard normal value
1 2σ
2
p ( x) = ⅇ
σ √2 π
x−μ
z=
σ
Exponential distribution
P ( x ) =λ ⅇ− λx
Finding a probability using the exponential distribution
P ( Arrival time< x )=1−ⅇ− λx
Chapter 8
Standard error of mean
σ
√n
z-value, μ and σ known
x́−μ
z=
σ /√n
Chapter 9
Confidence interval for μ, with σ known
σ
x́ ± Z
√n
Confidence interval for μ, σ unknown
s
x́ ± t
√n
Sample proportion
x
p=
n
Confidence interval for proportion
P ( 1− p )
p±z
√ n
Sample size for estimating mean
2
zσ
n= ( ) E
Chapter 10
z-value, σ not known
x́−μ
z=
s/√n
Type II Error
x́ −μ
z= c 1
σ / √n
Chapter 11
Two-sample test of means, known σ
x́ 1−x́ 2
z=
σ 21 σ 22
√ +
n1 n 2
Pooled variance
2
( n ,−1 ) s21 + ( n2−1 ) s 22
s=
p
n1 +n 2−2
Two-sample test of means, unknown but equal σ2s
x́ 1− x́ 2
t=
√ s 2P
( n1 + n1 )
1 2
Degrees of freedom for unequal variance test
2
s 21 s 22
ⅆf =
( +
n1 n 2 )
2 2
s 21 s 22
( ) ( )
+
n1 n2
n1−1 n2−1
Two-sample tests of means, unknown and unequal σ2s
x́1 −x́2
t=
(√ n1 + n1 )
1 2
Paired t test
d́
t=
s d ∕ √n
Chapter 12
Test for comparing two variances
s 21
F=
s 22
ANOVA
Sum of squares, total
SS Total=∑ ( x−x́ G )2
Sum of squares, error
SS E=∑ ( x−x́ C )2
Sum of squares, treatments
SSE=SS Total−SSE
Confidence interval for differences in treatment means
1 1
√
( x́ 1−x́ 2 ) ±t MSE n + n
1
(
2
)
TWO-WAY ANOVA
Sum of squares, blocks
SS B=k ∑ ( x́ b− x́ G )2
Sum of squares error, two-way ANOVA
SSE=SS Total−SST −SSB
Chapter 13
Correlation coefficient
∑ ( x− x́ ) ( y− ý )
r=
( n−1 ) s x s y
Test for significant correlation
r √n−2
t=
√1−r 2
ⅆf =n−2
Slope of the regression line
sy
b=r
sx
Test for a zero slope
b−0
t=
sb
Standard error of estimate
2
sy ⋅ X =
√∑ ( Y −Y^ )
n−2
Coefficient of determination
SS R SS E
r 2= =1−
SST SS Total
Confidence interval
2
√ 1 ( X− X́ )
Y^ ±t s y . x +
n ∑ ( X − X́ )2
Prediction interval
2
√ 1 ( X− X́ )
Y^ ±t s y . x 1+ +
n ∑ ( X − X́ )2
Chapter 14
Multiple standard error of estimate
2
SSE
sy=
√
∑ ( Y −Y^ ) =
n−( k +1 ) √
n−( k + 1 )
Coefficient of multiple determination
SS R
R 2=
SS Total
Adjusted coefficient of determination
SS E
n− ( k +1 )
R2aⅆj =1−
SS Total
n−1
Global test of hypothesis
SS R/k
F=
SS E /[n−( k +1 ) ]
Testing for a particular regression coefficient
bi−0
t=
s bi
Variance inflation factor
1
V IF=
1−R2j
Chapter 15
Test of hypothesis, one proportion
ρ−π
z=
π ( 1−π )
√ n
Pooled proportion
x 1+ x 2
pc =
n 1+ n2
Two-sample test of proportions
p 1− p2
z=
pc ( 1− pc ) pc ( 1− pc )
√ n1
+
n2
Chi-square test statistic
2
( f 0−f e )
2
x =∑ [ fe ]
Contingency Table -Expected frequency
( Row Total )( ColumnTotal )
f e=
Grand Total
Degree of Freedom
ⅆf =( Rows−1)(Column−1)
Chapter 16
Wilcoxon rank-sum test
n1 ( n1 +n 2+1 )
W−
2
z=
n 1 n2 ( n1 +n 2+1 )
√ 12
Kruskal-Wallis test
2 2
( ∑ R 1) (∑ R p)
H=
12
n ( n+1 ) n1 [ +…+
nk ]−3 ( n+1 )
Spearman coefficient of rank correlation
6 Σ ⅆ2
r S=1−
n ( n 2−1 )
Hypothesis test, rank correlation
n−2
t=r s
√ 1−r 2s
Chapter 18
Mean Absolute Deviation
∑|Error|
MAD=
n
Durbin-Watson statistic (test correlation)
∑ ( et −e t−1 )2
d=
∑ ( e t )2
