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Formula Sheet for Statistical Methods (201-DDD-05)
Five number summary:
min, Q1 , median, Q3 , max
Q1 : median of smallest half
Q3 : median of largest half
Fourth spread
fs = Q 3 − Q 1
Outliers
xi is an outlier if its distance from the
closest fourth (Q1 or Q3 ) is > 1.5fs
Sample variance
1 X
s2 =
(xi − x̄)2
n−1
P
X
1
( xi ) 2
s2 =
x2i −
n−1
n
Sample standard deviation
√
s = s2
Rule for expected value
E(aX + b) = aE(X) + b
Rule for variance
V (aX + b) = a2 V (X)
Binomial distribution
X ∼ Bin(n, p):
n
px (1 − p)n−x for x = 0, 1, . . . , n
p(x) =
x
E(X) = np, V (X) = np(1 − p)
Hypergeometric distribution
n = sample size, N = population size,
M = number of successes in population
M N −M p(x) =
x
E(X) = n ·
n−x
N
n
M
N
, V (X) =
N −n
N −1
·n·
M
N
· 1−
M
N
Poisson distribution
X ∼ Poisson(λ):
e−λ λx
for x = 0, 1 . . .
x!
E(X) = λ, V (X) = λ
Permutations
n!
Pk,n =
(n − k)!
p(x) =
Combinations
n
n!
Ck,n =
=
k
k!(n − k)!
Percentiles
η = 100p th percentile of X (continuous r.v.):
P (X ≤ η) = p
Normal distribution
Addition rule
P (A ∪ B) = P (A) + P (B) − P (A ∩ B)
Multiplication rule
P (A ∩ B) = P (A)P (B|A)
Independent events
A and B are independent if P (B|A) = P (B),
equivalently P (A ∩ B) = P (A)P (B)
Law of Total Probability
X −µ
∼ N (0, 1)
σ
For Z ∼ N (0, 1) set Φ(z) = P (Z ≤ z)
If X ∼ N (µ, σ) then
Φ(zα ) = 1 − α
Statistics
X1 , . . . , Xn random sample:
1X
X=
Xi (sample mean)
n
X
1
S2 =
(Xi − X)2 (sample variance)
n−1
A1 , . . . , Ak mutually exclusive & exhaustive:
P (B) = P (A1 ∩ B) + · · · + P (Ak ∩ B)
Special case: P (E) + P (E 0 ) = 1
De Morgan’s laws
(A ∪ B)0 = A0 ∩ B 0
(A ∩
B)0
=
A0
∪
Sampling distributions
X1 , . . . , Xn random sample,
Xi ∼ distribution with mean µ and std. dev. σ:
E(X) = µ, V (X) = σ 2 /n
CLT:
X−µ
√
σ/ n
∼ N (0, 1)
(n > 30)
B0
Regression and Correlation
Expected value for a discrete r.v.
P
E(X) = µX =
xp(x)
P
E(h(X)) =
h(x)p(x)
Expected value for a continuous r.v.
R∞
E(X) = µX = −∞
xf (x)dx
R∞
E(h(X)) = −∞ h(x)f (x)dx
Variance and standard deviation
2 = E(X 2 ) − E(X)2
V (X) = σX
p
σX = V (X)
Sxx = Σx2i −
(Σxi )2
n
(Σyi )2
n
(Σxi )(Σyi )
Sxy = Σ(xi yi ) −
n
SSE = Σyi2 − βˆ0 Σyi − βˆ1 Σxi yi
Syy = Σyi2 −
SST = Syy
β̂1 =
Sxy
Sxx
Σyi −β̂1 Σxi
n
√ Sxy ,
Sxx Syy
β̂0 =
r=
s2 =
SSE
n−2
r2 = 1 −
SSE
SST
Formula Sheet for Statistical Methods (201-DDD-05)
HYPOTHESIS TESTING
(α = significance level)
One mean
H 0 : µ = µ0
x − µ0
√ ∼ N (0, 1)
s/ n
t∗ =
x − µ0
√ ∼ tn−1
s/ n
(if n > 30)
(if data normally distr.)
s
x ± zα/2 √
n
(if data normally distr.)
x1 − x2 − ∆ 0
∼ N (0, 1)
z = r
∗
s2
2
n2
+
x1 − x2 − ∆ 0
r
∼ tν
x1 − x2 ± zα/2
(if data normally distr.)
s21
s22
(if data normally distr.)
1
Fα/2,ν2 ,ν1
Slope of regression line
H0 : β1 = β10
βˆ1 − β10
√
∼ tn−2
s/ Sxx
β̂1 ± tα/2,n−2 √
(if data normally distr.)
s
Sxx
Correlation coefficient
H0 : ρ = 0
√
r n−2
∼ tn−2
t∗ = √
1 − r2
(if data normally distr.)
Test of normality (Ryan-Joiner)
H0 : population distribution is normal
test statistic: correlation coefficient r from probability plot
p̂ − p0
∼ N (0, 1)
(if n large)
p0 (1−p0 )
n
p̂(1 − p̂)
n
Difference of two proportions
H0 : p1 − p2 = ∆ 0
p̂1 − p̂2 − ∆0
p̂1 (1−p̂1 )
n1
s
p̂1 − p̂2 ± zα/2
s21
∼ Fn1 −1,n2 −1
s22
2
One proportion
H0 : p = p0
z∗ = q
Ratio of two variances
H0 : σ12 = σ22
t∗ =
1
r
!
(if n1 > 30 and n2 > 30)
+
(if n1 > 30 and n2 > 30)
n1
n2
s
s21
s2
x1 − x2 ± tα/2,ν
+ 2
(if data normally distr.)
n1
n2
2
s2
s2
1
+ n2
n1
2
where ν = 2
(s1 /n1 )2
(s2 /n2 )2
+ n2 −1
n −1
p̂ ± zα/2
(if data normally distr.)
property of critical F -values: F1−α/2,ν1 ,ν2 =
s2
2
n2
+
s
z∗ = q
(n − 1)s2
∼ χ2n−1
σ02
(n − 1)s2
(n − 1)s2
,
χ2α/2,n−1 χ21−α/2,n−1
f∗ =
Difference of two means
H0 : µ1 − µ2 = ∆0
s2
1
n1
χ2 =
(if n > 30)
s
x ± tα/2,n−1 √
n
t∗ =
CONFIDENCE INTERVALS
(100(1 − α)% confidence level)
One variance
H0 : σ 2 = σ02
z∗ =
s2
1
n1
AND
+
p̂2 (1−p̂2 )
n2
∼ N (0, 1)
p̂1 (1 − p̂1 )
p̂2 (1 − p̂2 )
+
n1
n2
(if n1 , n2 large)
if r < rc , reject H0