Download Statistical Methods Formula Sheet: Descriptive Stats, Probability, Hypothesis Testing and more Cheat Sheet Statistics in PDF only on Docsity!
Formula Sheet for Statistical Methods (201-DDD-05)
Five number summary:
min, Q 1 , median, Q 3 , max
Q 1 : median of smallest half
Q 3 : median of largest half
Fourth spread
fs = Q 3 − Q 1
Outliers
xi is an outlier if its distance from the
closest fourth (Q 1 or Q 3 ) is > 1. 5 fs
Sample variance
s 2 =
n − 1
(xi − x¯) 2
s
2
n − 1
x
2 i −^
xi) 2
n
Sample standard deviation
s =
s 2
Permutations
Pk,n =
n!
(n − k)!
Combinations
Ck,n =
n
k
n!
k!(n − k)!
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
A 1 ,... , Ak mutually exclusive & exhaustive:
P (B) = P (A 1 ∩ B) + · · · + P (Ak ∩ B)
Special case: P (E) + P (E ′ ) = 1
De Morgan’s laws
(A ∪ B)
′ = A ′ ∩ B ′
(A ∩ B)
′ = A ′ ∪ B ′
Expected value for a discrete r.v.
E(X) = μX =
xp(x)
E(h(X)) =
h(x)p(x)
Expected value for a continuous r.v.
E(X) = μX =
−∞ xf (x)dx
E(h(X)) =
−∞ h(x)f (x)dx
Variance and standard deviation
V (X) = σ 2 X
= E(X
2 ) − E(X) 2
σX =
V (X)
Rule for expected value
E(aX + b) = aE(X) + b
Rule for variance
V (aX + b) = a 2 V (X)
Binomial distribution
X ∼ Bin(n, p):
p(x) =
n
x
p x (1 − p) n−x for x = 0, 1 ,... , n
E(X) = np, V (X) = np(1 − p)
Hypergeometric distribution
n = sample size, N = population size,
M = number of successes in population
p(x) =
M x
N −M n−x
N n
E(X) = n · M N
, V (X) =
N −n N − 1
· n · M N
M N
Poisson distribution
X ∼ Poisson(λ):
p(x) =
e −λ λ x
x!
for x = 0, 1...
E(X) = λ, V (X) = λ
Percentiles
η = 100p th percentile of X (continuous r.v.):
P (X ≤ η) = p
Normal distribution
If X ∼ N (μ, σ) then
X − μ
σ
∼ N (0, 1)
For Z ∼ N (0, 1) set Φ(z) = P (Z ≤ z)
Φ(zα) = 1 − α
Statistics
X 1 ,... , Xn random sample:
X =
n
Xi (sample mean)
S
2
n − 1
(Xi − X) 2 (sample variance)
Sampling distributions
X 1 ,... , 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)
Regression and Correlation
Sxx = Σx 2 i
(Σxi)^2 n
Syy = Σy 2 i −^
(Σyi) 2
n
Sxy = Σ(xiyi) −
(Σxi)(Σyi) n
SSE = Σy 2 i − βˆ 0 Σyi − βˆ 1 Σxiyi
SST = Syy
β^ ˆ 1 =^
Sxy Sxx
β^ ˆ 0 =^
Σyi− βˆ 1 Σxi n
r =
Sxy √ SxxSyy
, r 2 = 1 − SSE SST
s 2 = SSE n− 2
Formula Sheet for Statistical Methods (201-DDD-05)
HYPOTHESIS TESTING AND CONFIDENCE INTERVALS
(α = significance level) (100(1 − α)% confidence level)
One mean
H 0 : μ = μ 0
z ∗ =
x − μ 0
s/
n
∼ N (0, 1) (if n > 30)
t ∗ =
x − μ 0
s/
n
∼ tn− 1 (if data normally distr.)
x ± zα/ 2
s √ n
(if n > 30)
x ± tα/ 2 ,n− 1
s √ n
(if data normally distr.)
Difference of two means
H 0 : μ 1 − μ 2 = ∆ 0
z ∗ =
x 1 − x 2 − ∆ 0 √ s^21 n 1
s^22 n 2
∼ N (0, 1) (if n 1 > 30 and n 2 > 30)
t ∗ =
x 1 − x 2 − ∆ 0 √ s^21 n 1
s^22 n 2
∼ tν (if data normally distr.)
x 1 − x 2 ± zα/ 2
s^2 1 n 1
s^2 2 n 2
(if n 1 > 30 and n 2 > 30)
x 1 − x 2 ± tα/ 2 ,ν
s 2 1 n 1
s 2 2 n 2
(if data normally distr.)
where ν =
s^21 n 1
s^22 n 2
2
(s^21 /n 1 ) 2
n 1 − 1
(s^22 /n 2 ) 2
n 2 − 1
One proportion
H 0 : p = p 0
z ∗ =
ˆp − p 0 √ p 0 (1−p 0 ) n
∼ N (0, 1) (if n large)
pˆ ± zα/ 2
pˆ(1 − pˆ)
n
Difference of two proportions
H 0 : p 1 − p 2 = ∆ 0
z ∗ =
pˆ 1 − pˆ 2 − ∆ 0 √ pˆ 1 (1− pˆ 1 ) n 1
pˆ 2 (1− pˆ 2 ) n 2
∼ N (0, 1) (if n 1 , n 2 large)
pˆ 1 − pˆ 2 ± zα/ 2
p ˆ 1 (1 − ˆp 1 )
n 1
pˆ 2 (1 − pˆ 2 )
n 2
One variance
H 0 : σ^2 = σ^2 0
χ 2 =
(n − 1)s 2
σ^2 0
∼ χ 2 n− 1 (if data normally distr.)
(n − 1)s 2
χ 2 α/ 2 ,n− 1
(n − 1)s 2
χ 2 1 −α/ 2 ,n− 1
Ratio of two variances
H 0 : σ 2 1 =^ σ
2 2
f ∗ =
s^2 1 s 2 2
∼ Fn 1 − 1 ,n 2 − 1 (if data normally distr.)
property of critical F -values: F 1 −α/ 2 ,ν 1 ,ν 2
Fα/ 2 ,ν 2 ,ν 1
Slope of regression line
H 0 : β 1 = β 10
t ∗ =
β 1 − β 10
s/
Sxx
∼ tn− 2 (if data normally distr.)
βˆ 1 ±^ tα/ 2 ,n− 2
s √ Sxx
Correlation coefficient
H 0 : ρ = 0
t
∗
r
n − 2 √ 1 − r^2
∼ tn− 2 (if data normally distr.)
Test of normality (Ryan-Joiner)
H 0 : population distribution is normal
test statistic: correlation coefficient r from probability plot
if r < rc, reject H 0
