Basic Statistics Formulas
Population Measures
Mean µ=1
nXxi(1)
Variance σ2=1
nX(xi−x)2(2)
Standard Deviation σ=r1
nX(xi−x)2(3)
Sampling
Sample mean x=1
nXxi(4)
Sample variance s2
x=1
n−1X(xi−x)2(5)
Std. Deviation sx=r1
n−1X(xi−x)2(6)
z-score z=x−µ
σ(7)
Correlation r=
1
n−1
n
X
i=1 (xi−x)
sx(yi−y)
sy(8)
Linear Regression
Line ˆy=a+bx (9)
b=rsy
sx
, a =y−bx (10)
s=v
u
u
t
1
n−2
n
X
i=1
(yi−ˆy)2(11)
SEb=s
v
u
u
t
n
X
i=1
(xi−x)2
(12)
To test H0:b= 0,use t=b
SEb
(13)
CI =b±t∗S Eb(14)
Probability
P(Aor B) = P(A) + P(B)−P(Aand B) (15)
P(not A) = 1 −P(A) (16)
P(Aand B) = P(A)P(B) (independent) (17)
P(B|A) = P(Aand B)/P (A) (18)
0! = 1; n! = 1 ×2×3· · · × (n−1) ×n(19)
n
k=n!
k!(n−k)! (20)
Binomial Distribution :
P(X=k) = n
kpk(1 −p)n−k(21)
µ=np, σ =pnp(1 −p) (22)
One-Sample z-statistic
To test H0:µ=µ0usez=z−µ0
σ/√n(23)
Confidence Interval for µ=x±z∗σ
√n(24)
Margin of Error ME =z∗σ
√n(25)
Minimum sample size n≥z∗σ
M E 2
(26)
One-Sample t-statistic
SEM =sx
√n, t =x−µ
sx/√n(27)
Confidence Interval = x±t∗sx
√n(28)
Two-Sample t-statistic
t=x1−x2
ss2
1
n1
+s2
2
n2
(29)
Conf. Interval = (x1−x2)±t∗ss2
1
n1
+s2
2
n2
(30)
Sample Proportions
µˆp=p, σ ˆp=rp(1 −p)
n(31)
Conf. Int. = ˆp±z∗(SE) (32)
SE = rˆp(1 −ˆp)
n(33)
sample size n > z∗
M E 2
p∗(1 −p∗) (34)
To test H0:p=p0,use z=ˆp−p0
rp0(1 −p0)
n
(35)
Two-Sample Proportions
SE =sˆp1(1 −ˆp1)
n1
+ˆp2(1 −ˆp2)
n2
(36)
CI = ( ˆp1−ˆp2)±z∗(SE) (37)
To test H0:p1=p2,use (38)
z=ˆp1−ˆp2
sˆp(1 −ˆp)1
n1
+1
n2
(39)
ˆp=X1+X2
n1+n2
, Xi= successes (40)
Chi-Square Statistic
χ2=
n
X
i=1
(oi−ei)2
ei
(41)
oi= observed, ei= expected
Central Limit Theorem
sx→σ
√nas n→ ∞ (42)
2013 B.E. Shapiro. This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License (BY-NC-SA 3.0).
Last revised May 9, 2016.
