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statsistics equation, Study notes of Statistics

Reformed Theological Seminary (RTS)Statistics

statistics equation sheet for stats

Typology: Study notes

2025/2026

Uploaded on 04/21/2026

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Formulas for AP Statistics
I. Descriptive Statistics
1i
i
x
xx
nn
∑
=∑=
( ) ( )
2
2
1
11
i
xi
xx
s xx
nn
∑−
= ∑− =
−−
ˆ
y a bx= +
y a bx= +
1
1
ii
xy
xxyy
rn ss
−−


= ∑

−

y
x
s
br
s
=
II. Probability and Distributions
( ) ( ) ( ) ( )
PA B PA PB PA B∪= + − ∩
( ) ( )
( )
|PA B
PAB PB
∩
=
Probability Distribution Mean Standard Deviation
Discrete random variable, X
( ) ( )
2
X iX i
x Px
σµ
=∑−
If 𝑋𝑋 has a binomial distribution
with parameters n and p, then:
( ) ( )
1
nx
x
n
PX x p p
x
−

= = −


where
0, 1, 2, 3, ,xn=
X
np
µ
=
( )
1
Xnp p
σ
= −
If 𝑋𝑋 has a geometric distribution
with parameter p, then:
( ) ( )
1
1
x
PX x p p
−
= = −
where
1, 2, 3,x=
1
X
p
µ
=
1
X
p
p
σ
−
=
III. Sampling Distributions and Inferential Statistics
Standardized test statistic:
statistic parameter
standard error of the statistic
−
Confidence interval:
( )( )
statistic critical value standard error of statistic±
Chi-square statistic:
( )
2
2
observed expected
expected
χ
−
=
∑
pf3
pf4
pf5

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Formulas for AP Statistics

I. Descriptive Statistics

1 i

i

x

x x

n n

2

i x i

x x

s x x

n n

y^ ˆ = a + bx y = a + bx

i i x y

x x y y

r

n s s

 −^ ^ − 

y

x

s

b r

s

II. Probability and Distributions

P A ( ∪ B ) = P A ( ) + P B ( ) − P A ( ∩ B ) ( )

P A B

P A B

P B

Probability Distribution Mean Standard Deviation

Discrete random variable, X^ μ^ X =^ E^ (^ X^ )^ = ∑^ x P xi (^ i ) ( ) ( )

2 X i X i

σ = ∑ x −μ P x

If 𝑋𝑋 has a binomial distribution

with parameters n and p , then:

n x n x

P X x p p

x

where x = 0, 1, 2, 3, , n

X

μ = np ( 1 )

σ X = np − p

If 𝑋𝑋 has a geometric distribution

with parameter p , then:

1

x

P X x p p

−

where x = 1, 2, 3,

X p

X

p

p

III. Sampling Distributions and Inferential Statistics

Standardized test statistic:

statistic parameter

standard error of the statistic

Confidence interval: statistic ±( critical value )( standard error of statistic)

Chi-square statistic:

2

2 observed^ expected

expected

*Standard deviation is a measurement of variability from the theoretical population. Standard error is the estimate of the standard deviation. If the standard deviation of the statistic is assumed to be known, then the standard deviation should be used instead of the standard error.

III. Sampling Distributions and Inferential Statistics ( continued )

Sampling distributions for proportions:

Random

Variable

Parameters of

Sampling Distribution

Standard Error *

of Sample Statistic

For one

population:

p ˆ

μ (^) ˆ p = p ( ) ˆ

p

p p

n

σ

( ) ˆ

p

p p

s

n

For two

populations:

1 2

p ˆ^ − p ˆ p ˆ 1^ ˆ p^2^^1

μ (^) − = p − p (^ )^ (^ ) 1 2

1 1 2 2 ˆ ˆ 1 2

p p

p p p p

n n

σ (^) −

( ) ( ) 1 2

1 1 2 2 ˆ ˆ 1 2

p p

p p p p

s

− n n

When p 1 = p 2 is assumed:

( ) ˆ 1 ˆ (^21 )

s p − p pc pc n n

where 1 2

1 2

c

X X

p

n n

Sampling distributions for means:

Random

Variable

Parameters of Sampling Distribution

Standard Error

of Sample Statistic

For one

population:

X

X

X n

σ σ = X

s

s

n

For two

populations:

X 1 − X 2

X 1 (^) X 2 1 2

μ μ μ −

1 2

2 2 1 2

1 2

X X n n

σ σ σ −

1 2

2 2 1 2

1 2

X X

s s

s

− n n

Sampling distributions for simple linear regression:

Random

Variable

Parameters of Sampling Distribution

Standard Error *

of Sample Statistic

For slope:

b

b

b^ ,

x

n

σ σ σ

where

( )

2 i x

x x

n

σ

b x

s

s

s n

where

( )

2

i i

y y

s

n

and

( )

2

i x

x x

s

n

Table B t distribution critical values

Confidence level C

Probability p

t *

Table entry for p and C is the point t * with probability p lying above it and probability C lying