AP STATISTICS FORMULAS
Descriptive Statistics
n
x̄ =
∑i=1
n
σ =
∑ (x − μ)2
= Population
N
s=
∑ (x − x̄)2
= Sample
n−1
IQ R = Q3 − Q1
Q1 − (1.5 ⋅ IQ R) = Lower Outlier
Q3 + (1.5 ⋅ IQ R) = Higher Outlier
x−μ
σ
z=
r =
Correlation Coefficient = −1 ≤ r ≤ 1
xi − x̄
yi − ȳ
1
n − 1 ∑ ( sx )( sy )
Residual = yi − yî
ŷ = bx + a
Root-mean-square deviation =
b=r
( sx )
sy
∑ (yi − yî )2
n−2
a = ȳ − b x̄
R2 = 0 ≤ r 2 ≤ 1
Probability and Distributions
P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
P(A∖B) =
P(A ∩ B)
P(B)
Discrete random variable, X
μx = E(X ) =
x ⋅ P(xi )
∑ i
E(X + Y ) = E(X ) + E(Y )
2
σx−y
= σx2 + σy2
σx =
E(xi − μx )2 ⋅ P(xi ) =
Var(X )
Var(X + Y ) = Var(X ) + Var(Y )
If X has a binomial distribution with parameters n and p, then:
P(X = x) =
n
p x(1 − p)n−x
(x)
μx = np
where x = 0, 1, 2 ,3, …, n
Var(X ) = np(1 − p)
σx =
np(1 − p) =
Var(X )
If X has a geometric distribution with parameter p, then:
P(X = x) =
n
p x (1 − p)n−x
(x)
μx =
where x = 0, 1, 2, 3, …, n
1
p
σx =
1−p
p
Sampling Distributions and Inferential Statistics
Random Variable
Parameters of Sampling Distribution
For one population: p̂
μp̂ − p
σp̂ =
For two populations:
Standard Error* of
Sample Statistic
p(1 − p)
n
μp1̂ −p̂2 = μp1̂ − μp̂2
p1̂ − p2̂
σp1̂ −p̂ 2 =
sp̂ =
p1(1 − p1) p2(1 − p2 )
+
n1
n2
sp1̂ −p̂ 2 =
̂ − p)̂
p(1
n
p1(1 − p1) p2(1 − p2 )
+
n1
n2
when p1 = p2 is assumed:
sp1̂ −p̂ 2 =
pĉ (1 − pĉ )
where
pĉ =
1
1
+
( n1 n 2 )
X1 + X2
n1 + n 2
Sampling distributions for means:
Random Variable
Parameters of Sampling Distribution
X̄
μX̄ = μ
sX̄ =
s
n
σ
σX̄ =
X̄1 − X̄2
Standard Error* of
Sample Statistic
n
μX¯1−X¯2 = μ1 − μ2
s12 s22
+
n1 n2
sX¯1−X¯2 =
σ12 σ22
+
n1
n2
σX¯1−X¯2 =
Sampling distributions for simple linear regression:
Random Variable
Parameters of Sampling Distribution
For slope:
μb = β
b
σb =
Standard Error* of
Sample Statistic
sb =
s
sx n − 1
σ
σx n
where s =
σx =
∑ (yi − ȳi )2
n−2
∑ (xi − x̄)2
n
and sx =
∑ (xi − x̄ )2
n −1
Confidence Intervals:
Expected successes: np ≥ 10
z* ⋅
Margin of Error:
Expected failures: n(1 − p) ≥ 10
p̂ ± z* ⋅
Degrees of Freedom: n − 1
x̄ ± z* ⋅
s
: Underestimate (useless)
n
x̄ ± t* ⋅
s
n
s
n
s
n
: Closest Proportion
Significance Tests
if p < α reject H0
f p ≥ α fail to reject H0, suggest Ha
P(Z ≥ z) for Ha : μ > μ0
P(Z ≤ z) for Ha = μ < μ0
P(Z > | z | ) for Ha : μ ≠ μ0
z=
p̂ − p0
for proportions
p0 (1 − p0 )
t=
x̄ − μ0
sx
n
for means
n
α increase, power will increase and Type 2 Error risk will decrease
If interval excludes 0, then reject H0, accept Ha
If interval includes 0, fail to reject H0, cannot accept Ha
Central Limit Theorem:
If n ≥ 30, the distribution of sample means will be approximately normal.
statistic − parameter
standard error of the statistic
standarized test statistic:
confidence interval ± (critical value)(standard error of statistic)
Inference comparing two groups or populations
pĉ =
combined count of successes
combined sample size
(x̄1 − x̄2) ± t*
s12 s12
+
n1 n2
t=
(x̄1 − x̄2) − (μ1 − μ2)
s12
n1
( n1
s12
df =
s22
+n
2
)
s12
+n
2
2
s12
s22
1
1
2
( ) + n − 1 ( n )2
n1 − 1 n1
1
2
(obser ved − expected )2
Chi-square statistic: χ =
∑
expected
2
Expected =
row total ⋅ column total
table total
Random condition: The data came from a random sample from the population of interest,
or a randomized experiment.
Large counts condition: All expected counts are at least 5.
Independence rule: n < 0.10
More on Regression
ŷ = bx + a
d f = (n − 2)
t=
b − β0
SEb
Conditions for Inference
Linear - Actual relationship between x and y
Independence
Normal
Equal Variance
Random