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Frequently Used Statistics Formulas and Tables
Chapter 2
Class Width = highest value - lowest value (increase to next integer)
number classes
upper limit + lower limit
Class Midpoint = 2
Chapter 3 Chapter 3
n = sample size Limits for Unusual Data
N = population size Below:µσ - 2
f = frequency Above: 2µσ+
Σ=sum
w=weight Empirical Rule
About 68%: µσ- to µ+σ
∑x About 95%: µσ-2 to µ+2σ
Sample mean: x =
n About 99.7%: µσ-3 to µ+3σ
Population mean: µ = ∑x
N Sample coefficient of variation: CV = s 100%
wx
Weighted mean: x = ∑•() x
∑w Population coefficient of variation: CV = σ 100%
∑• µ
Mean for frequency table: x = ()fx
∑f
highest value + lowest value Sample standard deviation for frequency table:
Midrange = 2 22
n fx fx
s [ ∑•( ) ]−[ ∑•( ) ]
= nn
Range = Highest value - Lowest value ( −1)
xx
Sample z-score: z = −
∑−2 s
Sample standard deviation: s = ()xx
n−1 x−µ
∑−2 Population z-score: z = σ
()x
Population standard deviation: σ = µ
N
Interquartile Range: (IQR) =QQ−
31
Sample variance: s2
Modified Box Plot Outliers
2 lower limit: Q - 1.5 (IQR)
Population variance: σ 1
upper limit: Q3 + 1.5 (IQR)
Chapter 4 Chapter 5
Probability of the complement of event A Discrete Probability Distributions:
P(not A) = 1 - P(A)
Mean of a discrete probability distribution:
Multiplication rule for independent events =∑•
P(A and ) B = (P A) (P B)• µ [x Px( )]
Standard deviation of a probability distribution:
General multiplication rules 22
σµ=∑• −
x Px
P(A and ) B = (P A) (P B, •given A) [ ( )]
P(A and ) B = (P A) (P A, •given )B
Addition rule for mutually exclusive events Binomial Distributions
PA( or B) = PA( ) + P(B) r = number of successes (or x)
p = probability of success
General addition rule q = probability of failure
P(A or B) = P(A) + P(B)−P(A and B) q=1−p pq + = 1
Binomial probability distribution
n! rnr−
Permutation rule: P = Pr()= Cpq
nr (nr− )! nr
Mean: µ = np
Combination rule: C = n! Standard deviation: σ = npq
nr rn!( −r)!
Poisson Distributions
Permutation and Combination on TI 83/84
rx= number of successes (or )
µ = mean number of successes
n Math PRB nPr enter r (over a given interval)
Poisson probability distribution
n Math PRB nCr enter r e−µµr
Pr()=
r!
e ≈ 2.71828
Note: textbooks and formula
sheets interchange “r” and “x” µ = mean (over some interval)
for number of successes σµ=
2
σµ=
2
Chapter 6 Chapter 7
Normal Distributions Confidence Interval: Point estimate ± error
Raw score: xz=σµ+ Point estimate = Upper limit + Lower limit
2
Standard score: z = x−µ
σ Error = Upper limit - Lower limit
2
Mean of x distribution: µµ=
x Sample Size for Estimating
Standard deviation of x distribtuion: σ = σ means: 2
x z σ
n n = α/2
E
(standard error)
x −µ
xz proportions:
Standard score for : =
σ / n z 2
ˆˆ α/2
n = pq with preliminary estimate for p
E
Chapter 7
z 2
α/2
np0.25 without preliminary estimate for
One Sample Confidence Interval =
E
p np nq
for proportions ( ): ( >>5 and 5) variance or standard deviation:
*see table 7-2 (last page of formula sheet)
ˆˆ
pEppE
−<<+
pp
Ez (1− ) Confidence Intervals
where = α/2 n
Level of Confidence z-value ( z )
ˆ r α/2
p = n
70% 1.04
for means (µσ) when is known:
75% 1.15
xE−<µ<+xE
σ 80% 1.28
where Ez=
α/2 n 85% 1.44
for means (µσ) when is unknown:
90% 1.645
xE xE
−<µ<+ 95% 1.96
where Et s
= α/2 98% 2.33
n
df n
with . . 1
= −
99% 2.58
22
(ns−−1) (ns1)
22
for variance ( ) : < <
σσ
22
χχ
RL
with . .= 1−
df n
3
Chapter 8 Chapter 9
One Sample Hypothesis Testing Difference of means μ1-μ2 (independent samples)
Confidence Interval when σσ and are known
ˆ 12
pp
− ()x−−x E<(µµ−)<()x−+x E
p np nq z 12 1212
>>=
for ( 5 and 5):
pq/n 22
σσ
12
where Ez= +
α/2 nn
ˆ 12
q pprn
where −=1 =; /
x −µ σσ
Hypothesis Test when and are known
µσ z = 12
for ( known): −−µµ−
(xx)( )
σ / n = 1212
z 22
σσ
x −µ 12
µσ t df n +
for ( unknown): = with . .= 1− nn
sn 12
/
2
ns
( −1)
22 Confidence Interval when σσ and are unknown
σχ df n 12
for : =σ2 with . .= 1−
()x−−x E<(µµ−)<()x−+x E
12 1212
22
ss
12
Et= +
Chapter 9 α/2 nn
12
df n n
with . . = smaller of −−1 and 1
Two Sample Confidence Intervals 12
and Tests of Hypotheses
σσ
Hypothesis Test when and are unknown
Difference of Proportions (pp− ) 12
xxµ−µ)
12 (−−)(
t = 1212
22
ss
12
Confidence Interval: +
nn
12
ˆˆ ˆˆ
−−<−<−+
()pp E(pp)()ppE with df. . =smaller of n 1 a−−nd n 1
12 1212 12
ˆˆ ˆˆ
pq pq
where Ez 11 22 Matched pairs (dependent samples)
= +
α/2 nn
12 Confidence Interval
dE−+µ dE
ˆ ˆ ˆ ˆˆ ˆ < <
p rnp rn q pq p
/ ; / and 1 ; 1
= = −== − d
111222 1 12 2
s
d
Et=n−
where α/2 n with d.f. = 1
Hypothesis Test:
ˆˆ Hypothesis Test
pppp
−−−
()()
z = 1212
d −µd
t= df=n −
pq + pq s with . . 1
nn d
12 n
ed proportion is p
where the pool Two Sample Variances
+ 22
rr
12 Confidence Interval for σ and σ
= = − 12
p and qp1
+ 2 22
nn
12 σ
ss
11
1 11
• <<•
2 22
FF
σ
ss
22
ˆˆ right 2 left
= =
prnprn
/ ; /
111222
s2
Hypothesis Test Statistic: 1 where 22
F=ss≥
s2 12
2
numerator . . 1 and denominator . . 1
df=n−=df n−
12
4
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