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Frequently Used Statistics Formulas and Tables
Chapter 2
highest value - lowest value
Class Width = (increase to next integer)
number classes
upper limit + lower limit
Class Midpoint =
Chapter 3
sample size
population size
frequency
n
N
f
sum
w weight
Sample mean:
Population mean:
Weighted mean:
Mean for frequency table:
highest value + lowest value
Midrange
x
x
n
x
N
w x
x
w
f x
x
f
Range = Highest value - Lowest value
Sample standard deviation:
Population standard deviation:
Sample variance:
Population variance:
x x
s
n
x
N
s
Chapter 3
Limits for Unusual Data
Below : - 2
Above: 2
μ σ
μ + σ
Empirical Rule
About 68%: - to
About 95%: -2 to 2
About 99.7%: -3 to 3
μ σ μ σ
μ σ μ σ
μ σ μ σ
Sample coefficient of variation: 100%
Population coefficient of variation: 100%
Sample standard deviation for frequency table:
[ ( ) ] [ ( ) ]
s
CV
x
CV
n f x f x
s
n n
Sample z-score:
Population z-score:
x x
z
s
x
z
Interquartile Range: (IQR)
Modified Box Plot Outliers
lower limit: Q - 1.5 (IQR)
upper limit: Q + 1.5 (IQR)
= Q − Q
2
Chapter 4
Probability of the complement of event
Multiplication rule for independent events
General multiplication rules
A
P not A P A
P A and B P A P B
P A and B P A P B given A
Addition rule for mutually exclusive events
General addition rule
P A and B P A P A given B
P A or B P A P B
P A or B P A P B P A and B
! Permutation rule: ( )!
n r
n P n r
= −
! Combination rule: !( )!
n r
n C r n r
= −
Permutation and Combination on TI 83/
n Math PRB nPr enter r
n Math PRB nCr enter r
Note: textbooks and formula
sheets interchange “r” and “x”
for number of successes
Chapter 5
Discrete Probability Distributions:
2 2
Mean of a discrete probability distribution:
[ ( )]
Standard deviation of a probability distribution:
[ ( )]
x P x
x P x
Binomial Distributions
number of successes (or x)
probability of success
= probability of failure
1 = 1
Binomial probability distribution
( )
Mean:
Standard deviation:
r n r n r
r
p
q
q p p q
P r C p q
np
npq
μ
σ
−
=
=
= − +
=
=
=
Poisson Distributions
2
number of successes (or )
= mean number of successes (over a given interval)
Poisson probability distribution
( ) !
(over some interval)
r
r x
e P r r
e
mean
μ
μ
μ
μ
σ μ
σ μ
−
= = ≈ = = =
4
Chapter 8
One Sample Hypothesis Testing
for ( 5 and 5) :
where 1 ; ˆ /
for ( known):
for ( unknown): with.. 1
for : with.. 1
p p
p np nq z
pq n
q p p r n
x
z
n
x
t d f n
s n
n s
d f n
μ μ σ σ
μ μ σ
σ χ σ
Chapter 9
Two Sample Confidence Intervals
and Tests of Hypotheses
Difference of Proportions ( p 1 − p 2 )
Confidence Interval:
where
ˆ / ; ˆ / and ˆ 1 ˆ ; ˆ 1 ˆ
Hypothesis Test:
( ˆ^ ˆ ) ( )
where the pool
p p E p p p p E
p q p q
E z
n n
p r n p r n q p q p
p p p p
z
pq pq
n n
ed proportion is
and 1
p
r r
p q p
n n
p r n p r n
Chapter 9
Difference of means μ -μ (independent samples)
1 2 1 2 1 2 1 2 2 2 1 2 / 2 1 2
1 2 1 2 1 2 2 2 1 2 1 2
Confidence Interval when and are known
where
Hypothesis Test when and are known
x x E x x E
E z
n n
x x
z
n n
α
1 2
1 2 1 2 1 2 2 2 1 2 / 2 1 2
1 2
1 2
1 2
Confidence Interval when and are unknown
( ) ( ) ( )
with.. = smaller of 1 and 1
Hypothesis Test when and are unknown
( ) (
x x E x x E
s s E t n n
d f n n
x x t
α
− − < − < − +
= +
− −
− − = 1 2 2 2 1 2 1 2
1 2
)
with.. smaller of 1 and 1
s s n n
d f n n
= − −
Matched pairs (dependent samples)
Confidence Interval
where with d.f. = 1
Hypothesis Test
with.. 1
d d
d d
d E d E
s
E t n
n
d
t d f n
s
n
α
Two Sample Variances
2 2
2 2 1 2 2 2 2 1 1 1 2 2 2 2
2 1 2 2 2 1 2 2 1 2
Confidence Interval for and
1 1
Hypothesis Test Statistic: where
numerator.. 1 and denominator.. 1
right left
s s s F s F
s F s s s d f n d f n
• < < •
= ≥
= − = −
5
Chapter 10
Regression and Correlation
2 2 2 2
2
Linear Correlation Coefficient (r)
( )( )
( ) ( ) ( ) ( ) OR ( ) where z score for x and z score for y 1
explained variation Coefficient of Determination: total v
x y x y
n xy x y r n x x n y y
z z r z z n
r
∑ − ∑ ∑
∑ − ∑ ∑ − ∑
Σ = = = −
=
2
2 0 1
2 0 / 2 (^2 )
2
ariation
( ˆ) Standard Error of Estimate: s 2
or s 2
Prediction Interval: ˆ ˆ
1 ( ) where 1 ( ) ( )
Sample test statistic for
with 1 2
e
e
e
y y n y b y b xy n
y E y y E
n x x E t s n n x x
r r t r n
α
∑ −
− ∑ − ∑ − ∑ = −
− < < +
− = + + Σ − Σ
= − −
d f.. = n − 2
Least-Squares Line (Regression Line or Line of Best Fit)
0 1 0 1
1 2 2 1
2 0 2 2 0 1
ˆ (^) note that is the y-intercept and is the slope
( )( ) where or ( ) ( )
( )( ) ( )( ) where or ( ) ( )
y x
y b b x b b
n xy x y s b b r n x x s and y x x xy b b y b x n x x
= +
∑ − ∑ ∑ = = ∑ − ∑
∑ ∑ − ∑ ∑ = = − ∑ − ∑
0 0 0 0 2 / 2 (^2) 2
1 1 1 1
/ 2 (^2) 2
Confidence interval for y-intercept
1 where E = ( )
Confidence interval for slope
where E = ( )
e
e
b E b E
x t s n (^) x x n
b E b E s t x x n
α
α
− < < +
∑ ∑ −
− < < +
Chapter 11
2 (^ )^ (row total)(column total)
where
sample size
Tests of Independence.. ( 1)( 1)
Goodness of fit.. (number of categories) 1
O E
E
E
d f R C
d f
Chapter 12
One Way ANOVA
all groups
2 2
all groups
number of groups; total sample size
where.. 1
TOT
TOT
TOT
i TOT BET i
i W i i
TOT BET W
BET
BET BET
BET
W
W
k N
x
SS x
N
x x
SS
n N
x
SS x
n
SS SS SS
SS
MS d f k
d f
SS
MS
d
where..
W
W
d f N k
f
where.. numerator =.. 1
.. denominator =..
number of rows; number of columns row factor Row factor : error column factor Column factor :
BET BET W W
MS F d f d f k MS d f d f N k
r c MS F MS MS F MS
= = −
= −
= =
Two - Way ANOVA
error interaction Interaction : error
with degrees of freedom for row factor = 1 column factor = 1 interaction = ( 1)( 1) error = ( 1)
MS F MS
r c r c rc n
− − − − −
critical z-values for hypothesis testing
α = 0.
c-level = 0.
≠
z = - 1.645 z = 0 z = 1.
z = - 1.28 z = 0
z = 0 z = 1.
α = 0. 05
c-level = 0.9 5
≠
z = - 1. 645 z = 0
z = 0 z = 1. 645
Figure 8.
α = 0. 01
c-level = 0.9 9
≠
z = - 2 .575 z = 0 z = 2.
z = - 2.33 z = 0
z = 0 z = 2. 33
z = - 1.96 z = 0 z = 1.
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