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Frequently Used Statistics Formulas and Tables
Chapter 2
highest value - lowest valueClass Width = (increase to next integer)number classes
upper limit + lower limitClass Midpoint = 2
Chapter 3
sample size population size frequency
nNf
sumw weight
===
Σ ==
Sample mean:
Population mean:
( )Weighted mean:
( )Mean for frequency table:
highest value + lowest valueMidrange2
xxn
xNw xx
wf xx
f
µ
∑=
∑=
∑ •=
∑∑ •
=∑
=
2
2
2
2
Range = Highest value - Lowest value
( )Sample standard deviation: 1
( )Population standard deviation:
Sample variance:
Population variance:
x xsn
xN
s
µσ
σ
∑ −=
−
∑ −=
Chapter 3 Limits for Unusual Data Below : - 2 Above: 2
µ σµ σ+
Empirical Rule
About 68%: - to About 95%: -2 to 2About 99.7%: -3 to 3
µ σ µ σµ σ µ σµ σ µ σ
+++
2 2
Sample coefficient of variation: 100%
Population coefficient of variation: 100%
Sample standard deviation for frequency table:
[ ( ) ] [ ( ) ] ( 1)
sCVx
CV
n f x f xsn n
σµ
=
=
∑ • − ∑ •=
−
Sample z-score:
Population z-score:
x xzs
xz µσ
−=
−=
3 1
1
3
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 ( ) = 1 - ( )
Multiplication rule for independent events ( ) ( ) ( )
General multiplication rules ( ) ( ) ( , )
AP 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
nPn r
=−
!Combination rule:
!( )!n rnC
r n r=
−
Permutation and Combination on TI 83/84 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
µ
σ µ
= ∑ •
= ∑ • − Binomial Distributions
number of successes (or x) probability of success
= probability of failure1 = 1
Binomial probability distribution ( )
Mean:
Standard deviation:
r n rn r
rpqq 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
( )!
2.71828
(over some interval)
r
r x
eP rr
e
mean
µ
µ
µ
µ
σ µ
σ µ
−
=
=
≈
=
=
=
3
Chapter 6 Normal Distributions
Raw score:
Standard score:
x z
xz
σ µ
µσ
= +
−=
Mean of distribution:
Standard deviation of distribtuion:
(standard error)
Standard score for : /
x
x
x
xn
xx zn
µ µ
σσ
µσ
=
=
−=
Chapter 7
One Sample Confidence Interval
/ 2
for proportions ( ) : ( 5 and 5)
ˆ ˆ (1 ) where
ˆ
p np nq
p E p p Ep pE z
nrpn
α
> >
− < < +−
=
=
/ 2
/ 2
for means ( ) when is known:
where
for means ( ) when is unknown:
where
with . . 1
x E x E
E zn
x E x EsE tn
d f n
α
α
µ σ
µσ
µ σ
µ
− < < +
=
− < < +
=
= −
Chapter 7
Confidence Interval: Point estimate ± error Point estimate = Upper limit + Lower limit 2 Error = Upper limit - Lower limit 2
2/ 2
2/ 2
2/ 2
means:
proportions:
ˆ ˆ with preliminary estimate for
0.25 without preliminary estimate for
zn
E
zn pq p
E
zn p
E
α
α
α
σ =
=
=
Sample Size for Estimating
variance or standard deviation: *see table 7-2 (last page of formula sheet) Confidence Intervals
Level of Confidence z-value ( / 2zα ) 70% 1.04 75% 1.15 80% 1.28 85% 1.44 90% 1.645 95% 1.96 98% 2.33 99% 2.58
2 22 2
2 2
( 1) ( 1)for variance ( ) : <
with . . 1R L
n s n s
d f n
σ σχ χ− −
<
= −
4
Chapter 8
One Sample Hypothesis Testing
22 2
2
ˆfor ( 5 and 5) :
/
ˆ where 1 ; /
for ( known): /
for ( unknown): with . . 1/
( 1)for : with . . 1
p pp np nq zpq n
q p p r n
xzn
xt d f ns n
n s d f n
µµ σσ
µµ σ
σ χσ
−> > =
= − =
−=
−= = −
−= = −
Chapter 9
Two Sample Confidence Intervals and Tests of Hypotheses
1 2p p−Difference of Proportions ( )
1 2 1 2 1 2
1 1 2 2/ 2
1 2
1 1 1 2 2 2 1 1 2 2
1 2 1 2
1 2
Confidence Interval:
ˆ ˆ ˆ ˆ ( ) ( ) ( )
ˆ ˆ ˆ ˆ where
ˆ ˆ ˆ ˆ ˆ ˆ / ; / and 1 ; 1
Hypothesis Test:
ˆ ˆ( ) ( )
where the pool
p p E p p p p E
p q p qE zn n
p r n p r n q p q p
p p p pzpq pqn n
α
− − < − < − +
= +
= = = − = −
− − −=
+
1 2
1 2
1 1 1 2 2 2
ed proportion is
and 1
ˆ ˆ / ; /
p
r rp q pn n
p r n p r n
+= = −
+
= =
Chapter 9
21Difference of means μ -μ (independent samples)
1 2
1 2 1 2 1 22 21 2
/ 21 2
1 2
1 2 1 22 21 2
1 2
Confidence Interval when and are known ( ) ( ) ( )
where
Hypothesis Test when and are known( ) ( )
x x E x x E
E zn n
x xz
n n
α
σ σµ µσ σ
σ σµ µ
σ σ
− − < − < − +
= +
− − −=
+
1 2
1 2 1 2 1 2
2 21 2
/ 21 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 sE tn n
d f n n
x xt
α
σ σ
µ µ
σ σ
µ
− − < − < − +
= +
− −
− −= 1 2
2 21 2
1 2
1 2
)
with . . smaller of 1 and 1
s sn n
d f n n
µ−
+
= − −
Matched pairs (dependent samples)
/ 2
Confidence Interval < <
where with d.f. = 1
Hypothesis Test
with . . 1
d
d
d
d
d E d Es
E t nn
dt d f n
sn
α
µ
µ
− +
= −
−= = −
Two Sample Variances
2 2
2 21 2
2 2 21 1 12 2 2
2
22 211 22
2
1 2
Confidence Interval for and
1 1
Hypothesis Test Statistic: where
numerator . . 1 and denominator . . 1
right left
s sF Fs s
sF s ss
d f n d f n
σ σ
σσ
• < < •
= ≥
= − = −
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 y1
explained variationCoefficient of Determination: total v
x yx y
n xy x yrn x x n y y
z zr z z
n
r
∑ − ∑ ∑=
∑ − ∑ ∑ − ∑
Σ= = =
−
=
2
20 1
20
/ 2 2 2
2
ariation
ˆ( )Standard Error of Estimate: s2
or s2
ˆ ˆPrediction Interval:
( )1where 1( ) ( )
Sample test statistic for
with 1
2
e
e
e
y yn
y b y b xyn
y E y y E
n x xE t s
n n x x
rrt
rn
α
∑ −=
−∑ − ∑ − ∑
=−
− < < +
−= + +
Σ − Σ
=−−
. . 2d f n= −
Least-Squares Line (Regression Line or Line of Best Fit)
0 1 0 1
1 12 2
2
0 0 12 2
ˆ note that is the y-intercept and is the slope
( )( ) where or ( ) ( )
( )( ) ( )( ) where or ( ) ( )
y
x
y b b x b b
sn xy x yb b rsn x x
andy x x xyb b y b x
n x x
= +
∑ − ∑ ∑= =
∑ − ∑
∑ ∑ − ∑ ∑= = −
∑ − ∑
0
0 0 02
/ 2 22
1
1 1 1
/ 2 22
Confidence interval for y-intercept
1 where E = ( )
Confidence interval for slope
where E = ( )
e
e
b E b Ext s
n xxn
b E b Es
txx
n
α
α
ββ
ββ
− < < +
+∑
∑ −
− < < +
•∑
∑ −
Chapter 11 2
2 ( ) (row total)(column total) where sample size
Tests of Independence . . ( 1)( 1)
Goodness of fit . . (number of categories) 1
O E EE
d f R C
d f
χ −= ∑ =
= − −
= −
Chapter 12 One Way ANOVA
22
2 2
all groups
22
all groups
number of groups; total sample size
( )
( ) ( )
( )
where . . 1. .
TOTTOT
TOT
i TOTBET
i
iW i
i
TOT BET W
BETBET BET
BET
WW
k N
xSS x
N
x xSS
n N
xSS x
n
SS SS SS
SSMS d f kd f
SSMS
d
= =
∑= ∑ −
∑ ∑= −
∑= ∑ −
= +
= = −
=
∑
∑
where . .. . W
W
d f N kf
= −
where . . numerator = . . 1
. . denominator = . .
number of rows; number of columns row factor Row factor :
error column factor Column factor :
BETBET
W
W
MSF d f d f kMS
d f d f N k
r cMSF
MSMSF
MS
= = −
= −
= =
Two - Way ANOVA
error interaction Interaction :
error
with degrees of freedom for row factor = 1 column factor = 1 interaction = ( 1)( 1) error = ( 1)
MSFMS
rc
r crc n
−−
− −−
critical z-values for hypothesis testing
α = 0.10c-level = 0.90
≠
z = - 1.645 z = 0 z = 1.645
z = - 1.28 z = 0
z = 0 z = 1.28
α = 0.05c-level = 0.95
≠
z = - 1.645 z = 0
z = 0 z = 1.645
Figure 8.4
α = 0.01c-level = 0.99
≠
z = - 2.575 z = 0 z = 2.575
z = - 2.33 z = 0
z = 0 z = 2.33
z = - 1.96 z = 0 z = 1.96
0.0250.025
>
<< <
> >
0.050.05
0.10
0.10
0.005 0.005
0.01
0.010.05
0.05
Greek Alphabet
http://www.keyway.ca/htm2002/greekal.htm
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