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

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