Published on 4 Jun 2015

School

Department

Course

Professor

Statistics

Statistics

SAMPLES PROBABILITY THEORY

PROPERTIES OF SAMPLED DATA

Parameter Ungrouped Data Data Grouped in kClasses

Note: Standard Deviation = s=

Range of sample:

xmax – xmin

Mode: Most common

value of x

Median: Middle value or

mean of two middle values

Coefficient of variation:

V= (s/x

–)•100

Quartiles Q25, Q50, Q75:

Values that divide data set into

4 equal parts

Interquartile range: Q75 – Q25

• Rationalize xiby first subtracting a convenient number m

(often the median or mode) followed by division by a

suitable number c(often the class size)

• New data set uiyields mean x

–and standard deviation sx

of original set:

x

–= cu

–+ mand sx= c•su

Example

60–62 5 61 –2 –10 20

63–65 18 64 –1 –18 18

66–68 42 67 (m) 000

69–71 27 70 1 27 27

72–74 8 73 2 16 32

Notes: u

–is the mean of

the new set • x

–is the

mean of the original set

• sxis the standard

deviation of original set

Mean

Variance

Variance

(alternate)

THE CODE METHOD

variance

Class Frequency, fixiui= (xi– m)/cf

iuifiui2

ufun

xcum

scscfu

n

fu

nn

ii

xu ii ii

== =

=+= +=

=⋅ = −−−=

∑

∑∑

// .

(. ) .

()

().

15 100 0 15

3015 67 6745

11

297

22

FACTORIAL FUNCTIONS

nnn n!()( )()()()=−−12321

n Factorial

nnne

n

!(/)≅2

π

Stirling’s Approximation

n

r

n

rnr

n

=−

=

=

!

!( )! 0

0

01

Binomial Coefficients

Note: 0! = 1

PROBABILITY THEORY

Probability P(S) of “success” = number of events considered

successes (S) ⁄ total number of possible events (n)

Probability P(F) of “failure” = 1 – P(S)

• The probability P(A«B) that two independent events A and B

will both occur, simultaneously or in succession, is:

Example: The probability of drawing a queen

and a heart from two separate decks is:

• The combined probability P(A»B) that one or the other of two

mutually exclusive events Aand Bwill occur is:

Example: The probability of drawing a

queen or a jack from a single deck is:

• The combined probability P(A»B) that one or the other or

both of the two events Aand B will occur is:

Example: The probability of drawing a queen or a heart

or the queen of hearts

from a single deck is:

• The probability that event Boccurs given that event Ahas

already occurred is called the conditional probability P(B|A)

• The probability of the sequence occurring is:

Example: The probability of drawing a queen

and a jack in succession from a single deck is:

Rule 4 Rule 3 Rule 2 Rule 1

PA PB()()⋅

PA PB()()+

PA PB PA B()()( )+−∩

PAB PA PB A()()(|)= ⋅

PA B( )∪= + − ⋅

()

=

4

52

13

52

4

52

13

52

4

13

PAB( ) =⋅=

4

52

4

51

4

663

PA B( )∪= + =

4

52

4

52

2

13

PA B( )∩= ⋅=

4

52

13

52

1

52

PERMUTATIONS & COMBINATIONS

Rule 3 Rule 2 Rule 1

• The number of permutations (= arrange ments including

changes in order) of ndistinct objects

taken all at a time is:

• Number of permutations

of ndistinct objects taken

rat a time is:

• Number of combinations

(= arrangements omitting

changes in order) of ndistinct

objects taken rat a time is:

nr

Pnnr= −!()!

nr

Cn

r

n

rnr

=

=−

!

!( )!

nr r

Pn=

nr

Cnr

r

nr

rn

=+−

=+−

−

1

1

1

()!

!( )!

Without With Repetition

Repetition of Objects

Example: Given the following 3 objects (lJD), n= 3, r= 2

(two taken at a time)

32

32

3

32 6

3

23 2 3

P

C

=−=

=

=−=

=

!

()!

!

!( )!

●✤ ✤✪ ●✪

✤● ✪✤ ✪●

●✤ ●✪ ✪✤

32 2

32

39

321

23 1 6

P

C

==

=+−

−=

()!

!( )!

add

to the outcome at left

✤✤ ✪✪ ●●

Without

Repetition

With Repetition of Objects

xx

n

i

=∑

x

xf

n

ii

k

=

∑

1

sxx

n

i

22

1

=−

−

∑()

s

fx x

n

ii

k

2

2

1

1

=−

−

∑()

snx x

nn

ii

222

1

=−

−

∑∑ ()

()

s

nxf xf

nn

ii ii

kk

2

2

1

2

1

1

=

−

−

∑∑

()

nn

Pn=!

Population Totality of objects, individuals, and

events

Sample Part of population chosen for

measurement and observation

• Number of sample values is n

Class Subdivision of sample values xi

Class mark, xiMidpoint value of a class

Class size, cDifference between succes sive

lower or upper class limits

Class frequency, fNumber of sample values in a class

Relative frequency, f/n Class frequency divided by total

number of values n

Cumulative frequency Total number of values less than

given value

STATISTICS • 1-55080-620-31

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