Math for 800 06 statistics, probability, sets, and graphs-charts

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Hownumericaldata are distributed. FREQUENCY DISTRIBUTION

CUMULATIVE FREQUENCY
cumulative
frequency41319262931

{2, 0, 2, 0, 1, 1, 2, 1, 4, 0, 2, 2, 1, 1, 1} xFrequency 22 1403122341 Total15

The median of a set of numbers is the value that falls in the middle of the set.

You must first list the values in increasing or decreasing order. MEDIAN

Median
The median is not affected by extreme values.

ModeThe mode of a set of numbers is the value that appears most often.

sum of terms
average
number of terms

AVERAGE
(ARITHMETIC MEAN)

Just average the smallest and the largest terms. AVERAGE OF EVENLY SPACED TERMS

Arrange the terms in ascending or descending order:
•the number in the middle of the list,
•the average (arithmetic mean) of the two numbers in the middle of the list. AVERAGE OF EVENLY SPACED TERMS

4, 5, 6, 7, 8
4 8 12
6
2 2

 

10, 20, 30 , 40 , 50
10 50 60
30
2 2

 

12, 14, 16, 18, 20, 22
12 22 34
17
2 2

 

300, 400, 500, 600, 700, 800
300 800 1,100
550
2 2

 

sum of terms
average
number of terms

average number  of terms  sum of terms
SUM OF TERMS

What is the sum of the integers from 10 to 50,
inclusive?
averagenumber of terms  sum of terms
10 50
2

 50 10 1 
3041 1,230

The girl’s average score is 30. The boy’s
average score is 24. If there are twice as
many boys as girls, what is the overall
average?
1 30 2 24
3
26
average
average
  



Range
The difference between the highest and the lowest values.

Variance
A measure of the average distance between each of a set of data points and their mean value.

The variance is the square of the standard deviation. VARIANCE
2v

STEP 1:
Takethemeasures.
STEP 2:
FindtheMean.
STEP 3:
CalculatethedifferencesfromtheMean.

Abraham de Moivre(1667-1754). Carl Friedrich Gauss(1777-1855).
Bell shaped curve.

Symmetry about the center. mean = median = mode

EMPIRICAL RULE68− 95 − 99.7

The 20th percentile is the value (or score) below which 20 percent of the observations may be found.

DECILE AND PERCENTILEThe 20th percentile (2nd decile) is the value (or score) below which 20 percent of the observations may be found. D1D2D3D4D5D6D7D8D910%20%30%40%50%60%70%80%90%

QuartilesAre the three points that divide the data set into four equal groups, each representing a fourth of the population being sampled.

FIRST QUARTILE
Designated Q1= lower quartile = cuts off lowest 25% of data = 25th percentileQ1Q2Q3

SECOND QUARTILE
Designated Q2= median = cuts data set in half =
50th percentileQ1Q2Q3

INTERQUARTILE RANGE
- IQR
Is a measure of statistical dispersion,
being equal to the difference between
the upper and the lower quartiles.
3 1 IQR Q Q

BOX AND WHISKER PLOTMin:smallest observation (sample minimum), Q1:lower quartile, Q2:median, Q3:upper quartile, andMax:largest observation (sample maximum).

MAX: Q3+ 1.5 IQR (thehighestvalue)
MIN:Q1–1.5 IQR (thelowestvalue)

A boxplot may also indicate which observations, if any, might be considered outliers.

Hoursof excerciseper week
2
4
6
8
10
12
0
Q1
25%
Q3
75%
Q2 (median)
50%
MAX
MIN
IRQ = Q3 –Q1 = 6 –2 = 4

PoliticalBent
(0 = mostconservative, 100 = mostliberal)
10
20
30
40
50
60
0
Q1
Q3
Q2
MAX
MIN
70
80
90
100
Outliers
Outliers

Correlationcoefficients measure the strength of association between two variables.

PEARSON CORRELATIONCOEFFICIENT

LINEAR REGRESSION
Linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data.

Fundamental Counting Principle

COUNTING POSSIBILITIES REPETITION

n! (nfactorial)
means the product of all the integers from 1 to n inclusive.

!  1  2  3 ... 1
5! 5 4 3 2 1 120
7! 7 6 5 4 3 2 1 5,040
7! 7 6!
7! 7 6 5!
n  n n   n   n   
     
       
 
  
1! 1
0! 1



Permutation
An arrangement of items in some specific order.

! n n P  n
3-distinct color
patterns
3! 321 6

4! 4321 24
! n n P  n
4-distinct color patterns
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

21 21 P  21!
! n n P  n
51,090,942,171,709,440,000

How many different 4-letter arrangements can be formed with all the letters A, B, C, and D?
ABCD
ABDC
ACBD
ACDB
ADBC
ADCB
BACD
BADC
BCAD
BCDA
BDAC
BDCA
CABD
CADB
CBAD
CBDA
CDAB
CDBA
DABC
DACB
DBAC
DBCA
DCAB
DCBA444! 432124P  

There are three marbles: 1 blue, 1 red
and 1 green. In how many ways is it
possible to arrange marbles in a row?
3 3 3!
3 2 1
6
P 
  


There are three marbles: 1 blue, 1 red and 1 green. In how many ways is it possible to arrange marbles in a row if red marble have to be left to blue marble?

Permutationof nthings taken rat a time.
 ! !nrnPnr  

  6 4
6!
6 4 !
P 

6P4  360

  8 3
8!
8 3 !
P 

8P3  336

  7 3
7!
7 3 !
P 

7P3  210

How many different 3-letter arrangements
can be formed with all the letters A, B, C,
D, E, F, and G?
  7 3
7! 7!
210
7 3 ! 4!
P   

 
!
! n r
n
P
n r



Circular Permutation of
n things taken at a time.
 
!
1 !
n
n
n
 
 
5!
5 1 !
5
 
4! 24

 
14!
14 1 !
14
 
14!
13!
14


Permutation With Repetition
Is an arrangement of nitems, of which pare alike and qare alike, in some specific order.
! !! npq

How many nine-letter patterns can be
formed using all the letters of the word
Tennessee?
9!
3,780
4! 2! 2!
e n s

 

  52 5
52!
5! 52 5 !
C 
 
 
!
! ! n r
n
C
r n r

 
2,598,960

How many different 3-letter combinations
can be formed with all the letters A, B, C,
D, E, F, and G?
  7 3
7!
3! 7 3 !
C 
 
 
!
! ! n r
n
C
r n r

 
7 3
7!
35
3! 4!
C  


A variable whose value results from a measurement on some type of random process.

Is a numerical description of the outcome of an experiment. RANDOM VARIABLE

PROBABILITYFor situations in which the possible outcomes are all equally likely, the probability that an event E occurs, represented by “P(E)”, can be defined as:
FavorableoutcomesofEPETotalnumberofpossibleoutcomes 

Picking a card in a standard deck
1
52

SUM OF PROBABILITIESIf two or more events constitute all the outcomes, the sum of their probabilities is 1.

2010701001100100100100 

 
 
20
100
80
100
p red
p not red



Independent Events
Two events are said to be independent if the occurrence or nonoccurrence of either one in no way affects the occurrence of the other.

INDEPENDENT EVENTSTo find the probability of occurrence of both, find each probability separately and multiply consecutive probabilities.
PEandFPEPF

A dresser drawer contains one pair of socks with each
of the following colors: blue, brown, red, white and
black. Each pair is folded together in a matching set.
You reach into the sock drawer and choose a pair of
socks without looking. You replace this pair and then
choose another pair of socks. What is the probability
that you will choose the red pair of socks both times?
      1 2 p red and red  p red  p red
1 1
5 5
 
1
25


DEPENDENT EVENTSTo find the probability of occurrence of both, find each probability separately and multiply consecutive probabilities.
PEandFPEPF

A card is chosen at random from a standard
deck of 52 playing cards. Without replacing it, a
second card is chosen. What is the probability
that the first card chosen is a queen and the
second card chosen is a jack?
      1 2 p queen and jack  p queen  p jack
4 4
52 51
 
4
663


PE or F  PE PFPE and F
PE or F
PE and F
Probability that at least one
of the two events occurs.
Probability that events E
and F both occur.
ADDITION LAW OF
PROBABILITIES

A
A
A
A
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
9
9
9
10
10
10
10
J
J
J
J
Q
Q
Q
Q
K
K
K
K
What is the probability of picking a 9 or a club in an standard deck of cards?
DiamondsHeartsClubsSpades
P (9 or ) = P(9)+P() –P(9 and )
= 4/52 + 13/52 –1/52
= 16/52
= 4/13
9

If xis to be chosen at random from the set {1, 2, 3, 4} and yis to be chosen at random from the set {5, 6, 7}, what is the probability that xywill be even? x
1
2
3
4y
5
5
10
15
20
6
6
12
18
24
7
7
14
21
28
82123 

If a person rolls two dice, what is the
probability of getting a prime number as
the sum of the two dice?
15 5
36 12
2, 3, 5, 7, 11 

If a person rolls two dice, what is the
probability of getting an even number as
the sum of the two dice?
18 1
36 2


SETS
{1, 2, 3} = {2, 3, 1} The objects are called elements of the set.

The order in which the elements are listed in a set does not matter.

UNION OF TWO SETSThe union of two sets is a new set, each of whose elements are in either one or both of the original sets.

A = { 1, 3, 5, 7 } B = { 2, 3, 4, 5, 6 } A B = { 1, 2, 3, 4, 5, 6, 7 } 3, 5 1, 72, 4, 6 UNIONOF SETS

INTERSECTION OF TWO SETSThe intersection of two sets is a new set, whose elements are only those elements shared by the original sets.

A = { 1, 3, 5, 7 } B = { 2, 3, 4, 5, 6 } 3, 5 1, 72, 4, 6 A B = { 3, 5 }

Neitherset A onlyset B onlyboth sets A and Bneither A nor BSet ASet BVENN DIAGRAMSABabcdBoth

ADDITION RULE FOR TWO SETS
ABABAB

ADDITION RULE FOR TWO SETS
A
A
A
A
2
2
2
2
3
3
3
3
4
4
4
4
5
5
5
5
6
6
6
6
7
7
7
7
8
8
8
8
9
9
9
10
10
10
10
J
J
J
J
Q
Q
Q
Q
K
K
K
K
How many 9s or clubs are there in an standard deck of cards? |9 or |= |9| + ||–|9 and | = 4 + 13 –1= 16
DiamondsHearts
ClubsSpades
9

UNIONA = { 1, 3, 5, 7 } B = { 2, 3, 4, 5, 6 }5 3A B C= { 1, 2, 3, 4, 5, 6, 7, 8, 9 } c = { 5, 6, 7, 8, 9 } 172, 468, 9

INTERSECTIONA = { 1, 3, 5, 7 } B = { 2, 3, 4, 5, 6 }5 3A B C= { 5 } c = { 5, 6, 7, 8, 9 } 172, 468, 9

SET AA = { 1, 3, 5, 7 } B = { 2, 3, 4, 5, 6 } c = { 5, 6, 7, 8, 9 } 172, 468, 935

ONLY AA = { 1, 3, 5, 7 } B = { 2, 3, 4, 5, 6 } c = { 5, 6, 7, 8, 9 } 172, 468, 935 A B C= { 1 }

A OR B, NOT CA = { 1, 3, 5, 7 } B = { 2, 3, 4, 5, 6 } c = { 5, 6, 7, 8, 9 } 172, 468, 935 B C A= { 2, 4, 6, 8, 9 }

A AND CA = { 1, 3, 5, 7 } B = { 2, 3, 4, 5, 6 } c = { 5, 6, 7, 8, 9 } 172, 468, 935 A C = { 5, 7 }

A AND CB BUT NOT BA = { 1, 3, 5, 7 } B = { 2, 3, 4, 5, 6 } c = { 5, 6, 7, 8, 9 } 172, 468, 935 A C B = { 7 }

DEALING WITH GRAPHS & CHARTSBefore even reading the questions based on a graph or table, take 10 or 15 seconds to look it over.

Make sure you understand the information that is being displayed, the scales and the units of the quantities involved. DEALING WITH GRAPHS & CHARTS

Math for 800 06 statistics, probability, sets, and graphs-charts

Math for 800 06 statistics, probability, sets, and graphs-charts

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