Math for 800 06 statistics, probability, sets, and graphs-charts
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Transcript of Math for 800 06 statistics, probability, sets, and graphs-charts
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CONTENTS
STATISTICS
How
numerical
data are
distributed.
FREQUENCY DISTRIBUTION
CUMULATIVE FREQUENCY
cumulative
frequency
4
13
19
26
29
31
{2, 0, 2, 0, 1, 1, 2, 1, 4, 0, 2, 2, 1, 1, 1}
x Frequency
2 2
1 4
0 3
1 2
2 3
4 1
Total 15
MEASURES OF CENTRAL TENDENCY
Aritmetic Mean
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.
Mode
The mode of a set of numbers is the value
that appears most often.
sum of termsaverage
number of terms
AVERAGE(ARITHMETIC MEAN)
EVENLY SPACED
TERMS
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 8 126
2 2
10, 20, 30 , 40 , 50
10 50 6030
2 2
12, 14, 16, 18, 20, 22
12 22 3417
2 2
300, 400, 500, 600, 700, 800
300 800 1,100550
2 2
sum of termsaverage
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
WEIGHTED AVERAGE
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
MEASURES OF
CENTRAL
TENDENCY
MEASURES OF
DISPERSION
Range
The difference between the highest
and the lowest values.
Range
Standard Deviation
RANGE
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:
Take the
measures.
STEP 2:
Find the Mean.
STEP 3:
Calculate the
differences from
the Mean.
NORMAL DISTRIBUTION
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 PERCENTILE
The 20th percentile (2nd decile) is the value (or score) below which 20 percent of the observations may be found.
D1 D2 D3 D4 D5 D6 D7 D8 D9
10% 20% 30% 40% 50% 60% 70% 80% 90%
Quartiles
Are the three points that divide the data set into four equal groups, each representing a fourth of
the population being sampled.
QUARTILE
FIRST QUARTILE
Designated Q1 = lower quartile =
cuts off lowest 25% of data =
25th percentile
Q1 Q2 Q3
SECOND QUARTILE
Designated Q2 = median =
cuts data set in half =
50th percentile
Q1 Q2 Q3
THIRD QUARTILE
Q1 Q2 Q3
INTERQUARTILE RANGE - IQR
Is a measure of statistical dispersion,
being equal to the difference between
the upper and the lower quartiles.
3 1IQR Q Q
BOX AND WHISKER PLOT
Min: smallest observation (sample minimum),
Q1: lower quartile,
Q2: median,
Q3: upper quartile, and
Max: largest observation (sample maximum).
MAX: Q3 + 1.5 IQR (the highest value)
MIN: Q1 – 1.5 IQR (the lowest value)
A boxplot may also indicate which
observations, if any, might be considered
outliers.
Hours of excercise per week
2 4 6 8 10 120
Q1
25%
Q3
75%
Q2 (median)
50%
MAXMIN
IRQ = Q3 – Q1 = 6 – 2 = 4
Political Bent
(0 = most conservative, 100 = most liberal)
10 20 30 40 50 600
Q1 Q3Q2 MAXMIN
70 80 90 100
Outliers Outliers
Correlationcoefficients measure
the strength of association between
two variables.
PEARSON CORRELATIONCOEFFICIENT
CORRELATION
LINEAR REGRESSION
Linear regression
attempts to
model the
relationship
between two
variables by fitting
a linear equation
to observed data.
STATISTICS
COUNTING
METHODS
COUNTING POSSIBILITIES
COUNTING POSSIBILITIES
COUNTING POSSIBILITIES
Fundamental Counting Principle
COUNTING POSSIBILITIES
REPETITION
COUNTING POSSIBILITIES
REPETITION
n! (n factorial)
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 nP n
3-distinct color
patterns
3! 3 2 1 6
4! 4 3 2 1 24
!n nP n
4-distinct color patterns
123456789101112131415161718192021222324
21 21 21!P
!n nP 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
DCBA
4 44!
4 3 2 1
24
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?
3 33!
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?
Permutation of n
things taken r at a time.
!
!n r
nP
n r
6 4
6!
6 4 !P
6 4360P
8 3
8!
8 3 !P
8 3336P
7 3
7!
7 3 !P
7 3210P
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
nP
n r
Circular Permutation
Circular Permutation of
n things taken at a time.
!1 !
nn
n
5!
5 1 !5
4! 24
14!
14 1 !14
14!13!
14
Permutation With Repetition
Is an arrangement of n items, of
which p are alike and q are alike,
in some specific order.
!
! !
n
p q
4!
2!
8!
2!
10!
3!
11!
2! 3!
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
A Combination
52 5
52!
5! 52 5 !C
!
! !n r
nC
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
nC
r n r
7 3
7!35
3! 4!C
COUNTING
METHODS
PROBABILITY
Probability
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
PROBABILITY
For 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:
Favorable outcomes of E
P ETotal number of possible outcomes
COMMON PROBABILITIES
Picking a card in a standard deck
1
52
Throwing a die
1
6
Flipping a coin
1
2
SUM OF PROBABILITIES
If two or more events constitute
all the outcomes, the sum of
their probabilities is 1.
20 10 70 1001
100 100 100 100
NON-OCCURRENCE PROBABILITY
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 EVENTS
To find the probability of occurrence
of both, find each probability
separately and multiply consecutive
probabilities.
P E and F P E P F
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 2p red and red p red p red
1 1
5 5
1
25
DEPENDENT EVENTS
To find the probability of occurrence
of both, find each probability
separately and multiply consecutive
probabilities.
P E and F P E P F
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 2p 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 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 x is to be chosen at random from the set
{1, 2, 3, 4} and y is to be chosen at random
from the set {5, 6, 7}, what is the probability
that xy will be even?
x
1 2 3 4
y
5 5 10 15 206 6 12 18 247 7 14 21 28
8 2
12 3
If a person rolls two dice, what is the
probability of getting a prime number as
the sum of the two dice?
15 5
36 122, 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
PROBABILITY
SETS
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 SETS
The 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, 7 2, 4, 6
UNION OF SETS
INTERSECTION OF TWO SETS
The 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, 7 2, 4, 6
A B = { 3, 5 }
Neither
set A only
set B only
both sets A and B
neither A nor B
Set A
Set B
VENN DIAGRAMS
A B
a bc dBoth
ADDITION RULE FOR TWO SETS
A B A B A B
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 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
3 SETS
VENN
DIAGRAMS
UNION
A = { 1, 3, 5, 7 }
B = { 2, 3, 4, 5, 6 } 5
3
A B C =
{ 1, 2, 3, 4, 5, 6, 7, 8, 9 }
c = { 5, 6, 7, 8, 9 }
1
7
2, 4
6
8, 9
INTERSECTION
A = { 1, 3, 5, 7 }
B = { 2, 3, 4, 5, 6 } 5
3
A B C = { 5 }
c = { 5, 6, 7, 8, 9 }
1
7
2, 4
6
8, 9
SET A
A = { 1, 3, 5, 7 }
B = { 2, 3, 4, 5, 6 }
c = { 5, 6, 7, 8, 9 }
1
7
2, 4
6
8, 9
3
5
ONLY A
A = { 1, 3, 5, 7 }
B = { 2, 3, 4, 5, 6 }
c = { 5, 6, 7, 8, 9 }
1
7
2, 4
6
8, 9
3
5
A B C = { 1 }
A OR B, NOT C
A = { 1, 3, 5, 7 }
B = { 2, 3, 4, 5, 6 }
c = { 5, 6, 7, 8, 9 }
1
7
2, 4
6
8, 9
3
5
B C A=
{ 2, 4, 6, 8, 9 }
A AND C
A = { 1, 3, 5, 7 }
B = { 2, 3, 4, 5, 6 }
c = { 5, 6, 7, 8, 9 }
1
7
2, 4
6
8, 9
3
5
A C = { 5, 7 }
A AND CB BUT NOT B
A = { 1, 3, 5, 7 }
B = { 2, 3, 4, 5, 6 }
c = { 5, 6, 7, 8, 9 }
1
7
2, 4
6
8, 9
3
5
A C B = { 7 }
SETS
GRAPHS
DEALING WITH GRAPHS & CHARTS
Before 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
GRAPHS
SUMMARY
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