Download Geometry Test #1-1

Notes #4-___
Date:______
9.1 Basic Combinatorics (669)
Discrete math: distinct values, not continuous (daily attend)
I.
Simple Counting Problems –
You have a bag with a penny, dime, nickel and quarter.
How many ways can the sum be greater than 10¢ if…
A. With Replacement: you select a coin, write down the
value and put the coin back in the bag. You pull out a
coin again and add the value to the first.
1,1
5,1
10,1
25,1
1,5
5,5
10,5
25,5
1,10
5,10
10,10
25,10
1,25
5,25
10,25
25,25
12 ways
B. Without replacement: you select a coin and then
select another coin (or you select two at once)?
1,5
1,10
1,25
Multiple events.
Tree diagram
5,1
5,10
5,25
10,1
10,5
10,25
25,1
25,5
25,10
10 ways
II. Fundamental Counting Principle – the number of ways
that multiple events can occur is the product of the
outcomes (number of elements) of each those events.
Example: Your mom says you can have a pb & j, ham or
tuna sandwich on white or wheat with milk, oj or Pepsi.
How many different lunches could you have?
Ex.1 How many license plates can be made with:
a) 3 different letters & 4 different non-zero digits
b) 3 letters and 4 digits (0-9)
One event.
Factorial!
Define: 0! = 1
III. Permutation – an ordering (ranking) of outcomes.
Example: How many ways can you arrange a family of
five in a row for a photograph?
____
____ ____ ____ ____
A. How many ways can you arrange any two of them?
Also: P(n, r)
____ ____
Which leads to nPr =
n!
.
(n  r )!
Ex.2 How many ways can 8 runners be awarded gold,
silver and bronze medals?
B. Distinguishable (distinct) Permutations
You would expect 4!?
Example: Arrange the letters A, A, B, C.
#1 AABC then switch the As and you get #2 AABC.
How many ways can the As be arranged? ____ ____ 2!
So the formula is specifically
n!
4!
or generally
.
r1 !r2 !...
2!
Ex.3 Distinguishable arrangements of:
a) TENNESSEE
b) GREENEGGS
What happens if you
say the toppings in a
different order when
you buy the pizza?
IV. Combinations – a selection of outcomes.
Example: How many different 2 topping, small, deep
dish pizzas can we get if there are 10 toppings?
How many ways to
arrange 2
toppings?
How is a
combination
lock misleading?
nCr
n
n!
=  
. Why the extra r! compared to nPr?
r
r
!(
n

r
)!
 
Ex.4 How many ways can you choose a committee of 5
members from a student government of 9 girls & 6
boys? What if there had to be 3 girls and 2 guys?
Ex.5 How many ways can you select 4 of your dad’s 20
CDs for a road trip? In how many ways could you
listen to four of the 20 CDs on the trip?
Deck of Cards:
4 suits (2 red, 2 black)
Club, Diamond, Heart, Spade
20 even & 16 odd #s
In each suit:
* 3 face cards (J, Q & K)
* 4 letters (J, Q, K & A)
* 9 #s (2-10)
Ex.6 How many ways can you have:
a) 2 red face cards?
b) 3 even cards?
Factorial Notation: n! = 1 · 2 · 3 ··· n
0! = 1 (this value is defined as such)
4! = 1 · 2 · 3· 4 = 24
(n + 1)! = 1 · 2 · 3 ··· (n - 1) · n · (n + 1)
Ex.7 Evaluate:
Summary:
a)
3!8!
4!5!
c)
(2n  2)!
(2n  4)!
d)
(2n 1)!
(2n )!
e)
(2n 1)!
(2n )!
b)
2n !
n!
Notes #4-___
Date:______
9.2 The Binomial Theorem (678)
Expand:
(x + 1)0 =
1
(x + 1)1 =
1x + 1
(x + 1)2 =
1x2 + 2x + 1
(x + 1)3 =
1x3 + 3x2 + 3x + 1
Pascal’s Δ:
1
1
1
1
1
1
Row 0
1
2
3
Row 1
Row 2
1
3
1
4 6 4 1
5 10 10 5 1
Compare to:
nCr (0, 0) =
nCr (4,
nCr (1,
0) & nCr (1, 1) =
nCr (2,
{0, 1, 2}) =
nCr (3,
{0, 1, 2, 3}) =
{0, 1, 2, 3, 4}) = or
y = nCr (4, x) and use the table.
n
n!
Combination (Binomial coefficient): nCr =   
 r  r !(n  r )!
Ex.1 Evaluate 8C3 by hand.
Ex.2 Expand and simplify
a) (x – 3)4 =
x4 – 12x3 + 54x2 – 108x + 81
b) (2x – 3y)5 =
32x5 – 240x4y + 720x3y2 – 1080x2y3 + 810 xy4 – 243y5
Ex.3 Find the fifth term:
a) (2x + 1)9
4032x5
b) (x – 2)13
9
4
13C4(x) (-2)
= 11440x9
Ex.4 Find the coefficient of the term with a7 in the
expansion of (a – 3b)10.
-3,240
Ex.5 Use (b + g)8 to find the probability of having 4 boys
and 4 girls in a family of 8 children.
9! = 9(9 – 1)! or 9(8!)
n! = n(n – 1)! for n > 1
n  n 
Ex.6 Prove that    

 2   n2 
Summary:
(n + 1)! = (n + 1)n! for n > 0
Notes #4-___
Date:______
9.3 Probability (683)
Probability =
0%
# of outcomes in event
,
# of outcomes in sample space
Impossible
0 < P(E) < 1
P(E) =
Certain
n( E )
n( S )
100%
Ex1) What is the probability of rolling an even # on a
six-sided die?
Outcomes in event: 2, 4 & 6
Outcomes in sample space: 1, 2, 3, 4, 5 & 6
A matrix would
help.
(# outcomes)
(# outcomes)
Ex2) What is the probability of rolling a sum that is prime
on a single roll of two fair six-sided dice?
Deck of 52 Cards:
4 suits (2 red, 2 black)
Club, Diamond, Heart, Spade
20 even & 16 odd #s
In each suit:
* 3 face cards (J, Q & K)
* 4 letters (J, Q, K & A)
* 9 #s (2-10)
Ex3) What is the probability of drawing:
a) 2 red face cards?
cards in event Ccards drawn
52 Ccards drawn
b) 3 even cards?
Mutually exclusive: 2 events that have no common outcomes.
If A ∩ B = Ø then P(A or B) = P(A) + P(B).
Ex4) What is the probability of drawing:
a) a black 7 or a heart? b) even # or a king?
Not mutually exclusive: P(A or B) = P(A) + P(B) – P(A ∩ B)
Ex5) What is the probability of drawing:
a) a queen or a club?
b) red card or a six?
Complementary Event: P(E) + P(E') = 1 so P(E') = 1 – P(E).
Ex6) What is the probability of not drawing:
a) a queen or a club?
b) red card or a six?
Use a Venn
Diagram.
Ex7) 48% of the students at a school are girls and half of
them play sports. 51% of all the students play sports.
a) What % of the students who play sports are boys?
b) If a student is chosen at random, what is the
probability that he is a boy who doesn’t play sports.
Tree diagram helps.
Conditional Probability: the probability of an event that
depends on an earlier event.


P BA 
P( A and B)
P(A)P(B)
P( A B)
or
or
P(A)
P(A)
P(A)
Ex8) A shirt is drawn at random from one of two
identical drawers (drawer A has 3 t-shirts & 2
sweatshirts and drawer B has 2 t-shirts). What is the
probability that a t-shirt was drawn from drawer A?
Probability of a t-shirt from drawer A
Probability of a t-shirt

3 1
5 2
4
5
Binomial Distribution:
[P(E) + P(E')]
n
P(E): Probability event happens
P(E'): Probability it doesn’t happen
n: the # of trials
Ex10) 10% of African-Americans are carriers of the genetic
disease sickle-cell anemia. Find the P(of # carriers) in
a sample of 20 African Americans:
a) P(3)
b) P(at most 2)
3
17
20C17(.1) (.9)
Summary:
2
18
20C18(.1) (.9) +
1
19
20
20C19(.1) (.9) + (.9)
Notes #4-___
Date:______
9.4 Day 1: Sequences (696)
Sequence: Ordered list of numbers (ranked list): a1, a2, a3…
Number
Term
Said
1st
a1
“a” sub 1
Finite Sequence
Terms: 1 , 3 , 5 , 7
2 nd
a2
“a” sub 2
3 rd
a3
“a” sub 3
n th
an
“a” sub n
Infinite Sequence
2, 5, 8, 11, …
The subscript represents the number's place in the list:
a3 (a sub 3) is the third # in our list i.e. a3 = 5.
What type of
number can n be?
Ex.1 Find the first five terms of the sequence given by:
an = 5 + 2n(-1)n.
Ex.2 Write an expression for the apparent nth term of the
2 3 4 5
sequence: , , , , ... a n
1 2 3 4
Recursive definition: given initial term(s), terms are then
defined using the previous term.
Ex.3 a1 = -11
an = an-1 + 5 * an-1 is the number before an
Write the first five terms and find a100.
Ex.4 Write the first five terms:
a1 = 3
an = 2·an-1
Limits of Infinite Sequences
{an} = a1, a2, a3, a4, … an
If the lim an = a finite L then the sequence converges and L
n
is the limit of the sequence. Otherwise it diverges.
Ex.5 Does the sequence converge? If so, find the limit.
1 1 1 1
1
11 12 13 14
a) , , , ,..., ,...
b) , , , ,...
2 4 6 8 2n
1 2 3 4
c) 0.1, 0.2, 0.3, 0.4, …
e)
10n  2
45n
d) 10n – 10
f) 5n
A number (term : an) in an arithmetic sequence is equal to the
number before it (an-1) plus the common difference (d).
d1 = a2 – a1
d2 = a3 – a2
Ex.6 5 , 11 , 17 …
n
an
1
5
2
5+6
dn = an – an-1
Recursive definition:
a1 = 5
an = an-1 + 6
3
5 + 2(6)
4
5 + 3(6)
5
n
5 + 4(6) 5+(n-1)6
Explicit Formula: an = a1 + (n – 1)·d
Ex.7 Find a17 for -3, 4, 11, 18 …
Slope.
Ex.8 In an arithmetic sequence, a3 = 14 and a8 = 44, write the
first five terms and a formula.
a8 = a3 + 5d
Geometric Sequences
3, 6, 12, 24, 48, …
#1. Arithmetic? d1 & d2 = ?
#2. Geometric? r1 & r2 = ?
Common ratio:
n
1
2
3
an
3
6
12
an 3 · 1 3 · 2 3 · 2 · 2
an 3 · 20 3 · 21 3 · 22
r=
an
a
 2
a n 1
a1
4
5
n
24
48
?
3·2·2·2 3·2·2·2·2 3·2·2…
3 · 23
3 · 24
a1 · (r)n-1
Ex.9 Find a10 in 3, 6, 12, 24, 48, …
Ex.10 Find a formula for an and a10 for
a) 1, -1, 1, -1…
b) 4, 2, 1 …
Summary:
What comes next in the pattern?
#1) Fibonacci Sequence: 1, 1, 2, 3, 5, 8 …
#2) 31, 28, 31, 30, …
31 {Days in the months}
#3) J, F, M, A, …
M {Names of Months}
#4) 3, 3, 5, 4, 4, …
3 {# of letters in #s}
#5) Z, O, T, T, F, F, …
S {Whole numbers}
#6) A, E, F, H, …
I {straight letters}
#7) 8, 5, 4, 9, 1, …
7 {Alphabetically}
#8) 7, 8, 5, 5, 3, 4, …
4 {# of letters in months}
#9) S, M, T, W, …
T {Days of the Week}
#10) S, E, Q, U, …
E {the word sequence}
Notes #4-___
Date:______
9.4 Series (701)
Series: the sum of a list of terms: S5 = 2 + 5 + 8 + 11 + 14 = ?
Summation (Sigma) Notation
n
 ai = a1 + a2 + a3 + … + an
(i: index of summation)
i 1
Ex.1 Find the sum:
7
4
b)  ( 1)k (2k )
a)  (23i)
i2

k 1
i
 1
c)  5 
i  1  10 
Karl F. Gauss
(1777-1855)

d)  sin  n 
n 1
6
e)  cos n 
n0
Ex.2 Find the sum of the first 100 natural numbers.
1 + 2 + 3 + … + 98 + 99 + 100 = S100
100 + 99 + 98 + … + 3 + 2 + 1 = S100
101 + 101 + 101 + … + 101 + 101 + 101 = 2S100
100(101) = 2S100
Sn = a1 + (a1+d) + (a1+2d) +…+ (an – 2d) + (an – d) + an
+ Sn = an + (an – d) + (an – 2d) +…+ (a1+2d) + (a1+d) + a1
2Sn = (a1 + an) + (a1 + an) +…+ (a1 + an) + (a1 + an) + (a1 + an)
2Sn = n(a1 + an)
Sn =
n a1 + a n 
2
simplify
n
Sn =  [a1  ( n 1)  d ] 
i 1
n
(a  a )
2 1 n
Ex.3 Find S100 and write in sigma notation: 5, 8, 11, 14…
Ex.4 A theater has 30 seats in the 1st row and 2 more in each
subsequent row. How many seats are there if there are
78 seats in the last row?
Ex.5 Find a10 in 3, 6, 12, 24, 48, …
Ex.6 Find a formula for an and a10 for
a) 1, -1, 1, -1…
b) 4, 2, 1
Derive the Geometric sum formula:
Sn = a1 + a1·r + a1·r2 + … + a1·rn-2 + a1·rn-1
r·Sn = a1·r + a1·r2 + … + a1·rn-2 + a1·rn-1 + a1·rn
Sn – r·Sn = a1 – a1·rn
Sn(1 – r) = a1(1 – rn)
Sn 

a1 1 r n
1 r
=
k
 a1  r 
n 1
n 1
Ex.7 Find S10 for the sequences in Ex.5 & Ex.6.
Ex.8 a4 = 54 & a7 = 1458, find S7 if it is a geometric series.
If │r│ < 1 then the series converges: lim
n
Ex.9 Find S∞ for 4 + 2 + 1 + …
d i a
a1 1 r n
1 r
1
1 r
Ex.10 Find the sum:
n
F
1I
a) 16  GJ
2K
n0 H


b)  cos n 
n 1
Ex.11 Convert the repeating decimal to fraction form.
a) .797979…
b) -3.14141414…
.79 + .0079 + ...
a1 = .79 & r = .01
Formulas:
Sequences
Series
an = a1 + (n – 1)·d
Sn =  [a1  ( n 1)  d ] 
n-1
an = a1 · (r)
n
i 1
Sn 
S 
Summary:
d i
a1 1 r n
1 r
a1
1 r
n
n
(a  a )
2 1 n
 a1(r )n 1
i 1
Notes #4-___
Date:______
9.5 Mathematical Induction (711)
Recall:
Formula for compound interest:
F
r Int
A  PG
1 J
H
nK
A: the balance
P: the principle
r: annual interest rate (as a decimal)
n: compounded this many times (quarterly, monthly..)
t: time in years
Ex.1 How much money will you have in 6 years if you
invest $20 a month @ 5% compounded monthly.
F .05I  1.004
H 12 J
K
n = 12 · 6, a1 = 20 and r  G
1
There is a formula on
(746) that we also
used in 3.6, but the
geometric sum is
better.
S72 = 20 + 20(1.004) +…+ 20(1.004)71 + 20(1.004)72
20(11.00472 )
FV = S72 =
=
11.004
Ex.2 How much money will you have in 10 years if you
invest $50 quarterly @ 7% compounded quarterly?
Step #1: Show that P1 is true.
Step #2: Show that for any positive integer k, if Pk is true,
then Pk+1 is also true. (For sums: Sk+1 = Sk + ak+1)
Prove by Induction:
Ex.3 Sn = 5 + 7 + 9 + 11 + … + (3 + 2n) = n(n + 4)
S1 = 1(1 + 4) = 1(5) = 5 True
If Sk = k(k + 4) is true, then...
Sk+1 = (k + 1)[(k + 1) + 4] = (k + 1)(k + 5) = k2 +6k + 5
Sk = k(k + 4) and ak + 1 = [3 + 2(k + 1)] = 3 + 2k + 2 = 2k + 5
k2 +4k + 2k + 5 = k2 +6k + 5
Ex.4
1  1  1 ... 1  2n  1
2 4 8
2n
2n
21 1 21 1
S1 = 1 =
= True
2
2
2
2k 1
2k  1  1
If Sk =
is true, then... Sk+1 =
k
2
2k  1
2k 1
2 2k 1
1
1
Sk + ak + 1 = k +
=  k +
=
k 1
k 1
2
2
2
2
2
2k  1  2
2k  1  1
1
+
=
2k  1
2k  1
2k  1
Ex.5 5 is a factor of (42n – 1) or (42n – 1) is divisible by 5.
P1: 42(1) – 1 = 15 which is 5(3) True
If Pk: 42k – 1 = 5r then Pk+1: 42(k+1) – 1 = 5s
42(42k – 1) = 425r
42k+2 – 16 + 15 = 80r + 15
42(k+1) – 1 = 5(16r + 3)
42(k+1) – 1 = 5s
42k+2 – 16 = 80r
Ex.6 5n – 1 is divisible by 2 for all positive integers n.
P1: 51 – 1 = 5 – 1 = 4 which is divisible by 2. True
If Pk: 5k – 1 = 2r then Pk+1: 5(k+1) – 1 = 2s
5(5k – 1) = 52r
5k+1 – 5 = 10r
5k+1 – 5 + 4 = 10r + 4
5k+1 – 1 = 2(5r + 2)
5k+1 – 1 = 2s
Ex.7 n2 > 2n, n > 3
P3 = 32 > 2(3) or 9 > 6 which is true.
If Pk: k2 > 2k then Pk+1: (k + 1)2 > 2(k + 1)
k2 + 2k + 1 > 2k + 2
k2 > 1
On the TI-89:
Math 2nd 5
3: List
1: seq(expression, variable, low, high)
6: sum({a1, a2, a3, … an}) or sum(seq(expression…))
Ex.7 Find the first 5 terms of 3n! ÷ n2.
Mode
Graph
4: Sequence
Y= ♦ F1
u1= u1(n – 1) + 2
ui1= 7
an = an-1 + 2
a1 = 7
Ex.8 Graph: a1 = 3 & an = 2·an-1
Summary:
n
Sn =  [a1  ( n 1)  d ] 
an = a1 + (n – 1)·d
i 1
Sn 
n-1
an = a1 · (r)
S 
d i
a1 1 r n
n
(a  a )
2 1 n
n
 a1(r )n 1
1 r
i 1
a1
1 r
n(n 1)(2n 1)
2
 i = 12 + 22 + 3 2 + 42 + … + n 2 =
n
Ex.2
6
i 1
n(n 1) n
= 1 n
i = 1 + 2 + 3 + 4 + … + n =
n
23.
2
i 1
bg
n 2 (n 1) 2
i = 1 + 2 + 3 + 4 + … + n =
4
i 1
n
25.
2
3
3
3
3
3
3
Notes #4-___
Date:______
9.6 Statistics and Data - Graphical (717)
Statistics: the science of data (discrete or continuous)
Individual: usually people, objects, transactions or events
that form a population (set) we wish to study.
Variable: characteristic or property of an individual
Categorical (qualitative): variables that cannot be
measured on a natural numerical scale; they can only be
put into groups.
Quantitative: measurements that are recorded on a
naturally occurring numerical scale.
Ex.1 Classify each variable as Categorical or Quantitative:
Unemployment rates of the 50 states
Size of a car (compact, mid-size etc.)
Political Party
Temperature
A taste-tester’s ranking
River where each fish was captured
DDT concentration
Species
Length
Weight
Categorical
Size of car
Political Party
Taste-tester ranking
River
Species
Quantitative
Unemployment rate
Temperature
DDT concentration
Length
Weight
Ex.2 Construct a pie chart and a bar graph for each year.
Is smoking a cause of lung cancer? 1954 1999
Yes
41% 92%
No
31%
6%
No opinion
28%
2%
How do we find the
angles?
Stemplots (Stem-and-leaf)
A stem-and-leaf plot arranges data by separating the digits
into stems (the beginning digits) & leaves (the ending digits).
Ex.3 Make a stemplot for the % of student loans in default.
AZ 12.1
CA 11.4
CO 9.5
HI 12.8
ID 7.1
MT 6.4
NV 10.1
NM 7.5
OK 11.2
OR
7.9
UT
6.0
WA 8.4
6
7
8
9
10
11
12
04
159
4
5
1
24
18
Split Stem Stemplots: Stems split to spread out the data.
Ex.4 Make a split stemplot for the SAT scores from 2000.
CA
CT
HI
ME
MD
MA
NV
1015
1017
1007
1004
1016
1024
1027
NH
NJ
NY
RI
VT
VA
WV
1039
1011
1000
1005
1021
1009
1037
Back-to-back Stemplots
Ex.5 The table lists the % of graduates taking the SAT and
their average Math score. Create a back-to-back
stemplot with the states with less than 25% on the left.
Arizona
California
Colorado
Idaho
Nevada
New Mexico
Oregon
Texas
Utah
Washington
What is lost in a
frequency table?
22%
44%
29%
16%
24%
12%
50%
45%
6%
37%
520
484
511
501
486
524
486
462
536
494
Frequency Tables: the # of occurrences per interval.
Ex.6 Create frequency tables for the data in Ex.4 & 5.
How is this different
from a bar graph?
Histogram: displays the info of a frequency table.
Ex.7 Create histograms for the data in Ex.5.
Time Plots: a line graph (not linear) where discrete points are
connected to reveal trends in data over time.
Be sure to distinguish
between the two sets
of data.
Ex.8 Make time plots for the data below on the same set of
axes. Which decade did the largest decrease occur for
each? Give a possible explanation.
Median Age at
First Marriage
Year Male Female
1900 25.9 21.9
1910 25.1 21.6
1920 24.6 21.2
1930 24.3 21.3
1940 24.3 21.5
1950 22.8 20.3
1960 22.8 20.3
1970 23.2 20.8
1980 24.7 22.0
1990 26.1 23.9
2000 26.8 25.1
Summary:
Notes #4-___
Date: ______
9.7 Statistics and Data - Algebraic (730)
Statistics: the science of data. Collecting, classifying,
summarizing, organizing, analyzing and
interpreting numerical information
Measures of Central Tendency (describes “normal”):
Given a set of numbers {x1, x2, …, xn} the
* Mean
is X =
x1  x2 
n
 xn
=
1 n
  xi
n i 1
* Median is the middle number if n is odd.
or the mean of the two middle #s if n is even.
* Mode
the #s that occur most (may be more than one).
Ex.1 The mean age of 4 people in a car is 20. Imagine
the people in your head. What do you see?
A statistic is resistant
if it is not strongly
affected by outliers.
A 70 year old grandma and her grandkids at 2, 3 & 5?
Or four 20 year olds cruising?
Are these means in
Ex.1 resistant?
The mean yearly income for 5 people is $100,000.
Imagine the people in your head. What do you see?
Did you imagine a boss making $400,000 and her four
employees at $25,000? Or did you imagine 5 partners
making equal shares?
Ex.2 Find the mean, median and mode for Test #4-1 (S07).
47, 38, 14, 26, 33, 39, 67, 39, 16, 44, 44, 40, 33, 45,
36, 55, 28, 53, 43, 43, 35.
Do not confuse with
the range of a f(x).
Range: maximum – minimum
Ex.3 Find the range of Ex.2.
Ex.4 Find the mean, median, mode and range:
Quiz Scores for a class
Score
10 9 8 7 6 5 4
Frequency 4 4 4 4 2 2 5
3
3
2
0
1
2
0
2
Mean: [10(4) + 9(4) + … + 0(2)] / 32 = 189 / 32 ≈ 5.91
Median: 6.5 (16th # is a 7 and the 17th # is a 6 so 13/2 = 6.5)
Mode: 4 occurs the most
Range: 10 – 0 = 10
Q1 Lower quartile QL
Q2 Middle quart. QM
Q3 Upper quartile QU
A boxplot (box-and-whisker plot) divides data into quartiles
(4 parts). The 2nd quartile is the median. The 1st quartile is
the median of the data less than the 2nd quartile and the 3rd is
the median of the data greater than the median.
The five-number summary {min, Q1, median, Q3, max}.
Ex.5 Draw a boxplot for Test #4-1 (S07).
47, 38, 14, 26, 33, 39, 67, 39, 16, 44, 44, 40, 33, 45,
36, 55, 28, 53, 43, 43, 35.
The interquartile range IQR = Q3 – Q1. The IQR gives the
range of the middle half of the data and is more resistant than
the range.
Outlier: a number in a data set that is more than 1.5(IQR)
below Q1 or above Q3.
A modified boxplot separates outliers as isolated points.
Ex.6 Make a boxplot for the % of student loans in default.
AK
AZ
CA
CO
HI
ID
MT
19.7
12.1
11.4
9.5
12.8
7.1
6.4
NV 10.1
NM 7.5
OK 11.2
OR 7.9
UT 6.0
WA 8.4
WY 2.7
Standard Deviation and Variance (788)
Standard deviation is
strongly affected by
outliers.
Typically, two-thirds of the data values are within one
standard deviation (s when calculated from a sample or σ if
the entire population was used) of the mean. s will be larger.
You don’t need to
memorize these .
n
1
s
  ( X i  X )2
n 1 i  1
1 n

  ( X i  X )2
n i 1
Variance is σ2.
Ex.7 Find the standard deviation and the variance for Ex.6.
APPS
Stats/List E… (Enter)
Enter data into list1
F4 (Calculate)
1:1-Var Stats… (Enter)
Summary: