Statistically Analyzing the Short Tenn Profit of Basic Strategy, Multi-deck, Strip Blackjack
An Honors Thesis (HONRS 499)
by
Adam E. Coolman
Dr. Giray Okten
Ball State University
Muncie, IN
April 2002
May 4,2002
Abstract
After playing a given number of blackjack hands, actual earnings results vary, sometimes
drastically, from theoretical results. The enclosed, user friendly, computer program simulates a
user-entered number of blackjack hands (1-10000), calculates expected profit, standard
deviation, confidence intervals about the sample mean, and probability of profit. The statistical
breakdown gives the user a detailed description of the short-term blackjack earning potential any
player will want to view before hitting the multi-deck tables on the Las Vegas Strip.
This thesis paper delves into the many intricacies of the computer simulation and
statistical analysis processes such as blackjack rules on the Strip, basic strategy, and confidence
interval analysis. In addition, using included sample results, conclusions are made on the
relationship between actual and theoretical results, distribution of earnings per hand, effect of
short-term simulation on confidence intervals, probability of profit, and effects of playing
blackjack via basic strategy. forms of card counting, or an inferior method.
Acknowledgements
Best regards to Dr. Giray Okten for his mentoring, patience, and persistence. Throughout
the rigorous computer programming, simulation testing, and conclusion making process he made
sure I didn't stray from the correct path and helped me fight through each roadblock along the
way.
-
Table of Contents
I.
Statistically Analyzing the Short Term Earnings of Basic Strategy, Multi-deck, Strip
Blackjack
II.
Basic Strategy Tables (Las Vegas Strip)
III.
Dealer Strategy Table (Las Vegas Strip)
IV.
Sample Simulation Results for blackjack sessions of 25,50, 100,200,500, 1000,
2500, and 10000 hands
V.
Blackjack Simulation in the Short Run (CD-ROM)
Coolman 1
I. Statistically Analyzing the Short Term Earnings of Basic Strategy, Multi-deck, Strip Blackjack
Blackjack players, regardless of skill level, would pay big money to know their expected
earnings per hand. The mathematically inclined blackjack player knows that unless seemingly
infinite hands are played, such figures cannot be computed exactly. A non-infinite amount of
trials will produce variable expected earnings. For a finite number of trials, sample expected
earnings and standard deviation have a direct impact on the probability of profiting from playing
blackjack. This thesis demonstrates how a blackjack program I have developed determines, via
short-term simulation, sample expected earnings and standard deviation for a hand of blackjack
-
and uses that data to statistically analyze the short-term expected profitability and variability of
playing blackjack using basic strategy.
In his book, Winning Casino Blackjack for the Non-counter, world-renowned gambler
Avery Cardoza describes blackjack as a game where every decision affects a player's percentage
against the house. (117) A player's percentage against the house is their expected profit per unit
wagered. Today's blackjack player is ultimately concerned about maximizing this percentage.
For the non-cardcounter, this maximization is done using basic strategy.
In today's technological era, the average blackjack player is one who is well versed in
basic strategy. Basic strategy outlines the simulation-tested, profit-maximizing action that
should be taken for every possible player total - dealer upcard circumstance. Of the two initial
cards received by the dealer, the upcard is the card that can be seen by every player while the
-
hole card is concealed until all players have finished playing their hands. The correct basic
Coolman 2
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strategy action is determined by using conditional probabilities to compute the expected profit of
every possible action a player may take.
Depending on the composition of a player's cards and the dealer's upcard, the player may
have the opportunity to take any of the following actions on the hand: hit (take an additional
card), stand (take no further action on the hand), surrender, double, split, or take insurance.
Surrender allows a player to forfeit half his wager, a 1-2 payoff, and fold his hand before any
other action is taken. Doubling permits the player to double his initial bet after receiving the first
two cards. For doubling the bet, the player is forced to take one additional card and stand.
Splitting can be done when a player's initial two cards are the same value. All face cards and
tens are considered the same value. A player may place an equivalent bet, split the equal cards,
and play two separate hands with each card of similar value becoming the first card of the new
-
hands. Insurance, paying 2-1, is an option a player has when the dealer's upcard is an ace. If the
player takes insurance, a side bet of up to one-half the original bet is made betting the dealer has
a blackjack. Winning an insurance bet of one-half the original bet is a wash for the player.
Out of every possible player action, the one producing the largest expected profit, in the
long run, is deemed the correct basic strategy play for the tested player total - dealer upcard
circumstance. After every possible circumstance is tested, a complete basic strategy is formed.
The complete basic strategy, outlined in table form in Section II, reduces the house advantage to
its non-card counting minimum. Blackjack hands in my program use basic strategy to dictate the
way each blackjack hand is managed.
Basic strategy has its share of downfalls as well. As a memory-less approach to
blackjack, basic strategy limits its expected profits. Conditional probabilities for basic strategy
-
are calculated based on a full deck. Each hand is believed to have equivalent profit expectation.
Coolman 3
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In reality, profit-maximizing plays for a full deck may deviate as the distribution of the
remaining cards changes. For example, basic strategy fails to account for the fact that "some
card combinations which have the same total, but unlike compositions, require a different action
to optimize expectation." (Griffin, 17) In addition, by assuming each hand has equivalent
expectation, the basic strategist sees no reason to vary his neutral bet between hands.
The fallacies of basic strategy lead to The Fundamental Theorem of Card Counting,
stating, "variations in player expectation for a fixed strategy become increasingly spread out as
the deck is depleted." (Griffin, 22) As the distribution of the deck changes, card counters deviate
from basic strategy accordingly. The card counter takes advantage of bet minimization or
maximization by deviating from the neutral bet in relationship to the inherent favorable or
unfavorable characteristics of the underlying distribution of the remaining cards.
-
Standard blackjack rules give players the opportunity to hit, stand, split pairs, double
down, and take insurance. Regular and double down wins pay 2-1, blackjacks 3-2, insurance 21, and pushes 1-1. Casinos implement these basic rules and payoffs but also add their own mix
of restrictions on when doubling down or splitting is allowed. In addition, surrender, a 1-2
payout, is now becoming an increasingly popular non-standard option many casinos allow.
Dealers are required to hit until the total of their hand is greater than 16, at which point they are
required to stand. However, depending on house rules, the casino may rule that the dealer must
hit a soft 17, any total of 17 including an ace.
Las Vegas Strip multiple deck blackjack rules follow the basic standards, with a few
adjustments. Strip rules permit the player to split non-ace pairs up to three times, making four
total hands. Aces may only be split once with each ace being given only one additional card.
-
When aces are split, any resulting A-lO combination is considered a straight 21 rather than a
Coolman 4
,-
blackjack. Doubling is allowed on any two-card combination and is allowed after splitting.
Insurance is allowed but is never an option when using basic strategy because it is not the
optimal play in any circumstance. The Strip uses a shoe of six decks and the dealer must stand
on all totals of 17 or above, whether soft or hard. Section II contains basic strategy tables for the
Las Vegas Strip multiple deck blackjack rules used in the simulation program. (Silberstang, 91)
Section III contains the strategy table dealers must use on the Strip.
As stated earlier, basic strategy is calculated using probabilities based on a full deck and
the player assumes equal profit expectation on each hand. Factors such as the number of players
at a table, burning a card, shoe penetration, and using a new shoe to finish a previous shoe do not
affect the strategy of the player and are not factored into the simulation. Burning a card is an
action taken by the dealer when a new shoe begins or when a new dealer takes over. This action
-
requires the dealer to take the next playable card out of play. In order to deter card counters and
prevent unfinished hands, dealers reshuffle after a certain percentage of the cards have been
played. Shoe penetration defines the percentage of cards from the original shoe actually dealt for
play.
Long run expected profit per hand, using basic strategy, has been accurately calculated to
the ten-thousandths. Kenneth Smith's "Blackjack Basic Strategy Engine," and Eric Farmer's
"Basic Strategy Calculator" produce long run expected casino profit per hand of .0036 (.36%)
and .0033 (.33%) respectively when applying the Las Vegas Strip rules. These figures represent
the theoretical loss per hand a player expects to duplicate in the long run.
Duplicating the amount of trials used to compute these long run expected values would
be impossible for an individual to complete in a lifetime, let alone one session of game play.
-
Blackjack players everywhere deal with this all-too-real fact every time they sit down to play. In
Coolman 5
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the short-run, the house advantage can vary immensely from the theoretical advantage. Sample
statistical results from five simulations of 25, 50, 100, 200, 500, 1000, 2500, 5000, and 10,000
trials can be found in Section IV. My simulations yielded profits ranging from -.46 to .27 per
hand, a far cry from the theoretical value of -.0033 or -.0036. The long run house advantage is of
minor importance to the short-run player. The short-run player is concerned about the edge they
are experiencing, the standard deviation of per hand earnings, and ultimately how these
deviations affect the probability of profiting from playing a given number of hands. The
standard deviation of expected earnings is a measure of how dispersed the data are from the
sample mean. (Hogg, 14)
Blackjack is a game where long run simulation will not mirror short-term reality. Since
replicating theoretical expectation and deviation consistently is impossible in the short-run,
-
confidence intervals become the key analytical tool when examining expected returns over a
series of achievable trials. Using basic strategy, my computer program simulates a user-entered
number of blackjack hands (1-10000) and calculates the expected profit per hand, its standard
deviation, and both normal and Student's-t confidence intervals surrounding expected earnings.
For 1-200 simulations a bootstrap-t confidence interval is also calculated. For a stated level of
confidence, confidence intervals form bounds around the sample mean and enclose all values
that can be expected with the stated confidence level. The resulting bootstrap-t, normal, and
Student's-t confidence intervals give the user insight into the variability of short-term expected
earnings.
"Through the use of bootstrapping, accurate intervals can be obtained without having to
make assumptions about the underlying distribution." (Efron, 160) This method is particularly
-
useful when the application of the Central Limit Theorem, the support behind the normal
Coolman 6
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approximation, is suspect due to the small number of trials. Bootstrapping forms an idea about
the underlying distribution by constructing a given number of bootstrap samples. Bootstrap
samples are computed by sampling the empirical distribution with replacement. Each bootstrap
sample is then standardized like the normal distribution and the standardization of each resampling is sorted and used in confidence interval analysis.
As trials and bootstrap samples increased, my results showed the bootstrap characteristics
begin to approach those of the normal and Student's-t distribution. The average bootstrapping
interval for the five sets of 100 trials in Section IV was off 5.44 and 6.70% from the average
normal and Student's-t distribution, respectively. For the five sets of 200, those numbers
lowered to 4.26% and 4.90%. Increasing the bootstrapping samples further would have likely
continued the convergence toward establishing the normal or Student's-t distribution as the
_
correct underlying distribution. Meanwhile, as the sample standard deviation begins to approach
the theoretical standard deviation, by way of increased trials, the Student's-t distribution points
more and more to the normal distribution as an accurate distribution of expected profit per hand.
Therefore, in my program, the probability of profit calculation is made assuming the sample
standard deviation is equal to the theoretical deviation, whereby the normal distribution is the
best approximation for the distribution of expected earnings.
The confidence intervals produced by the computer program summarize expected profit
per session. They are calculated by mUltiplying the confidence interval bounds for the expected
profit per hand by the number of hands played in the session. The base earnings per session
confidence interval is calculated using a neutral bet of one. Calculating confidence intervals for
other neutral bets requires nothing other than multiplying the base interval bounds by the neutral
Coolman 7
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bet amount. The computer program outputs the base confidence interval as well as confidence
intervals for neutral bets of 5, 10,25,50, and 100.
The simulation program as a whole has the capabilities to show the power of multiple
trials. As trials decrease, confidence intervals and probability of profit calculations become
increasingly dependent on the sample expected earnings and standard deviation. The results
from the five simulations of 25 and 50 hands show the power increasing the number of trials has
on confidence analysis. The expected earnings intervals for both sessions are nearly the same
despite playing an additional 25 hands. Earlier we stated that the approximate theoretical profit
per hand is between -. 0033 and -. 0036, yet 10,000 trials is not enough to confidently close in on
the theoretical mean. The confidence intervals generated by the computer program can be easily
transformed into expected earnings per hand confidence intervals. This conversion is done by
-
dividing each interval bound by the number of hands per session. Using this conversion, my
results show that expected earnings per hand over 10,000 hands have maximum upper and lower
bounds of approximately -.04 and .06, a range of. 1. They also show that the average 99%
interval size for the five results of 25 trials is 1.28 while for 10,000 trials it is .057. As trials
increase, the variability of expected results lessens and the base confidence intervals increasingly
close in on the theoretical mean.
Results from simulations show the vast impact decreasing the house edge via forms of
card counting has on profit probability and confidence intervals. If a player was able to maintain
an advantage on the house of 3% over 10,000 trials, then according to my results and assuming a
similar sample standard deviation, the player can be 99% sure of a profit anywhere between $20
and $590. On the other hand, a player who plays without either using basic strategy or counting
-
cards has the potential to increase the house edge and experience lower expected profit and less
Coolman 8
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pleasant confidence intervals. Playing 1,000 hand poor enough to give the house an edge of
8.3% will result in losses between $2 and $163 with 98% confidence.
The realization that theoretical results cannot be consistently achieved over the short-term
gives the basic strategist hope of profitable short-term sessions. In theory, the expected loss per
hand would be duplicated consistently and result in a 0% chance of profit. However, playing
between 25-10,000 hands provides the basic strategist enough variability from the theoretical
results to enjoy profitable sessions. Keep in mind however, for a basic strategist the occurrence
of a profitable session is less likely then a losing one.
This thesis paper and enclosed computer program effectively provide the short-term
blackjack player insight into the realm of earnings possibilities. The scope of such insight is
directly proportional to the number of trials and dependent on the sample mean and standard
deviation. The next time someone plans on playing blackjack, they should run my computer
program a couple times to get a grasp of what earnings variability awaits them.
-
Coolman 9
Works Cited
Cardoza, A very Winning Casino Blackjack for the Non-counter. New York:
Cardoza Publishing, 1999.
Cooper, Carl, and Lance Humble The World's Greatest Blackjack Book. New York:
Doubleday, 1987.
Efron, Bradley, and Robert J. Tibshirani. An Introduction to the Bootstrap. New York: Chapman
& Hall, 1993.
-
Farmer, Eric. "Blackjack Game and Basic Strategy Calculator." 2001. Internet. 15 Apr. 2002
Griffin, Peter A. The Theory of Blackjack. Las Vegas: Huntington Press, 1999.
Hogg, Robert V., and Elliot A. Tanis. Probability and Statistical Inference. Upper Saddle
River: Prentice Hall
Silberstang, Edwin Winning Blackjack for the Serious Player. New York: Cardoza
Publishing, 1998.
Smith, Kenneth. "Blackjack Basic Strategy Engine." 2002. Internet. 10 Apr. 2002
-
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12.69
1134
10.34
9.54
8.88
279
1.08
(1810)
(16.82)
(15.88)
(15.12)
(12.62)
(8.76)
5.11
(15.32)
(13.18)
(11.86)
(10.89)
(10.11)
(7,52)
(3.54)
0.22
a.
1.12
,."
i:;iiHiI"
;~
99%
98%
97%
98%
95%
90%
75%
50%
)
;a;
(19.06)
(18.26)
(16.55)
(16.47)
(14.02)
(11.36)
(.,06)
..,
w," "
(1938)
(17.51)
(16.33)
(15-46)
(1475)
(1238)
(
)
5.08
19.38
17.51
16.33
15.46
1475
1238
959
.54
99%
98%
97%
98%
95%
90%
75%
50%
...
508
99%
98%
97%
98%
95%
90%
75%
50%
25.51
2357
2236
2145
2073
18.28
14,44
10.74
99%
98%
97%
98%
95%
90%
75%
50%
32.90
31.02
29.85
28.97
28.26
25.89
22.17
18.58
98%
97%
98%
95%
90%
75%
50%
22.09
2139
19.n
,.53
1704
1399
....
.:1,;
-
.,
(20 17)
(14.53)
(1327)
(.55)
(704)
(609)
(397)
(097)
1.53
23,63
2280
22.58
2258
22.36
19.16
14.23
10.22
99"
98%
97%
96%
95%
90%
75%
50%
(14.51)
(12.57)
(11.36)
(10.45)
(9.73)
(7.28)
("'4)
0.26
99%
(10 12)
(661)
(258)
(22')
(2.13)
0.12
4.25
.61
28.69
27.46
27.25
26.03
25.64
24.15
20.51
17.97
99%
98%
97%
98%
95%
90%
75%
50%
(5.90)
(4.02)
(2.85)
(197)
(128)
111
483
.42
.,'
99%
.1
(6.69)
(462)
(3.34)
(239)
(164)
0.7
4.73
.38
',"c,"
-
'44
2.29
098
(002)
(aBO)
"
20.17
18.10
1682
15.88
15.12
1262
8.78
511
• I
99%
98%
97%
98%
95%
90%
75%
50%
98%
97%
98%
95%
90%
75%
50%
" " " < ~ ~,
99%
98%
97%
98%
95%
90%
75%
50%
; ,'I'
(27.05)
(25.06)
(23.82)
(22.89)
(22.14)
(19.63)
(1568)
1186
.'.
382
1.67
0.45
(0.46)
(119)
(3.5)
(7.51)
(1123
26.32
2418
22 . .
2189
21.11
18.52
14.54
1078
3369
31.62
30.34
29.39
28.64
26.13
22.27
18.62
",......
Section IV
!
","",,·PJOiiIId
..
. 1001
_<10
99%
(31.88)
(. . 08)
(2733)
(2602)
(2.97)
(21 ....)
(1590)
lO56j
99%
,..:1
(3244)
(29.51)
(27.88)
(26.33)
(25.24)
(21.61)
(15.97)
(1059j
97%
96"
95"
90%
75%
SO%
(31.39)
(29.79)
(29.39)
(22.85)
(22.39)
(19.48)
(16.22)
10.Bl
22.56
21.31
18.9
1756
17,52
16.14
936
5.27
97%
96%
95"
90%
75"
SO"
99%
98"
97"
96%
95%
90%
75%
SO%
(20.48)
(19.55)
(1853)
(1823)
(1850)
(15.")
(10,07)
. (637j
2727
25,82
23.50
23.42
21.96
19.40
15.16
8.49
99%
98%
97%
96%
95%
90%
7.%
SO%
(2623)
(2354)
(21.86)
(2081)
(19.60)
(18.21)
(10.99)
578
29.23
26.54
24.86
23.61
22.60
19.21
13.88
678
99%
98%
97%
96%
95%
90%
75%
50%
(26.77)
(2396)
(22.20)
(20.90)
(19.M)
(18.37)
(1096)
5.79
9...
96%
97%
96%
(2762)
(2000)
(1925)
(19.02)
(1749)
(1517)
(981)
5.15
2925
25,73
24.74
23.69
22.92
1827
15.30
11.18
99%
98%
97%
96%
95%
90%
7'%
SO%
(23.23)
(20.84)
(19.02)
(17.81)
(16.8.)
(13.57)
(6.44)
(3SOj
30.23
27.64
26.02
2481
23.84
20.57
15.44
10SO
99%
98%
97%
98%
95"
90%
75%
50"
(2378)
(210..)
(1935)
(18.10)
(17.09)
(1373)
(8.51)
(353j
99%
(22.43)
(19.53)
(17.72)
(18.37)
(15.28)
(11.81)
(567)
0.34
37.-43
3453
32.72
31.37
30.28
2661
2087
15.34
99%
98%
97%
96%
95%
90%
75%
50%
(23.02)
(19.96)
(18.09)
(1668)
(15.56)
(11.79)
(5.95)
0.37
55.18
52.24
5041
4905
4795
4426
3847
3290
99%
98%
97%
96"
(5.75)
(2.69)
(078)
0.63
1.77
'.56
11.-45
1707
99%
96"
96"
"""
25.88
23.08
21.33
20.02
18.97
15.44
9.90
4.56
96"
97%
96%
95"
90%
75"
50%
!I!"J.
l,
,,"e;
-
99%
98%
97%
96%
95%
90%
7.%
50%
...
.1
(20.17)
(19.26)
(1902)
(1888)
(15.35)
(10.99)
(4.14)
122
*It
38.78
33.69
31.30
30.24
29.91
24.30
19.70
13.93
96"
97%
96"
95%
90%
75%
50%
95"
90%
75%
50%
(102)
1.71
200
460
592
781
1181
17 87
-,
",:-,
.'1·~
99%
98"
97%
98"
1~
52.10
SO.50
48.50
46.36
99%
98%
97%
96%
45.95
95"
43.38
37.43
32.50
90%
75%
50%
(5.16)
(2.24)
(0.• ')
0.95
2.05
5.74
11.53
17.10
95"
90"
75%
SO%
,i!
29.77
26.96
25.20
23.90
22.88
19.37
13.96
8.79
.
•
26.44
23.51
21.68
20.33
19.24
15.61
9.97
4.59
30.76
28.04
2635
25.10
24.09
20.73
15.51
10.53
1~
•. ,
3802
3498
33.09
3188
3056
26.79
20.95
1537
5575
5289
5078
4937
4823
4444
3855
3293
-
_,v
d
..
.,,·f'
99%
98%
97%
96%
95%
90%
75%
'
(.045.18)
(36.75)
32.26
(~.70)
31.76
(34.03)
(33.25)
2673
2681
2137
12.11
(30.10)
(23.01)
35.12
99%
98%
97%
98%
95%
90%
75%
(-46.73)
(42.79)
(40.32)
(36.48)
(36.99)
(32.01)
(2419)
16.67
99%
98%
97%
98%
95%
90%
75%
50%
(41.42)
1548
*
34.73
30.79
28.32
28.48
24.99
20.01
12.19
_I
99%
98%
97%
98%
95%
90%
75%
50%
(4749)
(43.75)
(3725)
(34.60)
33.23
(33.03)
(29.01)
(18.10)
27.36
22.13
16.94
10.88
(".oj
99%
98%
97%
98%
95%
90%
75%
50%
(35.01)
31.31
(32.72)
(29.70)
(29(5)
(2828)
(2328)
(17.36)
-",035L
30.93
2991
99%
98%
97%
98%
95%
90%
75%
50%
(24.20)
(20.99)
(20.84)
(lB.7B)
(lB.07)
(15.90)
54.71
44.95
-43.87
43 51
34 32
2688
99"k
(2190)
98'"
97%
98%
95%
90%
75%
50%
(2097)
30.89
30.47
29.75
'.;'
(3750)
(35.05)
(33.23)
(31.75)
(26.81)
(19.OS)
11.58)
o ~
29.91
28.46
25.06
16.11
10.29
~"
g.%
98%
97%
98%
95%
90%
75%
(4713)
(4309)
(4058)
(38.69)
(37.18)
(32.13)
(24.24)
16.69
99"k
(41 Bt)
39.81
98%
97%
98%
(3780)
(3530)
(3344)
35.80
33.30
95°k
(31.94)
(2693)
09tO)
"80
33.05
3123
2975
2481
90%
1705
9.58
75%
40.02
99%
98%
97%
98%
95%
90%
75%
50%
_ (9.87)
99"k
(3737)
98%
97%
98%
95%
90%
75%
50%
(33.35)
(3085)
(2899)
50%
.:1,
(39.02)
(35.19)
36.19
(3280)
33.80
(3101)
(2957)
(2474)
(17.15)
32.01
30.57
25.74
18.15
99%
98%
97%
98%
95%
90%
75%
50%
(36.98)
(33.08)
(30.60)
1085
::~
(3"
4008
3760
(28.77)
35.77
(27.30)
(2235)
(1(.58)
7.10j
34.30
29.35
21.58
14.10
(18.92)
(17.25)
(1897)
(15.24)
(4.19)
3395
3.23
23.95
99%
98%
97%
98%
95%
90%
75%
50%
(280()
(24.11)
(2165)
(19.82)
(1835)
(13.39)
(5.81)
1.99
2518
20 13
1224
(2"'6)
(1720)
(2749)
(2247)
(14.63)
(7.12
31"
29.94
24.93
17.10
9.60
(040
3848
"0(
3222
30.76
25.86
1820
1087
",'
. . 37
(0.35
37.85
.....
35.99
29.47
21.63
14.12
,..H
.,' 0
53.19
51.17
49.69
48.61
47.12
41.10
.,
(39.40)
(35.46)
(33.04)
(31.22)
(29.76)
'* j
'*1,;1',;
::::
..
35.13
31.09
28.56
26.69
I
3942
35.50
99%
98%
97%
9B%
95%
90%
75%
60%
(98SL
..
8tudInt'. t DlllrlMdion
Nonnel DlltiIbutiDn
BOO!!t!!ppInp (200 -.npl!!)
53.D<4
49.11
.
(885
"
".35
363.
3081
2312
99"k
98%
97%
98%
95%
90%
75%
50%
(28.43)
(24.41)
(21.90)
(200()
(1854)
(1351)
(566)
188
53(3
4941
48.90
(5.04
43.54
38.S1
30.88
23.14
_
Section IV
•
Student's I DIstribution
NoI'l1llI~uon
99%
98%
97%
96%
95%
90%
75%
50%
QQh!ll!enlle ;
Rapiiiijd ..
500'
. ""',;I:IM .. ,
(94.93)
(88.69)
(84.78)
(81.87)
(79.52)
(71.64)
(59.27)
(47.37)
tlet .. ;,
""33.93
27.69
23.78
20.87
18.52
10.64
(1.73)
(13.63
~;
',_";;l,)\\h;" ;k
"
99%
98%
97%
96%
95%
90%
75%
50%
(95.18)
(88.88)
(84.94)
(82.00)
(79.64)
(71.72)
(59.31)
(47.38)
34.18
27.88
23.94
21.00
18.64
10.72
(1.69)
(13.62
. OllflikindEl'lldliMlllli,"; ;;~;'lb~
"
99%
98%
97%
96%
95%
90%
75%
50%
(79.37)
(73.23)
(69.39)
(66.52)
(64.22)
(56.46)
(44.30)
(32.59)
47.37
4123
37.39
34.52
32.22
24.46
12.30
0.59
99%
98%
97%
96%
95%
90%
75%
50%
(79.61)
(73.41)
(69.54)
(66.66)
(64.33)
(56.54)
(44.33)
(32.61)
99%
98%
97%
98%
95%
90%
75%
50%
(50.52)
(44.27)
(40.35)
(37.44)
(35.09)
(27.20)
(14.81)
(2.89)
78.52
7227
68.35
65.44
63.09
55.20
42.81
30.89
99%
98%
97%
96%
95%
90%
75%
50%
(50.76)
(44.46)
(40.51)
(37.58)
(35.21)
(27.28)
(14.85)
(2.91)
99%
98%
97%
96%
95%
90%
75%
50%
(50.08)
(43.87)
(39.98)
(37.09)
(34.76)
(26.92)
(14.62)
(2.78)
78.08
71.87
67.98
6509
62.76
54.92
42.62
30.78
99%
98%
97%
96%
95%
90%
75%
50%
(50.32)
(44.06)
(40.14)
(37.22)
(34.88)
(26.99)
(14.65)
(2.79)
99%
98%
97%
96%
95%
90%
75%
50%
(25.79)
(19.57)
(15.67)
(12.76)
(10.42)
(2.56)
9.79
21.66
102.79
96.57
92.67
89.76
87.42
79.56
67.21
55.34
99%
98%
97%
98%
95%
90%
75%
50%
;
.0
BlltiOoL:h
....
,~,.;,
47.61
41.41
37.54
34.66
32.33
24.54
12.33
0.61
""'di
78.76
72.46
66.51
65.58
63.21
55.28
42.85
30.91
"
'<,,"
78.32
72.06
68.14
65.22
62.66
54.99
42.65
30.79
.,:1,; ,
(26.04)
103.04
(19.75)
96.75
(15.82)
92.82
(12.90)
89.90
(10.54)
87.54
(2.63)
79.63
9.75
67.25
21.65
55.35
,-,
Section IV
I
StudIInt'at Dlalrlbullon
',( i'/'IIIII=i:"
99%
98%
97')'.
96%
95%
90%
75%
50%
(172.39)
(163.73)
(156.31)
(154,27)
(151.02)
(140.08)
(122,92)
(106,41)
99%
98%
97%
96%
95%
90%
75%
50"10
(163,96)
(155,25)
(149.79)
(145.73)
(142,45)
(131.45)
(114,18)
(97.56)
99%
98%
97%
96%
95%
90%
75%
50%
(14O,SO)
(131.93)
(126,49)
(122,44)
(119,18)
(108,22)
(91,02)
(74,46)
I
" 'tsat=;
'e'
6,39
(2,27)
(7,69)
(11.73)
(14,98)
(25,92)
(43,08)
(59,59)
)#'1>2:",,;
99"10
(172,56)
(163,86)
(158.42)
(154.36)
(151.10)
(140,13)
(122.94)
(106,42)
98%
97%
96%
95%
90%
75%
50%
,~;
15,96
7.25
1.79
(2.27)
(5,55)
(16,55)
(33,82)
(50,44)
:,.i.:
,. :liCC"T;ii:
99%
98%
97%
96%
95%
90%
75%
50%
(164.13)
(155,38)
(149.90)
(145.82)
(142,53)
(131.50)
(114,20)
(97.57)
99%
98%
97%
96%
95%
90%
75%
50%
(140.78)
(132.06)
(126.SO)
(122,54)
(119.26)
(106.27)
(91.04)
(74.47)
99%
98%
97%
96%
95%
90%
75%
(103.99)
(95.11)
(89.54)
(85,40)
(82,07)
(70.87)
(53.31)
(36.42)
_ _ ,1'1«1«1,1,,:1
, lei
99"/.
-
98%
97%
96%
95%
90"10
75%
50%
38.60
29.93
24,49
20,44
17,18
6,22
(10,98)
(27,54)
78.82
69.97
64.43
SO,31
56,98
45.81
28.28
11.41
50"10
50%
,-
107,14
98,31
92.79
88.67
85.35
74,20
56,70
39,87
99%
98%
97%
96%
95%
90"/.
75%
50%
78,99
70,11
64,54
SO,4O
57,07
45,87
28,31
11.42
~'
: .. _",;'
1:"IU:nlllllli",:ll'+
(75.14)
(66,31)
(SO.79)
(56,67)
(53,35)
(42,20)
(24.70)
(7.87)
38.78
30,06
24,SO
20,54
17,26
6,27
(10,96)
(27,53)
::I!t':,
;
99%
98%
97%
96%
95%
90%
75%
16,13
7,38
1,90
(2,18)
(5,47)
(16.50)
(33,SO)
(50,43)
I
':: T!"";VU:>~" '
(103.82)
(94,97)
(89,43)
(85.31)
(8198)
(70,81)
(53,28)
(38,41)
6,56
(2.14)
(7,58)
(11.64)
(14.90)
(25,87)
(43,06)
(59.58)
(75,32)
(66,45)
(SO.90)
(56,76)
(53,43)
(42.25)
(24.73)
(7.87)
wuHlH
107,32
98,45
92.90
88.76
85,43
74,25
56.73
39.87
_
Section IV
II
Stu_.t DI8IrIbuUon
_ I Dl8lrlbuUon
···M
(204.86)
(191.12)
(182.52)
(176.11)
(170.94)
(153.59)
(126.35)
(100.15)
99%
98%
97%
96%
95%
90%
75%
SO%
78.86
65.12
56.52
50.11
44.94
27.59
0.35
(25.85
99%
98%
97%
96%
95%
75%
50%
(204.97)
(191.20)
(182.58)
(176.17)
(171.00)
(153.62)
(126.37)
(100.15)
124.60
110.94
102.38
96.00
90.87
73.60
46.52
20.45
99%
98%
97'%
96%
95%
90%
75%
50%
(157.71)
(144.02)
(135.45)
(129.06)
(123.92)
(106.64)
(79.53)
(53.45)
124.71
111.02
102.45
96.06
90.92
73.64
46.53
20.45
(160.24)
(146.27)
(137.52)
(131.00)
(125.75)
(108.11)
(80.42)
128.24
114.27
105.52
99.00
93.75
76.11
48.42
99%
98%
97%
96%
95%
(53.n)
21.n
.,:
(160.35)
(146.35)
(137.59)
(131.06)
(125.81)
(108.14)
(80.43)
(53.77)
128.35
114.35
105.59
99.06
93.81
76.14
48.43
21.77
~ ~ 1 ,';' "
50%
'i!iWi
99%
98%
97%
96%
95%
90%
75%
50%
,
'~L
;
~c;. y,;·.H";Hi!!ii~FH<:i"'h
(115.97)
(102.13)
(93.45)
(87.00)
(81.79)
(64.30)
(36.85)
(10.44)
99%
98%
97%
96%
95%
90%
75%
50%
I -~
·;HI
99"10
98%
97%
96%
;;(:i;
95%
90%
75%
50%
90%
75%
50%
!~I
169.97
158.13
147.45
141.00
135.79
118.30
90.85
64.44
99%
98%
97%
96%
95%
90%
75%
176.75
163.02
154.42
148.02
142.86
125.52
98.30
72.12
990/.
50%
_ _ i··· ., _
98%
97%
96%
95%
90"1.
75%
50%
78.97
65.20
56.58
SO.17
45.00
27.62
0.37
(25.85
'#\ ",
... ·1";'.,'
(116.08)
(102.21)
(93.52)
(87.06)
(81.84)
(64.33)
(36.87)
(10.44)
lLh'~;>
.;~
(106.75)
(93.02)
(84.42)
(78.02)
(72.86)
(55.52)
(28.30)
(2.12)
H
-
90%
~""'+
(157.60)
(143.94)
(135.38)
(129.00)
(123.87)
(106.60)
(79.52)
(5345)
99%
98%
97%
96%
95%
90%
75%
-
.
-i'ii
170.08
156.21
147.52
141.06
135.84
118.33
90.87
64.44
<Ilif'l
(106.85)
(93.10)
(84.49)
(78.08)
(72.91)
(55.55)
(28.32)
(2.12)
176.85
163.10
154.49
148.08
142.91
125.55
98.32
72.12
_
Section IV
h
_101_
s~r.tOhdri~
'H8c.,,; ,,"";;;
99%
98%
97%
96%
95%
90%
75%
50%
(298.42)
(279.10)
(267m)
(258.00)
(250,74)
(226.34)
(188.06)
(151.22)
100.42
8UO
69,01
60,00
52.74
28.34
(9.94)
(46.78)
;II,"
'i·"
(298,50)
(279,16)
(267,05)
(258,04)
(250.78)
(226.37)
(188.07)
(151.22)
99%
98%
97%
96%
95%
90%
75%
50%
..
~,ic
99"10
98%
97%
96%
95%
90%
75%
50%
::;.
97%
96%
95%
90%
75%
50%
(264.14)
(244.61)
(232.37)
(223.27)
(215.92)
(191.26)
(152.55)
(115.30)
'"
(262.70)
(243.21)
(231.01)
(221.92)
(214.59)
(189.98)
(151.35)
(114.18)
139.14
119.61
107.37
98.27
90.92
66.26
27,55
(9.70)
99%
98%
97%
96%
95%
90%
75%
50%
(264.21)
(244.66)
(232.42)
(223.31)
(215.96)
(191.28)
(152.56)
(115.30)
139.70
120.21
108.01
98.92
91.59
66.98
28.35
(8.82)
99%
98%
97%
96%
95%
(262.78)
(243.27)
(231.05)
(221.96)
(214.63)
(190.00)
(151.36)
(114.19)
" ,i:
t
;
~
::;.
97%
96%
, Hit!;dHr
-
95%
90%
75%
50%
~21005)
(190.39)
(178.07)
(188.90)
(161.50)
(136.66)
(97.88)
(60.17)
90%
75%
50%
;
i',·;_",
~l;,iH
196.05
176.39
164.07
154.90
147.50
122.66
63.88
46.17
99%
98%
97%
96%
95%
90%
75%
50%
198.29
178.65
166.35
157.19
149.80
124.99
88.06
48.80
(210.13)
(190.45)
(178.12)
(188,94)
(161.54)
(136.69)
(97.69)
(60.17)
"
-":'~i!:",
(207.29)
(187.65)
(175.35)
(166.19)
(158.80)
(133.99)
(95.06)
(57.80)
:;':i
139.21
119.66
107.42
98.31
90.96
66.28
27.56
(9.70)
·:.JI",?,,1:; ;
;::IliIllU_";;' ,"
99%
98%
97%
96%
95%
90%
75%
50%
100.50
8U6
69,05
60.04
52.78
28.37
(9.93)
(46.78)
99%
98%
97%
96%
95%
90%
75%
50%
. ,:'L:
:"",,"'
(207,37)
(187.71)
(175.39)
(166.23)
(158.64)
(134.02)
(95.07)
(57.60)
139.78
120.27
108.05
98.96
91.63
67,00
28.36
(8.81)
;".:"
196.13
176.45
164.12
154.94
147.54
122.69
83,69
46.17
" :;:Ht
198.37
178.71
166.39
157.23
149.64
125.02
86.07
48.60
-.
Section IV
1
No"",,1 D1atribullon
SIUdent'a I DhoIrtbulion
""
99%
98%
97%
98%
95%
90%
75%
50%
(384.83)
(357.19)
(339.89)
(327.00)
(316.61)
(281.70)
(226.93)
(174.21)
'"
185.83
158.19
140.89
128.00
117.61
82.70
27.93
(24.79
"i ';l:~rnmiHHnmnEP~lnF;llPH;lii;l;l,lHllmHmHJWEli;;('i
n; "
99%
98%
97%
98%
95%
90%
75%
50%
".'
~
99%
98%
97%
98%
95%
90%
75%
50%
·i.
-
"I0000J
. 'r " PiHifOilu"mM_1
SiTgr'
I'~
i,
1.1002_
110.0164_
99%
98%
97%
96%
95%
90%
75%
50%
(271.29)
(243.65)
(226.35)
(213.46)
(203.08)
(166.18)
(113.41)
'(60.70)
"
(156.39)
(128.94)
(111.75)
(98.95)
(88.63)
(53.98)
0.44
52.60
90%
75%
50%
~, , t
185.89
158.24
140.92
128.03
117.64
82.72
27.93
(24.78
JUnmm1i!
,~,
222.05
194.34
176.99
164.07
153.66
118.66
83.75
10.90
99%
98%
97%
98%
95%
90%
75%
299.29
271.65
254.35
241.46
231.08
198.18
141.41
88.70
99%
98%
410.38
99%
382.94
365.75
352.95
342.63
307.98
253.58
201.20
(339.92)
(327.03)
(316.64)
(281.72)
(226.93)
(174.22)
96%
95%
(HUidE
(350.05)
(322.34)
(304.99)
(292.07)
(281.66)
(248.66)
(191.75)
(138.90)
l~:~
99"A.
98%
97"A.
50%
97%
98%
95%
90%
75%
50%
98%
97%
98%
95%
90%
75%
50%
, " ~,i!(i;HJ":l.HI~ ,;~'~ii>ii<:
(350.10)
(322.39)
(305.03)
(292.10)
(281.66)
(246.66)
(191.75)
(138.91)
'~"
222.10
194.39
1n.03
164.10
153.68
116.66
83.75
10.91
(271.34)
(243.70)
(226.38)
(213.49)
(203.10)
(166.19)
(113.41)
' (60.71)
299.34
271.70
254.38
241.49
231.10
198.19
141.41
88.71
(156.44)
(128.981
(111.78)
(98.98)
(88.65)
(53.98)
0.44
52.79
382.98
365.78
352.98
342.65
307.98
253.58
201.21
410.44
.;
99"A.
98%
97%
98%
95%
90%
75%
50%
21.07
48.76
88.11
79.02
89.43
124.41
179.30
232.13
592.93
565.24
547.89
534.98
524.57
489.59
434.70
381.87
,,; ;~~~~;?:;m:!~;?F!;UffHmmnmmWnFtFmHmmnHHlmjWF!!r1~!
97%
96%
95%
21.01
48.72
66.07
78.99
89.41
90%
75%
50%
179.30
232.12
99"A.
98%
124.39
592.99
565.28
547.93
535.01
524.59
489.61
434.70
381.88