design notes and statistical tables
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Design Notes&
Statistical Tables
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CONTENTS
1. Highway Capacity Design (Older Version).....012. Highway Design ..033. Highway Capacity Design (HCM2000)..114. Traffic Incident Analysis 315. Traffic Signal Design Summary .336. Geometric Design Equations...377. Marshall Mix Design Criteria..388. Roadway Design..399. Key to Structural Catalogue ....4310.Statistical Tables .53
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1. Highway Capacity Design (HCM 1986)Determination of Level of Service
Vv =
PHF
where, v = flow rate for 15 minutes total flow for both directions in vphV = full hour volume total for both directions in vph
PHF = Peak Hour Factor
Service flow rate for two-lane roads on general terrain segments is given by the equation
SFi = 2800 x (v/c)i x fd x fw x fHV
where, SFi is the total service flow rate in both directions for prevailing
roadway and traffic conditions for level of service i in vph
(v/c)i is the ratio of flow rate to ideal capacity for level of service i
fd is the adjustment factor for directional distribution of traffic
fw is the adjustment factor for narrow lanes and restricted shoulder widths
fHV is the adjustment factor for the presence of heavy vehicles in the
traffic stream where,
fHV = 1/ [1+ PT (ET 1) + PB (EB 1) ]
and, PT = Proportion of trucks in the traffic stream
expressed as a decimal
PB = Proportion of buses in the traffic stream
expressed as a decimal
ET = Passenger Car Equivalent for trucks
EB = Passenger Car Equivalent for buses
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Service flow rate for two lane roads on specific grade segments for any upgrade speed is
given by the equation:
SFi = 2800 x (v/c)i x fd x fw x fg x fHV
and, fg = 1/ [ 1+ ( Pp Ip ) ]
fHV = 1/ [ 1+ PHV ( EHV 1 )
Ip = 0.02 ( E E0 )
EHV = 1+ ( 0.25 + PT/HV)( E 1)
where, SFi = Service flow rate for LOS i or speed i, total vph in both directions for
prevailing roadway and traffic conditions
(v/c)i = volume/capacity ratio for LOS i or speed i
fd = adjustment factor for directional distribution of traffic
fw = adjustment factor for narrow lanes and restricted shoulder widths
fg = adjustment factor for the effects of grades on passenger cars
fHV = adjustment factor for presence of heavy vehicles in the upgradetraffic stream
Pp = proportion of passenger cars in the upgrade traffic stream, expressedas a decimal
Ip = impedance factor for passenger cars
PHV = total proportion of heavy vehicles in the upgrade traffic stream
EHV = passenger car equivalent for specific mix on heavy vehicles presentin the upgrade traffic stream
E = base passenger car equivalent for a given percent of grade and agiven speed
E0 = base passenger car equivalent for zero percent grade and a given
upgrade speed
PT/HV = proportion of trucks among heavy vehicles
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2. Highway DesignTable 2.1 Equivalent Passenger Car Unit (PCU) Factors
For two-lane, two-way roads
Vehicle Type PCU FactorFlat Terrain Rolling Terrain Mountainous Terrain
PassengerCar 1.0 1.0 1.0
Small Bus 2.0 3.4 6.0
Large Bus 2.2 5.0 10.0
Light Truck 2.0 4.0 7.0
Medium Truck 2.2 5.0 10.0
Heavy Truck 2.2 5.0 10.0
Motor Cycle 0.5 0.5 0.5
Bicycle 1.0 1.0 1.0
Animal Drawn Carts 4.0 10 24.0
Source: Road Development Authority, Traffic and Planning Division
Table 2.2 Equivalent Passenger Car Unit (PCU) Factors
For multilane roads
Vehicle Type PCU Factor
Flat Terrain Rolling Terrain Mountainous Terrain
PassengerCar 1.0 1.0 1.0
Small Bus 1.5 3.0 5.0Large Bus 1.7 4.0 8.0
Light Truck 1.7 4.0 8.0
Medium Truck 1.7 4.0 8.0
Heavy Truck 1.7 4.0 8.0
Motor Cycle 0.5 0.5 0.5
Bicycle 1.0 1.0 1.0
Animal Drawn Carts 4.0 10.0 24.0
Source: Road Development Authority, Traffic and Planning Division
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Table 2.3 Level of Service criteria for general two-lane highway segmen
v/c ratio ( ratio of flow rate to an ideal capacity of 2800 pcph in both direct
LOS
Percent
TimeDelay
Level terrain Rolling terrain
AvgbSpeed
Percent no passing zones AvgbSpeed
Percent no passing zones AvgbSpeed0 20 40 60 80 100 0 20 40 60 80 100
A 30 94 0.15 0.12 0.09 0.07 0.05 0.04 92 0.15 0.10 0.07 0.05 0.04 0.03 90
B 45 89 0.27 0.24 0.21 0.19 0.17 0.16 87 0.26 0.23 0.19 0.17 0.15 0.13 87
C 60 84 0.43 0.39 0.36 0.34 0.33 0.32 82 0.42 0.39 0.35 0.32 0.30 0.28 79
D 75 81 0.64 0.62 0.60 0.59 0.58 0.57 79 0.62 0.57 0.52 0.48 0.46 0.43 73
E > 75 73 1.00 1.00 1.00 1.00 1.00 1.00 65 0.97 0.94 0.92 0.91 0.90 0.90 57
F 100 < 45 - - - - - -
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Table 2.4 Adjustment factors for directional distribution (fd) for two-lane highways
Directional
Distribution100/0 90/10 80/20 70/30 60/40 50/50
AdjustmentFactor
0.71 0.75 0.83 0.89 0.94 1.0
Source: Table 8-4, Highway Capacity Manual, Transport Research Board, National Research Council,Washington D.C., USA
Table 2.5 Adjustment factors for combined effect of narrow lanes and restricted
shoulder width (fw)
Usable
shoulderwidth
3.7 m lanes 3.4 m lanes 3.0 m lanes 2.7 m lanes
LOSA-D
LOSE
LOSA-D
LOSE
LOSA-D
LOSE
LOSA-D
LOSE
1.8 1.0 1.0 0.93 0.94 0.84 0.87 0.70 0.76
1.2 0.92 0.97 0.85 0.92 0.77 0.85 0.65 0.74
0.6 0.81 0.93 0.75 0.88 0.68 0.81 0.57 0.70
0 0.70 0.88 0.65 0.82 0.58 0.75 0.49 0.66
Note: where shoulder width is different on either side of the road use the average width
Source: Table 8-5, Highway Capacity Manual, Transport Research Board, National Research Council,
Washington D.C., USA
Table 2.6 Average passenger car equivalent for trucks and buses on two-lanehighways over different terrain segments
Vehicle Type LOSType of Terrain
Level Rolling Mountainous
Trucks ET
A 2.0 4.0 7.0
B & C 2.2 5.0 10.0
D & E 2.0 5.0 12.0
Buses EBA 1.8 3.0 5.7
B & C 2.0 3.4 5.7
D & E 1.6 2.9 6.5
Source: Table 8-6, Highway Capacity Manual, Transport Research Board, National Research Council,
Washington D.C., USA
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Table 2.7 Values of v/c ratiosa
vs speed, percent grade and percent no passing
zone for specific grades
Percent GradeAverage
Upgrade Speed (Kmph)
Percent no passing zones
0 20 40 60 80 100
3
89 0.27 0.23 0.19 0.17 0.14 0.12
85 0.42 0.38 0.33 0.31 0.29 0.2781 0.64 0.59 0.55 0.52 0.49 0.47
73 1.00 0.95 0.91 0.88 0.86 0.84
69 1.00 0.98 0.97 0.96 0.95 0.94
65 1.00 1.00 1.00 1.00 1.00 1.00
4
89 0.25 0.21 0.18 0.16 0.13 0.11
85 0.40 0.36 0.31 0.29 0.27 0.25
81 0.61 0.56 0.52 0.49 0.47 0.45
73 0.97 0.92 0.88 0.85 0.83 0.81
69 0.99 0.96 0.95 0.94 0.93 0.92
65 1.00 1.00 1.00 1.00 1.00 1.00
5
89 0.21 0.17 0.14 0.12 0.10 0.08
85 0.36 0.31 0.27 0.24 0.22 0.20
81 0.57 0.49 0.45 0.41 0.39 0.37
73 0.93 0.84 0.79 0.75 0.72 0.70
69 0.97 0.90 0.87 0.85 0.83 0.82
65 0.98 0.96 0.95 0.94 0.93 0.92
57 1.00 1.00 1.00 1.00 1.00 1.00
6
89 0.12 0.10 0.08 0.06 0.05 0.04
85 0.27 0.22 0.18 0.16 0.14 0.13
81 0.48 0.40 0.35 0.31 0.28 0.26
73 0.49 0.76 0.68 0.63 0.59 0.55
69 0.93 0.84 0.78 0.74 0.70 0.6765 0.97 0.91 0.87 0.83 0.81 0.78
57 1.00 0.96 0.95 0.93 0.91 0.90
49 1.00 0.99 0.99 0.98 0.98 0.98
7
89 0.00 0.00 0.00 0.00 0.00 0.00
85 0.13 0.10 0.08 0.07 0.05 0.04
81 0.34 0.27 0.22 0.18 0.15 0.12
73 0.77 0.65 0.55 0.46 0.40 0.35
69 0.86 0.75 0.67 0.60 0.54 0.48
65 0.93 0.82 0.75 0.69 0.64 0.59
57 1.00 0.91 0.87 0.82 0.79 0.76
49 1.00 0.95 0.92 0.90 0.88 0.86
a Ratio of flow rate to ideal capacity of 2800 pcph assuming car operation is unaffected by grade
Note: Interpolate for intermediate values of percent no passing zones round percent grade to the next
higher integer value
Source: Table 8-7, Highway Capacity Manual, Transport Research Board, National Research Council,Washington D.C., USA
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Table 2.8 Adjustment factors for directional distribution for specific grades (fd)
Percent of Traffic on Upgrade Adjustment Factor (fd)
100 0.58
90 0.64
80 0.7070 0.78
60 0.87
50 1.00
40 1.20
30 1.50
Source: Table 8-8, Highway Capacity Manual, Transport Research Board, National Research Council,Washington D.C., USA
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Table 2.9 Passenger car equivalent for specific grades
on two-lane highways E and E0
Grade % Length of grade
Meters
Average Upgrade Speed (kmph)
73 64 48
0 All 1.4 1.3 1.3
3
400 1.7 1.6 1.5800 2.0 1.8 1.7
1200 2.3 2.0 1.9
1600 2.6 2.3 2.1
2400 3.4 2.9 2.5
3200 4.6 3.7 2.9
4800 7.3 5.6 3.8
6400 11.3 7.7 4.9
4
400 1.8 1.7 1.6
800 2.2 2.0 1.9
1200 2.7 2.3 2.1
1600 3.2 2.7 2.4
2400 4.7 3.8 3.1
3200 6.9 5.3 3.8
4800 12.5 9.0 5.5
6400 22.8 13.8 7.4
5
400 2.0 1.8 1.7
800 2.5 2.2 2.0
1200 3.1 2.7 2.4
1600 4.0 3.3 2.8
2400 6.3 4.9 3.8
3200 10.2 7.5 4.8
4800 22.0 14.6 7.8
6400 55.0 25.0 11.5
6
400 2.1 1.9 1.8
800 2.8 2.4 2.2
1200 3.7 3.1 2.7
1600 4.9 4.0 3.3
2400 8.5 6.4 4.7
3200 15.3 10.7 6.3
4800 38.0 23.9 11.3
6400 90.0 45.0 18.1
7
400 2.2 2.0 1.9
800 3.2 2.7 2.4
1200 4.3 3.6 3.01600 6.1 4.8 3.8
2400 11.5 8.4 5.8
3200 22.8 15.4 8.2
4800 66.0 38.5 16.1
6400 A a 28.0
a - Speed not attainable on grade specified. Note: Round percent Grade to next higher integer value
Source: Table 8-9, Highway Capacity Manual, Transport Research Board, National Research Council,
Washington D.C., USA
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Table 2.10 Equivalent factors for different axle loads
Wheel Load
(single and dual)(10
3Kg)
Axle Load
(103
Kg)Equivalent Factor
1.5 3.0 0.012.0 4.0 0.04
2.5 5.0 0.11
3.0 6.0 0.25
3.5 7.0 0.50
4.0 8.0 0.91
4.5 9.0 1.55
5.0 10.0 2.50
5.5 11.0 3.83
6.0 12.0 5.67
6.5 13.0 8.13
7.0 14.0 11.37.5 15.0 15.5
8.0 16.0 20.7
8.5 17.0 27.2
9.0 18.0 35.2
9.5 19.0 44.9
10.0 20.0 56.5Source: Road Note 31, Fourth Edition 1993- Transport Research Laboratory, UK.
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3. Highway Capacity Design (HCM 2000)
f
VSFFS
HV
f
FM 0125.0+=
ffBFFSFFS ALS =
ffPHF
V
HVG
pv**
=
( ) ( )1111
++=
EPEPf
RRTT
HV
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fvFFSATS npp = 0125.0
fBPTSFPTSFnpd+=
( )e vBPTSF p000879.01100 =
c
vcv p=
LPHF
VVkmT t
= 25.015
LVVkmT t*60=
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ATS
VkmTTT
15
15 =
ffPHF
V
HVG
dv**
=
ffPHF
V
HVG
o
ov**
=
( ) ( ) ( ) ( )1111*11
+++=
EPEPPEPPf
RRTTC TTCTTC
HV
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( ) fvvFFSATS npoddd += 0125.0
fBPTSFPTSF npdd +=
= eav
BPTSFb
d
d 1100
( )LLLLL deplutd ++=
L
L
f
LfLLPTSFPTSF
t
de
pl
plpldud
pl
++++
=2
1
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( )
L
L
LfLfLfLPTSF
PTSFt
de
depl
deplplplud
pl
+++
=
2'
'
2
1
f
L
f
LLL
LATSATS
pl
de
pl
pl
du
tdpl
++++
=
1
2
*
( )
++
++
=
L
LLff
L
f
LL
LATSATS
de
dede
plpl
de
pl
pl
u
tdpl
'
'
11
2
*
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TTTTTT
PTSFTTPTSFTTPTSFTTPTSF
n
nnc
+++
+++=
...
*...**
21
2211
TTTTTT
VkmTVkmTVkmTATS
n
nc
+++
+++=
...
...
21
21
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4. TRAFFIC INCIDENT ANALYSISShock Wave Analysis
The velocity of a shock-wave that is created when the state of a traffic stream changes
from (qi, ki) to (qj, kj) is
ij
ij
ijkk
qqw
=
For forward and reverse shock-waveswij >0 and wij < 0 respectively.
Total delay
The total upstream delay is given by
)()( 1
3
2
1txAkk
i
ii=
whereAi (x,t) is the area in the (x,t) plane that has a density ki
The delay can also be computed when q2=0 to be equal to
( )1331
2
2 qq
qq
Where is the length of the incident
(when q2=0)
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5. TRAFFIC SIGNAL DESIGN SUMMARY1. Identify lane geometry2. Identify hourly volumes3. Calculate pedestrian crossing times (walking speed 1.0-1.5m/s)4. Calculate amber and all red times (reaction time- 1.0sec, Table 15)5. Check through plus right turns (assume 2 R.Ts during amber when separate
turning lane is available)
Gi +Ii) calculate new cycle time based onnew Gi + Iis. Recalculate effective green times.
16.Calculate green and red times.
Table 11 Right turn equivalencies
1.Right turn lane available 1 R.T. = 1.05 through
2. Right turn lane not available
Opposing Volume/Hour 0-299 300-599 600-999 >1000i. No separate turn phase
equivalent through per R.T.1.2 2.0 4.0 6.0
ii. Separate turn phase 1.2 1.2 1.2 1.2
Reference: TRB (1980)
Table 12 Right and left turning movement multiplicative factors
Pedestrian Flow per Hour Multiplicative Factor
Low 0-99 1.00
Moderate 100-599 1.25
Heavy 600-1199 1.50Very Heavy > 1200 2.00
Reference: TRB (1980)
Table 13 Lane width adjustment factors
Lane Width Multiplicative Factor
2.45 3.00 m 1.10
3.05 3.95 m 1.00
3.97 4.85 m 0.90
4.88 5.76 m 0.80
Reference: TRB (1980)
>
7
4=
10cycleperpedestrianifsec
10cycleperpedestrianifsec
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Table 14 Level of Service
Maximum Sum of Critical Lane Volumes
Level of Service Two Phase Three Phase Four or more
Phases
A 1000 950 900
B 1200 1140 1080C 1400 1340 1270
D 1600 1530 1460
E 1800 1720 1650
Reference: TRB (1980)
Table 15 Minimum amber periods
Speed Limit
(km/hr)
Minimum Amber Period
(sec.)
VL 50 3
50 < VL 60 4
60 < VL 70 570 < VL 80 6
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6. Geometric Design EquationsStopping Sight Distance (SSD)
2546.3
2
VVtSSD R +=
Where,
SSD is the Stopping Sight Distance in meters
V is the Design Speed of the vehicle in km/h
tRis the total Reaction Time
is the coefficient of friction
Minimum Radius of a Horizontal Curve
( )maxmax
2
min127 fe
VR
+=
Critical Conditions,
Sliding;
( )feV
R+
=127
2
Overturning;
( )H
Te
VR
2127
2
+=
For Vertical Curves
Length of Curve (L)Sight Distance (S)
C
ASL
2
=
C, a constant for crest curves
[ ]221200 hhC +=
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7. MARSHALL MIX DESIGN CRITERIATable 1 Minimum VMA requirement
Nominal Maximum Particle Size Minimum VMA(mm) (U.S.) (percent)
63 2.5 inch 11
50 2.0 inch 11.5
37.5 1.5 inch 12
25 1.0 inch 13
19 0.75 inch 14
12.5 0.5 inch 15
9.5 0.375 inch 16
4.75 No. 4 sieve 18
2.36 No. 8 sieve 21
1.18 No. 16 sieve 23.5
Table 2 Mix Design Criteria
Mix Criteria Light Traffic
(106 ESALS)
min max min max min max
Compaction
(number of blows oneach end of the sample)
35 50 75
Stability (minimum)2224 N
(500 lbs.)
3336 N
(750 lbs.)
6672 N
(1500 lbs.)
Flow (0.25mm/0.01inch) 8 20 8 18 8 16
Percent air voids 3 5 3 5 3 5
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Chapter 5 Roadway Design
2 Page Revised: 10/21/2008
Figure 1: Design chart for flexible pavements based on using mean values for each input
AASHTO Guide for Design of Pavement Structures, 1993
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Determination of Layer Thicknesses
1
11
a
SND
22
122
ma
SNSND
( )
33
213
3ma
SNSNSND
+
D1
D2
D3SN3
SN2
SN1
Roadbed Course
Surface Course
Base Course
Subbase Course
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Structural Coefficients
Layer Structural Coefficient
Single Surface Dressing 0.1
Sand Seal, hot bitumen application 0.05-0.01
DBST 0.2
Asphalt concrete (high stability) 0.4
Hot rolled asphalt (surfacing) 0.35-0.40
Flexible bituminous surfacing 0.3
Bitumen macadam road base 0.3
Bitumen bound base, Asphalt treated base 0.3
Penetration macadam 0.2
Dense graded crushed stone base (CBR 110) 0.14
Water bound and dry bound macadam 0.14
Single sized aggregate base 0.125
Granular base (CBR 80) 0.12
Granular base (CBR 30) 0.11
Granular sub base (CBR 20) 0.095
Capping layer (CBR 15) 0.09
Granular sub base/ Capping layer (CBR8) 0.07
Source: AASHTO 1993
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KEY TO STRUCTURAL CATALOGUE
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CHART 1 GRANULAR ROADBASE / SURFACE DRESSING
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CHART 2 COMPOSITE ROAD BASE (UNBOUND & CEMENTED) / SURFACE DRESSING
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CHART 3 GRANULAR ROADBASE / SEMI-STRUCTURAL SURFACE
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CHART 4 COMPOSITE ROADBASE / SEMI - STRUCTURAL SURFACE
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CHART 5 GRANULAR ROADBASE / STRUCTURAL SURFACE
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CHART 6 COMPOSITE ROADBASE / STRUCTURAL SURFACE
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CHART 7 BITUMINOUS ROADBASE / SEMI-STRUCTURAL SURFACE
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CHART 8 CEMENTED ROADBASE / SURFACE DRESSING
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z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09-3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002
-3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003
-3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005
-3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007
-3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010
-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014
-2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019
-2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026
-2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036
-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048
-2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064
-2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084
-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110
-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143
-2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183
-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233
-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294
-1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367
-1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455
-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559
-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681
-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823
-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985
-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170
-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379
-0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611
-0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867
-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148
-0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451
-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776
-0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483
-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859
-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247
0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641
Standard Normal Cumulative Probability Table
Cumulative probabilities for NEGATIVE z-values are shown in the following table:
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z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.090.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359
0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753
0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141
0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517
0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879
0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224
0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549
0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852
0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133
0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389
1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621
1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830
1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015
1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177
1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319
1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441
1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545
1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633
1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706
1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767
2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817
2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857
2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890
2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916
2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936
2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952
2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964
2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974
2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981
2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986
3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.99903.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993
3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995
3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997
3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998
Standard Normal Cumulative Probability Table
Cumulative probabilities for POSITIVE z-values are shown in the following table:
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Statistical Tables for Students Binomial
Table 1 Binomial distrib ution prob ability fun ction
p
x 0.01 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
n=1 0 .9900 .9500 .9000 .8500 .8000 .7500 .7000 .6500 .6000 .5500 .5000 1
1 .0100 .0500 .1000 .1500 .2000 .2500 .3000 .3500 .4000 .4500 .5000 0
n=2 0 .9801 .9025 .8100 .7225 .6400 .5625 .4900 .4225 .3600 .3025 .2500 2
1 .0198 .0950 .1800 .2550 .3200 .3750 .4200 .4550 .4800 .4950 .5000 1
2 .0001 .0025 .0100 .0225 .0400 .0625 .0900 .1225 .1600 .2025 .2500 0
n=3 0 .9703 .8574 .7290 .6141 .5120 .4219 .3430 .2746 .2160 .1664 .1250 3
1 .0294 .1354 .2430 .3251 .3840 .4219 .4410 .4436 .4320 .4084 .3750 2
2 .0003 .0071 .0270 .0574 .0960 .1406 .1890 .2389 .2880 .3341 .3750 1
3 .0001 .0010 .0034 .0080 .0156 .0270 .0429 .0640 .0911 .1250 0
n=4 0 .9606 .8145 .6561 .5220 .4096 .3164 .2401 .1785 .1296 .0915 .0625 4
1 .0388 .1715 .2916 .3685 .4096 .4219 .4116 .3845 .3456 .2995 .2500 32 .0006 .0135 .0486 .0975 .1536 .2109 .2646 .3105 .3456 .3675 .3750 2
3 .0005 .0036 .0115 .0256 .0469 .0756 .1115 .1536 .2005 .2500 1
4 .0001 .0005 .0016 .0039 .0081 .0150 .0256 .0410 .0625 0
n=5 0 .9510 .7738 .5905 .4437 .3277 .2373 .1681 .1160 .0778 .0503 .0313 5
1 .0480 .2036 .3281 .3915 .4096 .3955 .3602 .3124 .2592 .2059 .1563 4
2 .0010 .0214 .0729 .1382 .2048 .2637 .3087 .3364 .3456 .3369 .3125 3
3 .0011 .0081 .0244 .0512 .0879 .1323 .1811 .2304 .2757 .3125 2
4 .0005 .0022 .0064 .0146 .0284 .0488 .0768 .1128 .1563 1
5 .0001 .0003 .0010 .0024 .0053 .0102 .0185 .0313 0
n=6 0 .9415 .7351 .5314 .3771 .2621 .1780 .1176 .0754 .0467 .0277 .0156 6
1 .0571 .2321 .3543 .3993 .3932 .3560 .3025 .2437 .1866 .1359 .0938 5
2 .0014 .0305 .0984 .1762 .2458 .2966 .3241 .3280 .3110 .2780 .2344 4
3 .0021 .0146 .0415 .0819 .1318 .1852 .2355 .2765 .3032 .3125 34 .0001 .0012 .0055 .0154 .0330 .0595 .0951 .1382 .1861 .2344 2
5 .0001 .0004 .0015 .0044 .0102 .0205 .0369 .0609 .0938 1
6 .0001 .0002 .0007 .0018 .0041 .0083 .0156 0
n=7 0 .9321 .6983 .4783 .3206 .2097 .1335 .0824 .0490 .0280 .0152 .0078 7
1 .0659 .2573 .3720 .3960 .3670 .3115 .2471 .1848 .1306 .0872 .0547 6
2 .0020 .0406 .1240 .2097 .2753 .3115 .3177 .2985 .2613 .2140 .1641 5
3 .0036 .0230 .0617 .1147 .1730 .2269 .2679 .2903 .2918 .2734 4
4 .0002 .0026 .0109 .0287 .0577 .0972 .1442 .1935 .2388 .2734 3
5 .0002 .0012 .0043 .0115 .0250 .0466 .0774 .1172 .1641 2
6 .0001 .0004 .0013 .0036 .0084 .0172 .0320 .0547 1
7 .0001 .0002 .0006 .0016 .0037 .0078 0
n=8 0 .9227 .6634 .4305 .2725 .1678 .1001 .0576 .0319 .0168 .0084 .0039 8
1 .0746 .2793 .3826 .3847 .3355 .2670 .1977 .1373 .0896 .0548 .0313 72 .0026 .0515 .1488 .2376 .2936 .3115 .2965 .2587 .2090 .1569 .1094 6
3 .0001 .0054 .0331 .0839 .1468 .2076 .2541 .2786 .2787 .2568 .2188 5
4 .0004 .0046 .0185 .0459 .0865 .1361 .1875 .2322 .2627 .2734 4
5 .0004 .0026 .0092 .0231 .0467 .0808 .1239 .1719 .2188 3
6 .0002 .0011 .0038 .0100 .0217 .0413 .0703 .1094 2
7 .0001 .0004 .0012 .0033 .0079 .0164 .0313 1
8 .0001 .0002 .0007 .0017 .0039 0
0.99 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 xp
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Statistical Tables for Students Binomial
p
x 0.01 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
n=9 0 0.9135 0.6302 0.3874 0.2316 0.1342 0.0751 0.0404 0.0207 0.0101 0.0046 0.0020 9
1 0.0830 0.2985 0.3874 0.3679 0.3020 0.2253 0.1556 0.1004 0.0605 0.0339 0.0176 8
2 0.0034 0.0629 0.1722 0.2597 0.3020 0.3003 0.2668 0.2162 0.1612 0.1110 0.0703 7
3 0.0001 0.0077 0.0446 0.1069 0.1762 0.2336 0.2668 0.2716 0.2508 0.2119 0.1641 6
4 0.0006 0.0074 0.0283 0.0661 0.1168 0.1715 0.2194 0.2508 0.2600 0.2461 55 0.0008 0.0050 0.0165 0.0389 0.0735 0.1181 0.1672 0.2128 0.2461 4
6 0.0001 0.0006 0.0028 0.0087 0.0210 0.0424 0.0743 0.1160 0.1641 3
7 0.0003 0.0012 0.0039 0.0098 0.0212 0.0407 0.0703 2
8 0.0001 0.0004 0.0013 0.0035 0.0083 0.0176 1
9 0.0001 0.0003 0.0008 0.0020 0
n=10 0 0.9044 0.5987 0.3487 0.1969 0.1074 0.0563 0.0282 0.0135 0.0060 0.0025 0.0010 10
1 0.0914 0.3151 0.3874 0.3474 0.2684 0.1877 0.1211 0.0725 0.0403 0.0207 0.0098 9
2 0.0042 0.0746 0.1937 0.2759 0.3020 0.2816 0.2335 0.1757 0.1209 0.0763 0.0439 8
3 0.0001 0.0105 0.0574 0.1298 0.2013 0.2503 0.2668 0.2522 0.2150 0.1665 0.1172 7
4 0.0010 0.0112 0.0401 0.0881 0.1460 0.2001 0.2377 0.2508 0.2384 0.2051 6
5 0.0001 0.0015 0.0085 0.0264 0.0584 0.1029 0.1536 0.2007 0.2340 0.2461 5
6 0.0001 0.0012 0.0055 0.0162 0.0368 0.0689 0.1115 0.1596 0.2051 4
7 0.0001 0.0008 0.0031 0.0090 0.0212 0.0425 0.0746 0.1172 3
8 0.0001 0.0004 0.0014 0.0043 0.0106 0.0229 0.0439 29 0.0001 0.0005 0.0016 0.0042 0.0098 1
10 0.0001 0.0003 0.0010 0
n=11 0 0.8953 0.5688 0.3138 0.1673 0.0859 0.0422 0.0198 0.0088 0.0036 0.0014 0.0005 11
1 0.0995 0.3293 0.3835 0.3248 0.2362 0.1549 0.0932 0.0518 0.0266 0.0125 0.0054 10
2 0.0050 0.0867 0.2131 0.2866 0.2953 0.2581 0.1998 0.1395 0.0887 0.0513 0.0269 9
3 0.0002 0.0137 0.0710 0.1517 0.2215 0.2581 0.2568 0.2254 0.1774 0.1259 0.0806 8
4 0.0014 0.0158 0.0536 0.1107 0.1721 0.2201 0.2428 0.2365 0.2060 0.1611 7
5 0.0001 0.0025 0.0132 0.0388 0.0803 0.1321 0.1830 0.2207 0.2360 0.2256 6
6 0.0003 0.0023 0.0097 0.0268 0.0566 0.0985 0.1471 0.1931 0.2256 5
7 0.0003 0.0017 0.0064 0.0173 0.0379 0.0701 0.1128 0.1611 4
8 0.0002 0.0011 0.0037 0.0102 0.0234 0.0462 0.0806 3
9 0.0001 0.0005 0.0018 0.0052 0.0126 0.0269 2
10 0.0002 0.0007 0.0021 0.0054 1
11 0.0002 0.0005 0
n=12 0 0.8864 0.5404 0.2824 0.1422 0.0687 0.0317 0.0138 0.0057 0.0022 0.0008 0.0002 12
1 0.1074 0.3413 0.3766 0.3012 0.2062 0.1267 0.0712 0.0368 0.0174 0.0075 0.0029 11
2 0.0060 0.0988 0.2301 0.2924 0.2835 0.2323 0.1678 0.1088 0.0639 0.0339 0.0161 10
3 0.0002 0.0173 0.0852 0.1720 0.2362 0.2581 0.2397 0.1954 0.1419 0.0923 0.0537 9
4 0.0021 0.0213 0.0683 0.1329 0.1936 0.2311 0.2367 0.2128 0.1700 0.1208 8
5 0.0002 0.0038 0.0193 0.0532 0.1032 0.1585 0.2039 0.2270 0.2225 0.1934 7
6 0.0005 0.0040 0.0155 0.0401 0.0792 0.1281 0.1766 0.2124 0.2256 6
7 0.0006 0.0033 0.0115 0.0291 0.0591 0.1009 0.1489 0.1934 5
8 0.0001 0.0005 0.0024 0.0078 0.0199 0.0420 0.0762 0.1208 4
9 0.0001 0.0004 0.0015 0.0048 0.0125 0.0277 0.0537 3
10 0.0002 0.0008 0.0025 0.0068 0.0161 2
11 0.0001 0.0003 0.0010 0.0029 1
12 0.0001 0.0002 0
n=13 0 0.8775 0.5133 0.2542 0.1209 0.0550 0.0238 0.0097 0.0037 0.0013 0.0004 0.0001 13
1 0.1152 0.3512 0.3672 0.2774 0.1787 0.1029 0.0540 0.0259 0.0113 0.0045 0.0016 12
2 0.0070 0.1109 0.2448 0.2937 0.2680 0.2059 0.1388 0.0836 0.0453 0.0220 0.0095 11
3 0.0003 0.0214 0.0997 0.1900 0.2457 0.2517 0.2181 0.1651 0.1107 0.0660 0.0349 10
4 0.0028 0.0277 0.0838 0.1535 0.2097 0.2337 0.2222 0.1845 0.1350 0.0873 9
5 0.0003 0.0055 0.0266 0.0691 0.1258 0.1803 0.2154 0.2214 0.1989 0.1571 8
6 0.0008 0.0063 0.0230 0.0559 0.1030 0.1546 0.1968 0.2169 0.2095 7
7 0.0001 0.0011 0.0058 0.0186 0.0442 0.0833 0.1312 0.1775 0.2095 6
8 0.0001 0.0011 0.0047 0.0142 0.0336 0.0656 0.1089 0.1571 5
9 0.0001 0.0009 0.0034 0.0101 0.0243 0.0495 0.0873 4
10 0.0001 0.0006 0.0022 0.0065 0.0162 0.0349 3
11 0.0001 0.0003 0.0012 0.0036 0.0095 2
12 0.0001 0.0005 0.0016 1
13 0.0001 0
0.99 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 x
p
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Statistical Tables for Students Binomial
p
x 0.01 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
n=14 0 0.8687 0.4877 0.2288 0.1028 0.0440 0.0178 0.0068 0.0024 0.0008 0.0002 0.0001 14
1 0.1229 0.3593 0.3559 0.2539 0.1539 0.0832 0.0407 0.0181 0.0073 0.0027 0.0009 13
2 0.0081 0.1229 0.2570 0.2912 0.2501 0.1802 0.1134 0.0634 0.0317 0.0141 0.0056 12
3 0.0003 0.0259 0.1142 0.2056 0.2501 0.2402 0.1943 0.1366 0.0845 0.0462 0.0222 11
4 0.0037 0.0349 0.0998 0.1720 0.2202 0.2290 0.2022 0.1549 0.1040 0.0611 10
5 0.0004 0.0078 0.0352 0.0860 0.1468 0.1963 0.2178 0.2066 0.1701 0.1222 9
6 0.0013 0.0093 0.0322 0.0734 0.1262 0.1759 0.2066 0.2088 0.1833 8
7 0.0002 0.0019 0.0092 0.0280 0.0618 0.1082 0.1574 0.1952 0.2095 7
8 0.0003 0.0020 0.0082 0.0232 0.0510 0.0918 0.1398 0.1833 6
9 0.0003 0.0018 0.0066 0.0183 0.0408 0.0762 0.1222 5
10 0.0003 0.0014 0.0049 0.0136 0.0312 0.0611 4
11 0.0002 0.0010 0.0033 0.0093 0.0222 3
12 0.0001 0.0005 0.0019 0.0056 2
13 0.0001 0.0002 0.0009 1
14 0.0001 0
n=15 0 0.8601 0.4633 0.2059 0.0874 0.0352 0.0134 0.0047 0.0016 0.0005 0.0001 15
1 0.1303 0.3658 0.3432 0.2312 0.1319 0.0668 0.0305 0.0126 0.0047 0.0016 0.0005 14
2 0.0092 0.1348 0.2669 0.2856 0.2309 0.1559 0.0916 0.0476 0.0219 0.0090 0.0032 13
3 0.0004 0.0307 0.1285 0.2184 0.2501 0.2252 0.1700 0.1110 0.0634 0.0318 0.0139 12
4 0.0049 0.0428 0.1156 0.1876 0.2252 0.2186 0.1792 0.1268 0.0780 0.0417 11
5 0.0006 0.0105 0.0449 0.1032 0.1651 0.2061 0.2123 0.1859 0.1404 0.0916 106 0.0019 0.0132 0.0430 0.0917 0.1472 0.1906 0.2066 0.1914 0.1527 9
7 0.0003 0.0030 0.0138 0.0393 0.0811 0.1319 0.1771 0.2013 0.1964 8
8 0.0005 0.0035 0.0131 0.0348 0.0710 0.1181 0.1647 0.1964 7
9 0.0001 0.0007 0.0034 0.0116 0.0298 0.0612 0.1048 0.1527 6
10 0.0001 0.0007 0.0030 0.0096 0.0245 0.0515 0.0916 5
11 0.0001 0.0006 0.0024 0.0074 0.0191 0.0417 4
12 0.0001 0.0004 0.0016 0.0052 0.0139 3
13 0.0001 0.0003 0.0010 0.0032 2
14 0.0001 0.0005 1
15 0
n=16 0 0.8515 0.4401 0.1853 0.0743 0.0281 0.0100 0.0033 0.0010 0.0003 0.0001 16
1 0.1376 0.3706 0.3294 0.2097 0.1126 0.0535 0.0228 0.0087 0.0030 0.0009 0.0002 15
2 0.0104 0.1463 0.2745 0.2775 0.2111 0.1336 0.0732 0.0353 0.0150 0.0056 0.0018 14
3 0.0005 0.0359 0.1423 0.2285 0.2463 0.2079 0.1465 0.0888 0.0468 0.0215 0.0085 13
4 0.0061 0.0514 0.1311 0.2001 0.2252 0.2040 0.1553 0.1014 0.0572 0.0278 12
5 0.0008 0.0137 0.0555 0.1201 0.1802 0.2099 0.2008 0.1623 0.1123 0.0667 11
6 0.0001 0.0028 0.0180 0.0550 0.1101 0.1649 0.1982 0.1983 0.1684 0.1222 10
7 0.0004 0.0045 0.0197 0.0524 0.1010 0.1524 0.1889 0.1969 0.1746 9
8 0.0001 0.0009 0.0055 0.0197 0.0487 0.0923 0.1417 0.1812 0.1964 8
9 0.0001 0.0012 0.0058 0.0185 0.0442 0.0840 0.1318 0.1746 7
10 0.0002 0.0014 0.0056 0.0167 0.0392 0.0755 0.1222 6
11 0.0002 0.0013 0.0049 0.0142 0.0337 0.0667 5
12 0.0002 0.0011 0.0040 0.0115 0.0278 4
13 0.0000 0.0002 0.0008 0.0029 0.0085 3
14 0.0001 0.0005 0.0018 2
15 0.0001 0.0002 1
16 0
n=17 0 0.8429 0.4181 0.1668 0.0631 0.0225 0.0075 0.0023 0.0007 0.0002 17
1 0.1447 0.3741 0.3150 0.1893 0.0957 0.0426 0.0169 0.0060 0.0019 0.0005 0.0001 16
2 0.0117 0.1575 0.2800 0.2673 0.1914 0.1136 0.0581 0.0260 0.0102 0.0035 0.0010 15
3 0.0006 0.0415 0.1556 0.2359 0.2393 0.1893 0.1245 0.0701 0.0341 0.0144 0.0052 14
4 0.0076 0.0605 0.1457 0.2093 0.2209 0.1868 0.1320 0.0796 0.0411 0.0182 13
5 0.0010 0.0175 0.0668 0.1361 0.1914 0.2081 0.1849 0.1379 0.0875 0.0472 12
6 0.0001 0.0039 0.0236 0.0680 0.1276 0.1784 0.1991 0.1839 0.1432 0.0944 11
7 0.0007 0.0065 0.0267 0.0668 0.1201 0.1685 0.1927 0.1841 0.1484 10
8 0.0001 0.0014 0.0084 0.0279 0.0644 0.1134 0.1606 0.1883 0.1855 9
9 0.0003 0.0021 0.0093 0.0276 0.0611 0.1070 0.1540 0.1855 8
10 0.0004 0.0025 0.0095 0.0263 0.0571 0.1008 0.1484 7
11 0.0001 0.0005 0.0026 0.0090 0.0242 0.0525 0.0944 6
12 0.0001 0.0006 0.0024 0.0081 0.0215 0.0472 5
13 0.0001 0.0005 0.0021 0.0068 0.0182 4
14 0.0001 0.0004 0.0016 0.0052 3
15 0.0001 0.0003 0.0010 2
16 0.0001 1
17 0
0.99 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 xp
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Statistical Tables for Students Binomial
p
x 0.01 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50
n=18 0 0.8345 0.3972 0.1501 0.0536 0.0180 0.0056 0.0016 0.0004 0.0001 18
1 0.1517 0.3763 0.3002 0.1704 0.0811 0.0338 0.0126 0.0042 0.0012 0.0003 0.0001 17
2 0.0130 0.1683 0.2835 0.2556 0.1723 0.0958 0.0458 0.0190 0.0069 0.0022 0.0006 16
3 0.0007 0.0473 0.1680 0.2406 0.2297 0.1704 0.1046 0.0547 0.0246 0.0095 0.0031 15
4 0.0093 0.0700 0.1592 0.2153 0.2130 0.1681 0.1104 0.0614 0.0291 0.0117 14
5 0.0014 0.0218 0.0787 0.1507 0.1988 0.2017 0.1664 0.1146 0.0666 0.0327 13
6 0.0002 0.0052 0.0301 0.0816 0.1436 0.1873 0.1941 0.1655 0.1181 0.0708 12
7 0.0010 0.0091 0.0350 0.0820 0.1376 0.1792 0.1892 0.1657 0.1214 11
8 0.0002 0.0022 0.0120 0.0376 0.0811 0.1327 0.1734 0.1864 0.1669 10
9 0.0004 0.0033 0.0139 0.0386 0.0794 0.1284 0.1694 0.1855 9
10 0.0001 0.0008 0.0042 0.0149 0.0385 0.0771 0.1248 0.1669 8
11 0.0001 0.0010 0.0046 0.0151 0.0374 0.0742 0.1214 7
12 0.0002 0.0012 0.0047 0.0145 0.0354 0.0708 6
13 0.0002 0.0012 0.0045 0.0134 0.0327 5
14 0.0002 0.0011 0.0039 0.0117 4
15 0.0002 0.0009 0.0031 3
16 0.0001 0.0006 2
17 0.0001 1
18 0
n=19 0 0.8262 0.3774 0.1351 0.0456 0.0144 0.0042 0.0011 0.0003 0.0001 19
1 0.1586 0.3774 0.2852 0.1529 0.0685 0.0268 0.0093 0.0029 0.0008 0.0002 18
2 0.0144 0.1787 0.2852 0.2428 0.1540 0.0803 0.0358 0.0138 0.0046 0.0013 0.0003 17
3 0.0008 0.0533 0.1796 0.2428 0.2182 0.1517 0.0869 0.0422 0.0175 0.0062 0.0018 16
4 0.0112 0.0798 0.1714 0.2182 0.2023 0.1491 0.0909 0.0467 0.0203 0.0074 15
5 0.0018 0.0266 0.0907 0.1636 0.2023 0.1916 0.1468 0.0933 0.0497 0.0222 14
6 0.0002 0.0069 0.0374 0.0955 0.1574 0.1916 0.1844 0.1451 0.0949 0.0518 13
7 0.0014 0.0122 0.0443 0.0974 0.1525 0.1844 0.1797 0.1443 0.0961 12
8 0.0002 0.0032 0.0166 0.0487 0.0981 0.1489 0.1797 0.1771 0.1442 11
9 0.0007 0.0051 0.0198 0.0514 0.0980 0.1464 0.1771 0.1762 10
10 0.0001 0.0013 0.0066 0.0220 0.0528 0.0976 0.1449 0.1762 9
11 0.0003 0.0018 0.0077 0.0233 0.0532 0.0970 0.1442 8
12 0.0004 0.0022 0.0083 0.0237 0.0529 0.0961 7
13 0.0001 0.0005 0.0024 0.0085 0.0233 0.0518 6
14 0.0001 0.0006 0.0024 0.0082 0.0222 5
15 0.0001 0.0005 0.0022 0.0074 4
16 0.0001 0.0005 0.0018 3
17 0.0001 0.0003 2
18 1
19 0
n=20 0 0.8179 0.3585 0.1216 0.0388 0.0115 0.0032 0.0008 0.0002 20
1 0.1652 0.3774 0.2702 0.1368 0.0576 0.0211 0.0068 0.0020 0.0005 0.0001 19
2 0.0159 0.1887 0.2852 0.2293 0.1369 0.0669 0.0278 0.0100 0.0031 0.0008 0.0002 18
3 0.0010 0.0596 0.1901 0.2428 0.2054 0.1339 0.0716 0.0323 0.0123 0.0040 0.0011 17
4 0.0133 0.0898 0.1821 0.2182 0.1897 0.1304 0.0738 0.0350 0.0139 0.0046 16
5 0.0022 0.0319 0.1028 0.1746 0.2023 0.1789 0.1272 0.0746 0.0365 0.0148 15
6 0.0003 0.0089 0.0454 0.1091 0.1686 0.1916 0.1712 0.1244 0.0746 0.0370 14
7 0.0020 0.0160 0.0545 0.1124 0.1643 0.1844 0.1659 0.1221 0.0739 13
8 0.0004 0.0046 0.0222 0.0609 0.1144 0.1614 0.1797 0.1623 0.1201 12
9 0.0001 0.0011 0.0074 0.0271 0.0654 0.1158 0.1597 0.1771 0.1602 11
10 0.0002 0.0020 0.0099 0.0308 0.0686 0.1171 0.1593 0.1762 10
11 0.0005 0.0030 0.0120 0.0336 0.0710 0.1185 0.1602 9
12 0.0001 0.0008 0.0039 0.0136 0.0355 0.0727 0.1201 8
13 0.0002 0.0010 0.0045 0.0146 0.0366 0.0739 7
14 0.0002 0.0012 0.0049 0.0150 0.0370 6
15 0.0003 0.0013 0.0049 0.0148 5
16 0.0003 0.0013 0.0046 4
17 0.0002 0.0011 3
18 0.0002 2
19 1
20 0
0.99 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 x
p
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Poisson Probability Sums:
r 0.1 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0 0.9048 0.8187 0.7788 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066
1 0.9953 0.9825 0.9735 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0.7725
2 0.9998 0.9989 0.9978 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0.9371
3 1.0000 0.9999 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0.9865
4 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9986 0.9977
5 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.99976 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
r 1 1.5 2 2.5 3 3.5 4 4.5 5
0 0.3679 0.2231 0.1353 0.0821 0.0498 0.0302 0.0183 0.0111 0.0067
1 0.7358 0.5578 0.4060 0.2873 0.1991 0.1359 0.0916 0.0611 0.0404
2 0.9197 0.8088 0.6767 0.5438 0.4232 0.3208 0.2381 0.1736 0.1247
3 0.9810 0.9344 0.8571 0.7576 0.6472 0.5366 0.4335 0.3423 0.2650
4 0.9963 0.9814 0.9473 0.8912 0.8153 0.7254 0.6288 0.5321 0.4405
5 0.9994 0.9955 0.9834 0.9580 0.9161 0.8576 0.7851 0.7029 0.6160
6 0.9999 0.9991 0.9955 0.9858 0.9665 0.9347 0.8893 0.8311 0.7622
7 1.0000 0.9998 0.9989 0.9958 0.9881 0.9733 0.9489 0.9134 0.8666
8 1.0000 0.9998 0.9989 0.9962 0.9901 0.9786 0.9597 0.9319
9 1.0000 0.9997 0.9989 0.9967 0.9919 0.9829 0.968210 0.9999 0.9997 0.9990 0.9972 0.9933 0.9863
11 1.0000 0.9999 0.9997 0.9991 0.9976 0.9945
12 1.0000 0.9999 0.9997 0.9992 0.9980
13 1.0000 0.9999 0.9997 0.9993
14 1.0000 0.9999 0.9998
15 1.0000 0.9999
16 1.0000
r 5.5 6 6.5 7 7.5 8 8.5 9 9.5
0 0.0041 0.0025 0.0015 0.0009 0.0006 0.0003 0.0002 0.0001 0.0001
1 0.0266 0.0174 0.0113 0.0073 0.0047 0.0030 0.0019 0.0012 0.0008
2 0.0884 0.0620 0.0430 0.0296 0.0203 0.0138 0.0093 0.0062 0.0042
3 0.2017 0.1512 0.1118 0.0818 0.0591 0.0424 0.0301 0.0212 0.0149
4 0.3575 0.2851 0.2237 0.1730 0.1321 0.0996 0.0744 0.0550 0.0403
5 0.5289 0.4457 0.3690 0.3007 0.2414 0.1912 0.1496 0.1157 0.0885
6 0.6860 0.6063 0.5265 0.4497 0.3782 0.3134 0.2562 0.2068 0.1649
7 0.8095 0.7440 0.6728 0.5987 0.5246 0.4530 0.3856 0.3239 0.2687
8 0.8944 0.8472 0.7916 0.7291 0.6620 0.5925 0.5231 0.4557 0.3918
9 0.9462 0.9161 0.8774 0.8305 0.7764 0.7166 0.6530 0.5874 0.5218
10 0.9747 0.9574 0.9332 0.9015 0.8622 0.8159 0.7634 0.7060 0.6453
11 0.9890 0.9799 0.9661 0.9467 0.9208 0.8881 0.8487 0.8030 0.7520
12 0.9955 0.9912 0.9840 0.9730 0.9573 0.9362 0.9091 0.8758 0.8364
13 0.9983 0.9964 0.9929 0.9872 0.9784 0.9658 0.9486 0.9261 0.8981
14 0.9994 0.9986 0.9970 0.9943 0.9897 0.9827 0.9726 0.9585 0.9400
15 0.9998 0.9995 0.9988 0.9976 0.9954 0.9918 0.9862 0.9780 0.966516 0.9999 0.9998 0.9996 0.9990 0.9980 0.9963 0.9934 0.9889 0.9823
17 1.0000 0.9999 0.9998 0.9996 0.9992 0.9984 0.9970 0.9947 0.9911
18 1.0000 0.9999 0.9999 0.9997 0.9993 0.9987 0.9976 0.9957
19 1.0000 1.0000 0.9999 0.9997 0.9995 0.9989 0.9980
20 1.0000 0.9999 0.9998 0.9996 0.9991
21 1.0000 0.9999 0.9998 0.9996
22 1.0000 0.9999 0.9999
23 1.0000 0.9999
24 1.0000
=
r
x
xp
0
);(
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r 10 11 12 13 14 15 16 17 18
0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
1 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
2 0.0028 0.0012 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000
3 0.0103 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.0000
4 0.0293 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.0001
5 0.0671 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003
6 0.1301 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.0010
7 0.2202 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.0029
8 0.3328 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.00719 0.4579 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154
10 0.5830 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304
11 0.6968 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.0549
12 0.7916 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.0917
13 0.8645 0.7813 0.6815 0.5730 0.4644 0.3632 0.2745 0.2009 0.1426
14 0.9165 0.8540 0.7720 0.6751 0.5704 0.4657 0.3675 0.2808 0.2081
15 0.9513 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867
16 0.9730 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.3751
17 0.9857 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.4686
18 0.9928 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.5622
19 0.9965 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.6509
20 0.9984 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.730721 0.9993 0.9977 0.9939 0.9859 0.9712 0.9469 0.9108 0.8615 0.7991
22 0.9997 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.8551
23 0.9999 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989
24 1.0000 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.9317
25 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554
26 1.0000 0.9999 0.9995 0.9987 0.9967 0.9925 0.9848 0.9718
27 0.9999 0.9998 0.9994 0.9983 0.9959 0.9912 0.9827
28 1.0000 0.9999 0.9997 0.9991 0.9978 0.9950 0.9897
29 1.0000 0.9999 0.9996 0.9989 0.9973 0.9941
30 0.9999 0.9998 0.9994 0.9986 0.9967
31 1.0000 0.9999 0.9997 0.9993 0.9982
32 1.0000 0.9999 0.9996 0.9990
33 0.9999 0.9998 0.9995
34 1.0000 0.9999 0.9998
35 1.0000 0.9999
36 0.9999
37 1.0000
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Chi-Square Distribution Table
20
The shaded area is equal to for 2 = 2
.
df 2.995
2.990
2.975
2.950
2.900
2.100
2.050
2.025
2.010
2.005
1 0.000 0.000 0.001 0.004 0.016 2.706 3.841 5.024 6.635 7.879
2 0.010 0.020 0.051 0.103 0.211 4.605 5.991 7.378 9.210 10.597
3 0.072 0.115 0.216 0.352 0.584 6.251 7.815 9.348 11.345 12.8384 0.207 0.297 0.484 0.711 1.064 7.779 9.488 11.143 13.277 14.860
5 0.412 0.554 0.831 1.145 1.610 9.236 11.070 12.833 15.086 16.750
6 0.676 0.872 1.237 1.635 2.204 10.645 12.592 14.449 16.812 18.548
7 0.989 1.239 1.690 2.167 2.833 12.017 14.067 16.013 18.475 20.278
8 1.344 1.646 2.180 2.733 3.490 13.362 15.507 17.535 20.090 21.955
9 1.735 2.088 2.700 3.325 4.168 14.684 16.919 19.023 21.666 23.589
10 2.156 2.558 3.247 3.940 4.865 15.987 18.307 20.483 23.209 25.188
11 2.603 3.053 3.816 4.575 5.578 17.275 19.675 21.920 24.725 26.757
12 3.074 3.571 4.404 5.226 6.304 18.549 21.026 23.337 26.217 28.300
13 3.565 4.107 5.009 5.892 7.042 19.812 22.362 24.736 27.688 29.819
14 4.075 4.660 5.629 6.571 7.790 21.064 23.685 26.119 29.141 31.319
15 4.601 5.229 6.262 7.261 8.547 22.307 24.996 27.488 30.578 32.801
16 5.142 5.812 6.908 7.962 9.312 23.542 26.296 28.845 32.000 34.26717 5.697 6.408 7.564 8.672 10.085 24.769 27.587 30.191 33.409 35.718
18 6.265 7.015 8.231 9.390 10.865 25.989 28.869 31.526 34.805 37.156
19 6.844 7.633 8.907 10.117 11.651 27.204 30.144 32.852 36.191 38.582
20 7.434 8.260 9.591 10.851 12.443 28.412 31.410 34.170 37.566 39.997
21 8.034 8.897 10.283 11.591 13.240 29.615 32.671 35.479 38.932 41.401
22 8.643 9.542 10.982 12.338 14.041 30.813 33.924 36.781 40.289 42.796
23 9.260 10.196 11.689 13.091 14.848 32.007 35.172 38.076 41.638 44.181
24 9.886 10.856 12.401 13.848 15.659 33.196 36.415 39.364 42.980 45.559
25 10.520 11.524 13.120 14.611 16.473 34.382 37.652 40.646 44.314 46.928
26 11.160 12.198 13.844 15.379 17.292 35.563 38.885 41.923 45.642 48.290
27 11.808 12.879 14.573 16.151 18.114 36.741 40.113 43.195 46.963 49.645
28 12.461 13.565 15.308 16.928 18.939 37.916 41.337 44.461 48.278 50.993
29 13.121 14.256 16.047 17.708 19.768 39.087 42.557 45.722 49.588 52.336
30 13.787 14.953 16.791 18.493 20.599 40.256 43.773 46.979 50.892 53.672
40 20.707 22.164 24.433 26.509 29.051 51.805 55.758 59.342 63.691 66.766
50 27.991 29.707 32.357 34.764 37.689 63.167 67.505 71.420 76.154 79.490
60 35.534 37.485 40.482 43.188 46.459 74.397 79.082 83.298 88.379 91.952
70 43.275 45.442 48.758 51.739 55.329 85.527 90.531 95.023 100.425 104.215
80 51.172 53.540 57.153 60.391 64.278 96.578 101.879 106.629 112.329 116.321
90 59.196 61.754 65.647 69.126 73.291 107.565 113.145 118.136 124.116 128.299
100 67.328 70.065 74.222 77.929 82.358 118.498 124.342 129.561 135.807 140.169
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d.f. 0.40 0.25 0.10 0.05 0.04 0.025 0.02 0.01 0.005 0.0025 0.001 0.0005
1 0.325 1.000 3.078 6.314 7.916 12.706 15.894 31.821 63.656 127.321 318.289 636.578
2 0.289 0.816 1.886 2.920 3.320 4.303 4.849 6.965 9.925 14.089 22.328 31.600
30.277 0.765 1.638 2.353 2.605 3.182 3.482 4.541 5.841 7.453 10.214 12.924
4 0.271 0.741 1.533 2.132 2.333 2.776 2.999 3.747 4.604 5.598 7.173 8.610
5 0.267 0.727 1.476 2.015 2.191 2.571 2.757 3.365 4.032 4.773 5.894 6.869
6 0.265 0.718 1.440 1.943 2.104 2.447 2.612 3.143 3.707 4.317 5.208 5.959
7 0.263 0.711 1.415 1.895 2.046 2.365 2.517 2.998 3.499 4.029 4.785 5.408
8 0.262 0.706 1.397 1.860 2.004 2.306 2.449 2.896 3.355 3.833 4.501 5.041
9 0.261 0.703 1.383 1.833 1.973 2.262 2.398 2.821 3.250 3.690 4.297 4.781
10 0.260 0.700 1.372 1.812 1.948 2.228 2.359 2.764 3.169 3.581 4.144 4.587
11 0.260 0.697 1.363 1.796 1.928 2.201 2.328 2.718 3.106 3.497 4.025 4.437
12 0.259 0.695 1.356 1.782 1.912 2.179 2.303 2.681 3.055 3.428 3.930 4.318
13 0.259 0.694 1.350 1.771 1.899 2.160 2.282 2.650 3.012 3.372 3.852 4.221
14 0.258 0.692 1.345 1.761 1.887 2.145 2.264 2.624 2.977 3.326 3.787 4.140
15 0.258 0.691 1.341 1.753 1.878 2.131 2.249 2.602 2.947 3.286 3.733 4.073
16 0.258 0.690 1.337 1.746 1.869 2.120 2.235 2.583 2.921 3.252 3.686 4.015
17 0.257 0.689 1.333 1.740 1.862 2.110 2.224 2.567 2.898 3.222 3.646 3.965
18 0.257 0.688 1.330 1.734 1.855 2.101 2.214 2.552 2.878 3.197 3.610 3.922
19 0.257 0.688 1.328 1.729 1.850 2.093 2.205 2.539 2.861 3.174 3.579 3.883
20 0.257 0.687 1.325 1.725 1.844 2.086 2.197 2.528 2.845 3.153 3.552 3.850
21 0.257 0.686 1.323 1.721 1.840 2.080 2.189 2.518 2.831 3.135 3.527 3.819
22 0.256 0.686 1.321 1.717 1.835 2.074 2.183 2.508 2.819 3.119 3.505 3.792
23 0.256 0.685 1.319 1.714 1.832 2.069 2.177 2.500 2.807 3.104 3.485 3.768
24 0.256 0.685 1.318 1.711 1.828 2.064 2.172 2.492 2.797 3.091 3.467 3.745
25 0.256 0.684 1.316 1.708 1.825 2.060 2.167 2.485 2.787 3.078 3.450 3.725
26 0.256 0.684 1.315 1.706 1.822 2.056 2.162 2.479 2.779 3.067 3.435 3.707
27 0.256 0.684 1.314 1.703 1.819 2.052 2.158 2.473 2.771 3.057 3.421 3.689
28 0.256 0.683 1.313 1.701 1.817 2.048 2.154 2.467 2.763 3.047 3.408 3.67429 0.256 0.683 1.311 1.699 1.814 2.045 2.150 2.462 2.756 3.038 3.396 3.660
30 0.256 0.683 1.310 1.697 1.812 2.042 2.147 2.457 2.750 3.030 3.385 3.646
31 0.256 0.682 1.309 1.696 1.810 2.040 2.144 2.453 2.744 3.022 3.375 3.633
32 0.255 0.682 1.309 1.694 1.808 2.037 2.141 2.449 2.738 3.015 3.365 3.622
33 0.255 0.682 1.308 1.692 1.806 2.035 2.138 2.445 2.733 3.008 3.356 3.611
34 0.255 0.682 1.307 1.691 1.805 2.032 2.136 2.441 2.728 3.002 3.348 3.601
35 0.255 0.682 1.306 1.690 1.803 2.030 2.133 2.438 2.724 2.996 3.340 3.591
36 0.255 0.681 1.306 1.688 1.802 2.028 2.131 2.434 2.719 2.990 3.333 3.582
37 0.255 0.681 1.305 1.687 1.800 2.026 2.129 2.431 2.715 2.985 3.326 3.574
38 0.255 0.681 1.304 1.686 1.799 2.024 2.127 2.429 2.712 2.980 3.319 3.566
39 0.255 0.681 1.304 1.685 1.798 2.023 2.125 2.426 2.708 2.976 3.313 3.558
40 0.255 0.681 1.303 1.684 1.796 2.021 2.123 2.423 2.704 2.971 3.307 3.55160 0.254 0.679 1.296 1.671 1.781 2.000 2.099 2.390 2.660 2.915 3.232 3.460
80 0.254 0.678 1.292 1.664 1.773 1.990 2.088 2.374 2.639 2.887 3.195 3.416
100 0 254 0 677 1 290 1 660 1 769 1 984 2 081 2 364 2 626 2 871 3 174 3 390
Column headings denote probabilities () above tabulated values.
TABLE of CRITICAL VALUES for STUDENT'S t DISTRIBUTIONS
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