II. The MultivariateNormal Distributionit is not enough to know that a sample could have come from a normal population; we must be clear that it is at the same time improbable that it has come from a population differing so much from the normal as to invalidate the use of the normal theory tests in further handling of the material.
E. S. Pearson, 1930 (quoted on page 1 in Tests of Normality, Henry C. Thode, Jr., 2002)
A.Review of the Univariate Normal Distribution
Normal Probability Distribution - expresses the probabilities of outcomes for a continuous random variable x with a particular symmetric and unimodal distribution. This density function is given bywhere = mean = standard deviation = 3.14159 e = 2.71828
but the probability is given byThis looks like a difficult integration problem! Will I have to integrate this function every time I want to calculate probabilities for some normal random variable?
Characteristics of the normal probability distribution are: -there are an infinite number of normal distributions, each defined by their unique combination of the mean and standard deviation - determines the central location and determines the spread or width -the distribution is symmetric about -it is unimodal - = Md = Mo -it is asymptotic with respect to the horizontal axis -the area under the curve is 1.0 -it is neither platykurtic nor leptokurtic -it follows the empirical rule:
Normal distributions with the same mean but different standard deviations:
Chart1
0.00548531950.00000389560.0000000001
0.01275263420.00002599960.0000000034
0.02785205770.00015076320.0000000765
0.0571444530.00075955310.0000013551
0.11014132150.00332471650.0000187022
0.19942789050.01264400760.0002010176
0.33921958390.04177808650.0016827291
0.54204500650.11993502070.0109706389
0.81367165630.29914183580.0557041154
1.14742005720.6482491640.2202826431
1.52004027681.22050748450.6784391678
1.89167778351.99651510061.6273433127
2.21155956192.83751667533.0400805535
2.42889939383.50378876974.4231191238
2.50599281723.75898922585.0119856345
2.42889939383.50378876974.4231191238
2.21155956192.83751667533.0400805535
1.89167778351.99651510061.6273433127
1.52004027681.22050748450.6784391678
1.14742005720.6482491640.2202826431
0.81367165630.29914183580.0557041154
0.54204500650.11993502070.0109706389
0.33921958390.04177808650.0016827291
0.19942789050.01264400760.0002010176
0.11014132150.00332471650.0000187022
0.0571444530.00075955310.0000013551
0.02785205770.00015076320.0000000765
0.01275263420.00002599960.0000000034
0.00548531950.00000389560.0000000001
m x
f(x)
Sheet1
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00.39904344222.50599281720.26602896153.75898922580.19952172115.0119856345
0.250.41170910682.42889939380.28540533283.50378876970.22608479944.4231191238
0.50.45216959892.21155956190.35242083642.83751667530.32893865223.0400805535
0.750.52863125461.89167778350.50087274561.99651510060.61449848491.6273433127
10.65787730451.52004027680.81933131321.22050748451.47397150330.6784391678
1.250.87152041121.14742005721.54261672130.6482491644.5396223050.2202826431
1.51.22899696970.81367165633.34289584510.299141835817.95199498490.0557041154
1.751.84486525670.54204500658.33784822880.119935020791.1523938050.0109706389
22.94794300650.339219583923.93599331320.0417780865594.27272977890.0016827291
2.255.01434376770.199427890579.08884867910.01264400764974.68845716070.0002010176
2.59.07924460990.1101413215300.77752492690.003324716553469.77263904550.0000187022
2.7517.49951127870.0571444531316.56358371720.0007595531737926.5734041250.0000013551
335.90398996990.02785205776632.9179428810.000150763213076162.99982720.0000000765
3.2578.41517185580.012752634238462.16619577360.0000259996297515683.4305660.0000000034
3.5182.304787610.0054853195256701.9601512720.00000389568691640725.506530.0000000001
00.51
111
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00.39904344222.50599281720.45216959892.21155956190.65787730451.5200402768
0.250.41170910682.42889939380.41170910682.42889939380.52863125461.8916777835
0.50.45216959892.21155956190.39904344222.50599281720.45216959892.2115595619
0.750.52863125461.89167778350.41170910682.42889939380.41170910682.4288993938
10.65787730451.52004027680.45216959892.21155956190.39904344222.5059928172
1.250.87152041121.14742005720.52863125461.89167778350.41170910682.4288993938
1.51.22899696970.81367165630.65787730451.52004027680.45216959892.2115595619
1.751.84486525670.54204500650.87152041121.14742005720.52863125461.8916777835
22.94794300650.33921958391.22899696970.81367165630.65787730451.5200402768
2.255.01434376770.19942789051.84486525670.54204500650.87152041121.1474200572
2.59.07924460990.11014132152.94794300650.33921958391.22899696970.8136716563
2.7517.49951127870.0571444535.01434376770.19942789051.84486525670.5420450065
335.90398996990.02785205779.07924460990.11014132152.94794300650.3392195839
3.2578.41517185580.012752634217.49951127870.0571444535.01434376770.1994278905
3.5182.304787610.005485319535.90398996990.02785205779.07924460990.1101413215
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m x
f(x)
m m m x
f(x)
Normal distributions with the same standard deviation but different means:
Chart3
0.00548531950.00084136450.0001005088
0.01275263420.00221647770.0003000298
0.02785205770.00548531950.0008413645
0.0571444530.01275263420.0022164777
0.11014132150.02785205770.0054853195
0.19942789050.0571444530.0127526342
0.33921958390.11014132150.0278520577
0.54204500650.19942789050.057144453
0.81367165630.33921958390.1101413215
1.14742005720.54204500650.1994278905
1.52004027680.81367165630.3392195839
1.89167778351.14742005720.5420450065
2.21155956191.52004027680.8136716563
2.42889939381.89167778351.1474200572
2.50599281722.21155956191.5200402768
2.42889939382.42889939381.8916777835
2.21155956192.50599281722.2115595619
1.89167778352.42889939382.4288993938
1.52004027682.21155956192.5059928172
1.14742005721.89167778352.4288993938
0.81367165631.52004027682.2115595619
0.54204500651.14742005721.8916777835
0.33921958390.81367165631.5200402768
0.19942789050.54204500651.1474200572
0.11014132150.33921958390.8136716563
0.0571444530.19942789050.5420450065
0.02785205770.11014132150.3392195839
0.01275263420.0571444530.1994278905
0.00548531950.02785205770.1101413215
m m m x
f(x)
Sheet1
000
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00.39904344222.50599281720.26602896153.75898922580.19952172115.0119856345
0.250.41170910682.42889939380.28540533283.50378876970.22608479944.4231191238
0.50.45216959892.21155956190.35242083642.83751667530.32893865223.0400805535
0.750.52863125461.89167778350.50087274561.99651510060.61449848491.6273433127
10.65787730451.52004027680.81933131321.22050748451.47397150330.6784391678
1.250.87152041121.14742005721.54261672130.6482491644.5396223050.2202826431
1.51.22899696970.81367165633.34289584510.299141835817.95199498490.0557041154
1.751.84486525670.54204500658.33784822880.119935020791.1523938050.0109706389
22.94794300650.339219583923.93599331320.0417780865594.27272977890.0016827291
2.255.01434376770.199427890579.08884867910.01264400764974.68845716070.0002010176
2.59.07924460990.1101413215300.77752492690.003324716553469.77263904550.0000187022
2.7517.49951127870.0571444531316.56358371720.0007595531737926.5734041250.0000013551
335.90398996990.02785205776632.9179428810.000150763213076162.99982720.0000000765
3.2578.41517185580.012752634238462.16619577360.0000259996297515683.4305660.0000000034
3.5182.304787610.0054853195256701.9601512720.00000389568691640725.506530.0000000001
00.51
111
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00.39904344222.50599281720.45216959892.21155956190.65787730451.5200402768
0.250.41170910682.42889939380.41170910682.42889939380.52863125461.8916777835
0.50.45216959892.21155956190.39904344222.50599281720.45216959892.2115595619
0.750.52863125461.89167778350.41170910682.42889939380.41170910682.4288993938
10.65787730451.52004027680.45216959892.21155956190.39904344222.5059928172
1.250.87152041121.14742005720.52863125461.89167778350.41170910682.4288993938
1.51.22899696970.81367165630.65787730451.52004027680.45216959892.2115595619
1.751.84486525670.54204500650.87152041121.14742005720.52863125461.8916777835
22.94794300650.33921958391.22899696970.81367165630.65787730451.5200402768
2.255.01434376770.19942789051.84486525670.54204500650.87152041121.1474200572
2.59.07924460990.11014132152.94794300650.33921958391.22899696970.8136716563
2.7517.49951127870.0571444535.01434376770.19942789051.84486525670.5420450065
335.90398996990.02785205779.07924460990.11014132152.94794300650.3392195839
3.2578.41517185580.012752634217.49951127870.0571444535.01434376770.1994278905
3.5182.304787610.005485319535.90398996990.02785205779.07924460990.1101413215
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f(x)
m m m x
f(x)
The Standard Normal Probability Distribution - the probability distribution associated with any normal random variable (usually denoted z) that has = 0 and = 1.
There are tables that can be used to obtain the results of the integration
for the standard normal random variable.
Some of the tables work from the cumulative standard normal probability distribution (the probability that a random value selected from the standard normal random variable falls between and some given value b > 0, i.e., P(- z b))There are tables that give the results of the integration (Table 1 of the Appendices in J&W).
Cumulative Standard Normal Distribution (J&W Table 1)
Sheet17
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Normal Distribution Graphs
0.50.251.5
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00.450.49999999980.54999999980.36-2.50.75210.60
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00.450.420.1586552598
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-30.00134996720.00297981860.022750062
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Normal Distribution Graphs
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
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Normal Distribution Table
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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m
x
f(x)
t-Distribution Table
0.0013499672
0.1
0.7521
0.7105
0.2251
0.1
0.15
0.29
0.43
0.29
0.05
0.08
0.19
0.23
0.27
0.38
0.44
0.5
0.42
0.21
0.09
0.07
0.03
0.02
0.01
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m
x
f(x)
Sheet3
0.00134996720.0010
0.10.080
0.75210.60
0.71050.550
0.22510.30
0.10.20
0.150.220.0000000025
0.290.260.0000007178
0.430.330.0000792151
0.290.250.0033749181
0.050.230.0568751551
0.080.330.3966381494
0.190.851.3749999995
0.230.430.3966381494
0.270.350.0568751551
0.380.290.0033749181
0.440.310.0000792151
0.50.330.0000007178
0.420.220.0000000025
0.210.150
0.090.050
0.070.030
0.030.020
0.020.010
0.010.0010
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0.00134996720.000000001
0.00297981860.000000019
0.00620967990.0000002871
0.01222443340.0000034008
0.0227500620.000031686
0.04005911360.0002326734
0.06680722880.0013499672
0.10564983910.0062096799
0.15865525980.022750062
0.22662727970.0668072288
0.330.1586552598
0.420.3085375326
0.450.4999999998
0.420.3085375326
0.330.1586552598
0.22662727970.0668072288
0.15865525980.022750062
0.10564983910.0062096799
0.06680722880.0013499672
0.04005911360.0002326734
0.0227500620.000031686
0.01222443340.0000034008
0.00620967990.0000002871
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0.0029798186
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0.0668072288
0.1056498391
0.1586552598
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0.45
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0.33
0.2266272797
0.1586552598
0.1056498391
0.0668072288
0.0400591136
0.022750062
0.0122244334
0.0062096799
0.0029798186
0.0013499672
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z
f(z)
Sheet11
0.0013499672
0.0029798186
0.0062096799
0.0122244334
0.022750062
0.0400591136
0.0668072288
0.1056498391
0.1586552598
0.2266272797
0.33
0.42
0.45
0.42
0.33
0.2266272797
0.1586552598
0.1056498391
0.0668072288
0.0400591136
0.022750062
0.0122244334
0.0062096799
0.0029798186
0.0013499672
m=5
x
f(x)
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m
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f(x)
m
x
z0.000.010.020.030.040.050.060.070.080.09z0.000.010.020.030.040.050.060.070.080.09
0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359:::::::::::
0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753:::::::::::
0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61411.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177
0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65171.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319
0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68791.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441
0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72241.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545
0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75491.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633
0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78521.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706
0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81331.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767
0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.83892.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817
1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.86212.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857
1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88302.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890
1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90152.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916
1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91772.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936
1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93192.50.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015
1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94412.60.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177
1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95452.70.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319
1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96332.80.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441
1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97062.90.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545
1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.97673.00.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633
2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.98173.10.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706
2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98573.20.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767
2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890
2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916
2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936
2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952
2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964
2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974
2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981
2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986
3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990
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degreesArea in Upper tail
of freedom0.100.050.0250.010.005
13.0786.31412.70631.82163.656
21.8862.9204.3036.9659.925
31.6382.3533.1824.5415.841
41.5332.1322.7763.7474.604
51.4762.0152.5713.3654.032
61.4401.9432.4473.1433.707
71.4151.8952.3652.9983.499
81.3971.8602.3062.8963.355
91.3831.8332.2622.8213.250
101.3721.8122.2282.7643.169
111.3631.7962.2012.7183.106
121.3561.7822.1792.6813.055
131.3501.7712.1602.6503.012
141.3451.7612.1452.6242.977
151.3411.7532.1312.6022.947
161.3371.7462.1202.5832.921
171.3331.7402.1102.5672.898
181.3301.7342.1012.5522.878
191.3281.7292.0932.5392.861
201.3251.7252.0862.5282.845
211.3231.7212.0802.5182.831
221.3211.7172.0742.5082.819
231.3191.7142.0692.5002.807
241.3181.7112.0642.4922.797
251.3161.7082.0602.4852.787
261.3151.7062.0562.4792.779
271.3141.7032.0522.4732.771
281.3131.7012.0482.4672.763
291.3111.6992.0452.4622.756
301.3101.6972.0422.4572.750
401.3031.6842.0212.4232.704
601.2961.6712.0002.3902.660
1201.2891.6581.9802.3582.617
1.2821.6451.9602.3262.576
degreesArea in Upper tail
of freedom0.100.050.0250.010.005
13.0786.31412.70631.82163.656
......
......
......
151.3411.7532.1312.6022.947
161.3371.7462.1202.5832.921
171.3331.7402.1102.5672.898
181.3301.7342.1012.5522.878
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Lets focus on a small part of the Cumulative Standard Normal Probability Distribution Table
Example: for a standard normal random variable z, what is the probability that z is between - and 0.43?
Sheet17
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Normal Distribution Graphs
0.50.251.5
-30.00134996720.00000000100.01
-2.750.00297981860.00000001900.02
-2.50.00620967990.000000287100.03
-2.250.01222443340.000003400800.04
-20.0227500620.00003168600.06
-1.750.04005911360.000232673400.09
-1.50.06680722880.00134996720.0000000010.12
-1.250.10564983910.00620967990.00000028710.15
-10.15865525980.0227500620.0000316860.19
-0.750.22662727970.06680722880.00134996720.240.52.5
-0.50.330.15865525980.0227500620.29-30.00134996720.0010
-0.250.420.30853753260.15865525980.33-2.750.10.080
00.450.49999999980.54999999980.36-2.50.75210.60
-0.250.420.30853753260.15865525980.33-2.250.71050.550
-0.50.330.15865525980.0227500620.29-20.22510.30
-0.750.22662727970.06680722880.00134996720.24-1.750.10.20
-10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025
-1.250.10564983910.00620967990.00000028710.15-1.250.290.260.0000007178
-1.50.06680722880.00134996720.0000000010.12-10.430.330.0000792151
-1.750.04005911360.000232673400.09-0.750.290.250.0033749181
-20.0227500620.00003168600.06-0.50.050.230.0568751551
-2.250.01222443340.000003400800.04-0.250.080.330.3966381494
-2.50.00620967990.000000287100.0300.190.851.3749999995
-2.750.00297981860.00000001900.02-0.250.230.430.3966381494
-30.00134996720.00000000100.01-0.50.270.350.0568751551
-0.750.380.290.0033749181
-10.440.310.0000792151
-1.250.50.330.0000007178
-1.50.420.220.0000000025
-1.750.210.150
-20.090.050
-2.250.070.030
-2.50.030.020
-2.750.020.010
-30.010.0010
-40.000031686
-3.750.0000884446
-3.50.0002326734
-3.250.0005770865
-30.00134996720.00057708650.000031686
-2.750.00297981860.00134996720.0000884446
-2.50.00620967990.00297981860.0002326734
-2.250.01222443340.00620967990.0005770865
-20.0227500620.01222443340.0013499672
-1.750.04005911360.0227500620.0029798186
-1.50.06680722880.04005911360.0062096799
-1.250.10564983910.06680722880.0122244334
-10.15865525980.10564983910.022750062
-0.750.22662727970.15865525980.0400591136
-0.50.330.22662727970.0668072288
-0.250.420.330.1056498391
00.450.420.1586552598
-0.250.420.450.2266272797
-0.50.330.420.33
-0.750.22662727970.330.42
-10.15865525980.22662727970.45
-1.250.10564983910.15865525980.42
-1.50.06680722880.10564983910.33
-1.750.04005911360.06680722880.2266272797
-20.0227500620.04005911360.1586552598
-2.250.01222443340.0227500620.1056498391
-2.50.00620967990.01222443340.0668072288
-2.750.00297981860.00620967990.0400591136
-30.00134996720.00297981860.022750062
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Normal Distribution Graphs
000
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Normal Distribution Table
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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m
x
f(x)
t-Distribution Table
0.0013499672
0.1
0.7521
0.7105
0.2251
0.1
0.15
0.29
0.43
0.29
0.05
0.08
0.19
0.23
0.27
0.38
0.44
0.5
0.42
0.21
0.09
0.07
0.03
0.02
0.01
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m
x
f(x)
Sheet3
0.00134996720.0010
0.10.080
0.75210.60
0.71050.550
0.22510.30
0.10.20
0.150.220.0000000025
0.290.260.0000007178
0.430.330.0000792151
0.290.250.0033749181
0.050.230.0568751551
0.080.330.3966381494
0.190.851.3749999995
0.230.430.3966381494
0.270.350.0568751551
0.380.290.0033749181
0.440.310.0000792151
0.50.330.0000007178
0.420.220.0000000025
0.210.150
0.090.050
0.070.030
0.030.020
0.020.010
0.010.0010
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Sheet4
0.00134996720.000000001
0.00297981860.000000019
0.00620967990.0000002871
0.01222443340.0000034008
0.0227500620.000031686
0.04005911360.0002326734
0.06680722880.0013499672
0.10564983910.0062096799
0.15865525980.022750062
0.22662727970.0668072288
0.330.1586552598
0.420.3085375326
0.450.4999999998
0.420.3085375326
0.330.1586552598
0.22662727970.0668072288
0.15865525980.022750062
0.10564983910.0062096799
0.06680722880.0013499672
0.04005911360.0002326734
0.0227500620.000031686
0.01222443340.0000034008
0.00620967990.0000002871
0.00297981860.000000019
0.00134996720.000000001
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Sheet5
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z,t
Sheet8
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0.000000001
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0
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f(z,t)
z,t
Sheet9
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m
x
f(x)
Sheet10
0.0013499672
0.0029798186
0.0062096799
0.0122244334
0.022750062
0.0400591136
0.0668072288
0.1056498391
0.1586552598
0.2266272797
0.33
0.42
0.45
0.42
0.33
0.2266272797
0.1586552598
0.1056498391
0.0668072288
0.0400591136
0.022750062
0.0122244334
0.0062096799
0.0029798186
0.0013499672
0
z
f(z)
Sheet11
0.0013499672
0.0029798186
0.0062096799
0.0122244334
0.022750062
0.0400591136
0.0668072288
0.1056498391
0.1586552598
0.2266272797
0.33
0.42
0.45
0.42
0.33
0.2266272797
0.1586552598
0.1056498391
0.0668072288
0.0400591136
0.022750062
0.0122244334
0.0062096799
0.0029798186
0.0013499672
m=5
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f(x)
Sheet12
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m
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f(x)
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f(x)
m
x
z0.000.010.020.030.040.050.060.070.080.09z0.000.010.020.030.040.050.060.070.080.09
0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359:::::::::::
0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753:::::::::::
0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61411.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177
0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65171.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319
0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68791.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441
0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72241.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545
0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75491.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633
0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78521.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706
0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81331.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767
0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.83892.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817
1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.86212.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857
1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88302.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890
1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90152.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916
1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91772.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936
1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93192.50.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015
1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94412.60.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177
1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95452.70.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319
1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96332.80.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441
1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97062.90.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545
1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.97673.00.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633
2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.98173.10.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706
2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98573.20.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767
2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890
2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916
2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936
2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952
2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964
2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974
2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981
2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986
3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990
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degreesArea in Upper tail
of freedom0.100.050.0250.010.005
13.0786.31412.70631.82163.656
21.8862.9204.3036.9659.925
31.6382.3533.1824.5415.841
41.5332.1322.7763.7474.604
51.4762.0152.5713.3654.032
61.4401.9432.4473.1433.707
71.4151.8952.3652.9983.499
81.3971.8602.3062.8963.355
91.3831.8332.2622.8213.250
101.3721.8122.2282.7643.169
111.3631.7962.2012.7183.106
121.3561.7822.1792.6813.055
131.3501.7712.1602.6503.012
141.3451.7612.1452.6242.977
151.3411.7532.1312.6022.947
161.3371.7462.1202.5832.921
171.3331.7402.1102.5672.898
181.3301.7342.1012.5522.878
191.3281.7292.0932.5392.861
201.3251.7252.0862.5282.845
211.3231.7212.0802.5182.831
221.3211.7172.0742.5082.819
231.3191.7142.0692.5002.807
241.3181.7112.0642.4922.797
251.3161.7082.0602.4852.787
261.3151.7062.0562.4792.779
271.3141.7032.0522.4732.771
281.3131.7012.0482.4672.763
291.3111.6992.0452.4622.756
301.3101.6972.0422.4572.750
401.3031.6842.0212.4232.704
601.2961.6712.0002.3902.660
1201.2891.6581.9802.3582.617
1.2821.6451.9602.3262.576
degreesArea in Upper tail
of freedom0.100.050.0250.010.005
13.0786.31412.70631.82163.656
......
......
......
151.3411.7532.1312.6022.947
161.3371.7462.1202.5832.921
171.3331.7402.1102.5672.898
181.3301.7342.1012.5522.878
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Example: for a standard normal random variable z, what is the probability that z is between 0 and 2.0?2.0
Chart2
0.0013499672
0.0029798186
0.0062096799
0.0122244334
0.022750062
0.0400591136
0.0668072288
0.1056498391
0.1586552598
0.2266272797
0.33
0.42
0.45
0.42
0.33
0.2266272797
0.1586552598
0.1056498391
0.0668072288
0.0400591136
0.022750062
0.0122244334
0.0062096799
0.0029798186
0.0013499672
0
z
f(z)
Sheet17
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Normal Distribution Graphs
0.50.251.5
-30.00134996720.00000000100.01
-2.750.00297981860.00000001900.02
-2.50.00620967990.000000287100.03
-2.250.01222443340.000003400800.04
-20.0227500620.00003168600.06
-1.750.04005911360.000232673400.09
-1.50.06680722880.00134996720.0000000010.12
-1.250.10564983910.00620967990.00000028710.15
-10.15865525980.0227500620.0000316860.19
-0.750.22662727970.06680722880.00134996720.240.52.5
-0.50.330.15865525980.0227500620.29-30.00134996720.0010
-0.250.420.30853753260.15865525980.33-2.750.10.080
00.450.49999999980.54999999980.36-2.50.75210.60
-0.250.420.30853753260.15865525980.33-2.250.71050.550
-0.50.330.15865525980.0227500620.29-20.22510.30
-0.750.22662727970.06680722880.00134996720.24-1.750.10.20
-10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025
-1.250.10564983910.00620967990.00000028710.15-1.250.290.260.0000007178
-1.50.06680722880.00134996720.0000000010.12-10.430.330.0000792151
-1.750.04005911360.000232673400.09-0.750.290.250.0033749181
-20.0227500620.00003168600.06-0.50.050.230.0568751551
-2.250.01222443340.000003400800.04-0.250.080.330.3966381494
-2.50.00620967990.000000287100.0300.190.851.3749999995
-2.750.00297981860.00000001900.02-0.250.230.430.3966381494
-30.00134996720.00000000100.01-0.50.270.350.0568751551
-0.750.380.290.0033749181
-10.440.310.0000792151
-1.250.50.330.0000007178
-1.50.420.220.0000000025
-1.750.210.150
-20.090.050
-2.250.070.030
-2.50.030.020
-2.750.020.010
-30.010.0010
-40.000031686
-3.750.0000884446
-3.50.0002326734
-3.250.0005770865
-30.00134996720.00057708650.000031686
-2.750.00297981860.00134996720.0000884446
-2.50.00620967990.00297981860.0002326734
-2.250.01222443340.00620967990.0005770865
-20.0227500620.01222443340.0013499672
-1.750.04005911360.0227500620.0029798186
-1.50.06680722880.04005911360.0062096799
-1.250.10564983910.06680722880.0122244334
-10.15865525980.10564983910.022750062
-0.750.22662727970.15865525980.0400591136
-0.50.330.22662727970.0668072288
-0.250.420.330.1056498391
00.450.420.1586552598
-0.250.420.450.2266272797
-0.50.330.420.33
-0.750.22662727970.330.42
-10.15865525980.22662727970.45
-1.250.10564983910.15865525980.42
-1.50.06680722880.10564983910.33
-1.750.04005911360.06680722880.2266272797
-20.0227500620.04005911360.1586552598
-2.250.01222443340.0227500620.1056498391
-2.50.00620967990.01222443340.0668072288
-2.750.00297981860.00620967990.0400591136
-30.00134996720.00297981860.022750062
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Normal Distribution Graphs
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Normal Distribution Table
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m
x
f(x)
t-Distribution Table
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m
f(x)
Sheet3
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Sheet4
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Sheet5
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Sheet6
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Sheet7
0.00134996720.00000000100.01
0.00297981860.00000001900.02
0.00620967990.000000287100.03
0.01222443340.000003400800.04
0.0227500620.00003168600.06
0.04005911360.000232673400.09
0.06680722880.00134996720.0000000010.12
0.10564983910.00620967990.00000028710.15
0.15865525980.0227500620.0000316860.19
0.22662727970.06680722880.00134996720.24
0.330.15865525980.0227500620.29
0.420.30853753260.15865525980.33
0.450.49999999980.54999999980.36
0.420.30853753260.15865525980.33
0.330.15865525980.0227500620.29
0.22662727970.06680722880.00134996720.24
0.15865525980.0227500620.0000316860.19
0.10564983910.00620967990.00000028710.15
0.06680722880.00134996720.0000000010.12
0.04005911360.000232673400.09
0.0227500620.00003168600.06
0.01222443340.000003400800.04
0.00620967990.000000287100.03
0.00297981860.00000001900.02
0.00134996720.00000000100.01
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0
f(z,t)
z,t
Sheet8
0.001349967200.01
0.002979818600.02
0.006209679900.03
0.012224433400.04
0.02275006200.06
0.040059113600.09
0.06680722880.0000000010.12
0.10564983910.00000028710.15
0.15865525980.0000316860.19
0.22662727970.00134996720.24
0.330.0227500620.29
0.420.15865525980.33
0.450.54999999980.36
0.420.15865525980.33
0.330.0227500620.29
0.22662727970.00134996720.24
0.15865525980.0000316860.19
0.10564983910.00000028710.15
0.06680722880.0000000010.12
0.040059113600.09
0.02275006200.06
0.012224433400.04
0.006209679900.03
0.002979818600.02
0.001349967200.01
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0
f(z,t)
z,t
Sheet9
00.01
00.02
00.03
00.04
00.06
00.09
0.0000000010.12
0.00000028710.15
0.0000316860.19
0.00134996720.24
0.0227500620.29
0.15865525980.33
0.54999999980.36
0.15865525980.33
0.0227500620.29
0.00134996720.24
0.0000316860.19
0.00000028710.15
0.0000000010.12
00.09
00.06
00.04
00.03
00.02
00.01
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0
f(z,t)
z,t
Sheet10
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0
f(z,t)
z,t
Sheet11
m
x
f(x)
Sheet12
0
z
f(z)
Sheet13
m=5
x
f(x)
Sheet14
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m
m
m
x
f(x)
Sheet15
z0.000.010.020.030.040.050.060.070.080.09
0.0-0.00000.00400.00800.01200.01600.01990.02390.02790.03190.0359
0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753
0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141
0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517
0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879
0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224
0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549
0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852
0.80.28810.29100.29390.29670.29950.30230.30510.30780.31060.3133
0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389
1.00.34130.34380.34610.34850.35080.35310.35540.35770.35990.3621
1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830
1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015
1.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177
1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319
1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441
1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545
1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633
1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706
1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767
2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817
2.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857
2.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890
2.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916
2.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936
2.50.49380.49400.49410.49430.49450.49460.49480.49490.49510.4952
2.60.49530.49550.49560.49570.49590.49600.49610.49620.49630.4964
2.70.49650.49660.49670.49680.49690.49700.49710.49720.49730.4974
2.80.49740.49750.49760.49770.49770.49780.49790.49790.49800.4981
2.90.49810.49820.49820.49830.49840.49840.49850.49850.49860.4986
3.00.49870.49870.49870.49880.49880.49890.49890.49890.49900.4990
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Sheet16
degreesArea in Upper tail
of freedom0.100.050.0250.010.005
13.0786.31412.70631.82163.656
21.8862.9204.3036.9659.925
31.6382.3533.1824.5415.841
41.5332.1322.7763.7474.604
51.4762.0152.5713.3654.032
61.4401.9432.4473.1433.707
71.4151.8952.3652.9983.499
81.3971.8602.3062.8963.355
91.3831.8332.2622.8213.250
101.3721.8122.2282.7643.169
111.3631.7962.2012.7183.106
121.3561.7822.1792.6813.055
131.3501.7712.1602.6503.012
141.3451.7612.1452.6242.977
151.3411.7532.1312.6022.947
161.3371.7462.1202.5832.921
171.3331.7402.1102.5672.898
181.3301.7342.1012.5522.878
191.3281.7292.0932.5392.861
201.3251.7252.0862.5282.845
211.3231.7212.0802.5182.831
221.3211.7172.0742.5082.819
231.3191.7142.0692.5002.807
241.3181.7112.0642.4922.797
251.3161.7082.0602.4852.787
261.3151.7062.0562.4792.779
271.3141.7032.0522.4732.771
281.3131.7012.0482.4672.763
291.3111.6992.0452.4622.756
301.3101.6972.0422.4572.750
401.3031.6842.0212.4232.704
601.2961.6712.0002.3902.660
1201.2891.6581.9802.3582.617
1.2821.6451.9602.3262.576
degreesArea in Upper tail
of freedom0.100.050.0250.010.005
13.0786.31412.70631.82163.656
......
......
......
151.3411.7532.1312.6022.947
161.3371.7462.1202.5832.921
171.3331.7402.1102.5672.898
181.3301.7342.1012.5522.878
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Again, looking at a small part of the Cumulative Standard Normal Probability Distribution Table, we find the probability that a standard normal random variable z is between - and 2.00?
Sheet17
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Normal Distribution Graphs
0.50.251.5
-30.00134996720.00000000100.01
-2.750.00297981860.00000001900.02
-2.50.00620967990.000000287100.03
-2.250.01222443340.000003400800.04
-20.0227500620.00003168600.06
-1.750.04005911360.000232673400.09
-1.50.06680722880.00134996720.0000000010.12
-1.250.10564983910.00620967990.00000028710.15
-10.15865525980.0227500620.0000316860.19
-0.750.22662727970.06680722880.00134996720.240.52.5
-0.50.330.15865525980.0227500620.29-30.00134996720.0010
-0.250.420.30853753260.15865525980.33-2.750.10.080
00.450.49999999980.54999999980.36-2.50.75210.60
-0.250.420.30853753260.15865525980.33-2.250.71050.550
-0.50.330.15865525980.0227500620.29-20.22510.30
-0.750.22662727970.06680722880.00134996720.24-1.750.10.20
-10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025
-1.250.10564983910.00620967990.00000028710.15-1.250.290.260.0000007178
-1.50.06680722880.00134996720.0000000010.12-10.430.330.0000792151
-1.750.04005911360.000232673400.09-0.750.290.250.0033749181
-20.0227500620.00003168600.06-0.50.050.230.0568751551
-2.250.01222443340.000003400800.04-0.250.080.330.3966381494
-2.50.00620967990.000000287100.0300.190.851.3749999995
-2.750.00297981860.00000001900.02-0.250.230.430.3966381494
-30.00134996720.00000000100.01-0.50.270.350.0568751551
-0.750.380.290.0033749181
-10.440.310.0000792151
-1.250.50.330.0000007178
-1.50.420.220.0000000025
-1.750.210.150
-20.090.050
-2.250.070.030
-2.50.030.020
-2.750.020.010
-30.010.0010
-40.000031686
-3.750.0000884446
-3.50.0002326734
-3.250.0005770865
-30.00134996720.00057708650.000031686
-2.750.00297981860.00134996720.0000884446
-2.50.00620967990.00297981860.0002326734
-2.250.01222443340.00620967990.0005770865
-20.0227500620.01222443340.0013499672
-1.750.04005911360.0227500620.0029798186
-1.50.06680722880.04005911360.0062096799
-1.250.10564983910.06680722880.0122244334
-10.15865525980.10564983910.022750062
-0.750.22662727970.15865525980.0400591136
-0.50.330.22662727970.0668072288
-0.250.420.330.1056498391
00.450.420.1586552598
-0.250.420.450.2266272797
-0.50.330.420.33
-0.750.22662727970.330.42
-10.15865525980.22662727970.45
-1.250.10564983910.15865525980.42
-1.50.06680722880.10564983910.33
-1.750.04005911360.06680722880.2266272797
-20.0227500620.04005911360.1586552598
-2.250.01222443340.0227500620.1056498391
-2.50.00620967990.01222443340.0668072288
-2.750.00297981860.00620967990.0400591136
-30.00134996720.00297981860.022750062
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Normal Distribution Graphs
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
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000
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Normal Distribution Table
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
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m
x
f(x)
t-Distribution Table
0.0013499672
0.1
0.7521
0.7105
0.2251
0.1
0.15
0.29
0.43
0.29
0.05
0.08
0.19
0.23
0.27
0.38
0.44
0.5
0.42
0.21
0.09
0.07
0.03
0.02
0.01
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m
x
f(x)
Sheet3
0.00134996720.0010
0.10.080
0.75210.60
0.71050.550
0.22510.30
0.10.20
0.150.220.0000000025
0.290.260.0000007178
0.430.330.0000792151
0.290.250.0033749181
0.050.230.0568751551
0.080.330.3966381494
0.190.851.3749999995
0.230.430.3966381494
0.270.350.0568751551
0.380.290.0033749181
0.440.310.0000792151
0.50.330.0000007178
0.420.220.0000000025
0.210.150
0.090.050
0.070.030
0.030.020
0.020.010
0.010.0010
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Sheet4
0.00134996720.000000001
0.00297981860.000000019
0.00620967990.0000002871
0.01222443340.0000034008
0.0227500620.000031686
0.04005911360.0002326734
0.06680722880.0013499672
0.10564983910.0062096799
0.15865525980.022750062
0.22662727970.0668072288
0.330.1586552598
0.420.3085375326
0.450.4999999998
0.420.3085375326
0.330.1586552598
0.22662727970.0668072288
0.15865525980.022750062
0.10564983910.0062096799
0.06680722880.0013499672
0.04005911360.0002326734
0.0227500620.000031686
0.01222443340.0000034008
0.00620967990.0000002871
0.00297981860.000000019
0.00134996720.000000001
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Sheet5
0000
0000
0000
0000
0000
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0000
0000
0000
0000
0000
0000
0000
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0000
0000
0000
0000
0000
0000
0000
0000
0000
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0
f(z,t)
z,t
Sheet6
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
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0
f(z,t)
z,t
Sheet7
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
00
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0
f(z,t)
z,t
Sheet8
0
0
0
0
0
0
0.000000001
0.0000002871
0.000031686
0.0013499672
0.022750062
0.1586552598
0.5499999998
0.1586552598
0.022750062
0.0013499672
0.000031686
0.0000002871
0.000000001
0
0
0
0
0
0
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0
f(z,t)
z,t
Sheet9
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
m
x
f(x)
Sheet10
0.0013499672
0.0029798186
0.0062096799
0.0122244334
0.022750062
0.0400591136
0.0668072288
0.1056498391
0.1586552598
0.2266272797
0.33
0.42
0.45
0.42
0.33
0.2266272797
0.1586552598
0.1056498391
0.0668072288
0.0400591136
0.022750062
0.0122244334
0.0062096799
0.0029798186
0.0013499672
0
z
f(z)
Sheet11
0.0013499672
0.0029798186
0.0062096799
0.0122244334
0.022750062
0.0400591136
0.0668072288
0.1056498391
0.1586552598
0.2266272797
0.33
0.42
0.45
0.42
0.33
0.2266272797
0.1586552598
0.1056498391
0.0668072288
0.0400591136
0.022750062
0.0122244334
0.0062096799
0.0029798186
0.0013499672
m=5
x
f(x)
Sheet12
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
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m
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f(x)
Sheet13
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000
000
000
000
000
000
000
000
000
000
000
000
000
000
Sheet14
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
Sheet15
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
000
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000
000
000
000
000
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Sheet16
000
000
000
000
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000
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000
000
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000
000
000
000
000
000
000
000
000
000
000
000
000
f(x)
m
x
z0.000.010.020.030.040.050.060.070.080.09z0.000.010.020.030.040.050.060.070.080.09
0.00.50000.50400.50800.51200.51600.51990.52390.52790.53190.5359:::::::::::
0.10.53980.54380.54780.55170.55570.55960.56360.56750.57140.5753:::::::::::
0.20.57930.58320.58710.59100.59480.59870.60260.60640.61030.61411.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.9177
0.30.61790.62170.62550.62930.63310.63680.64060.64430.64800.65171.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.9319
0.40.65540.65910.66280.66640.67000.67360.67720.68080.68440.68791.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.9441
0.50.69150.69500.69850.70190.70540.70880.71230.71570.71900.72241.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.9545
0.60.72570.72910.73240.73570.73890.74220.74540.74860.75170.75491.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.9633
0.70.75800.76110.76420.76730.77040.77340.77640.77940.78230.78521.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.9706
0.80.78810.79100.79390.79670.79950.80230.80510.80780.81060.81331.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.9767
0.90.81590.81860.82120.82380.82640.82890.83150.83400.83650.83892.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.9817
1.00.84130.84380.84610.84850.85080.85310.85540.85770.85990.86212.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.9857
1.10.86430.86650.86860.87080.87290.87490.87700.87900.88100.88302.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890
1.20.88490.88690.88880.89070.89250.89440.89620.89800.89970.90152.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916
1.30.90320.90490.90660.90820.90990.91150.91310.91470.91620.91772.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936
1.40.91920.92070.92220.92360.92510.92650.92790.92920.93060.93192.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952
1.50.93320.93450.93570.93700.93820.93940.94060.94180.94290.94412.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964
1.60.94520.94630.94740.94840.94950.95050.95150.95250.95350.95452.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974
1.70.95540.95640.95730.95820.95910.95990.96080.96160.96250.96332.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981
1.80.96410.96490.96560.96640.96710.96780.96860.96930.96990.97062.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986
1.90.97130.97190.97260.97320.97380.97440.97500.97560.97610.97673.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990
2.00.97720.97780.97830.97880.97930.97980.98030.98080.98120.98173.10.99900.99910.99910.99910.99920.99920.99920.99920.99930.9993
2.10.98210.98260.98300.98340.98380.98420.98460.98500.98540.98573.20.99930.99930.99940.99940.99940.99940.99940.99950.99950.9995
2.20.98610.98640.98680.98710.98750.98780.98810.98840.98870.9890
2.30.98930.98960.98980.99010.99040.99060.99090.99110.99130.9916
2.40.99180.99200.99220.99250.99270.99290.99310.99320.99340.9936
2.50.99380.99400.99410.99430.99450.99460.99480.99490.99510.9952
2.60.99530.99550.99560.99570.99590.99600.99610.99620.99630.9964
2.70.99650.99660.99670.99680.99690.99700.99710.99720.99730.9974
2.80.99740.99750.99760.99770.99770.99780.99790.99790.99800.9981
2.90.99810.99820.99820.99830.99840.99840.99850.99850.99860.9986
3.00.99870.99870.99870.99880.99880.99890.99890.99890.99900.9990
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degreesArea in Upper tail
of freedom0.100.050.0250.010.005
13.0786.31412.70631.82163.656
21.8862.9204.3036.9659.925
31.6382.3533.1824.5415.841
41.5332.1322.7763.7474.604
51.4762.0152.5713.3654.032
61.4401.9432.4473.1433.707
71.4151.8952.3652.9983.499
81.3971.8602.3062.8963.355
91.3831.8332.2622.8213.250
101.3721.8122.2282.7643.169
111.3631.7962.2012.7183.106
121.3561.7822.1792.6813.055
131.3501.7712.1602.6503.012
141.3451.7612.1452.6242.977
151.3411.7532.1312.6022.947
161.3371.7462.1202.5832.921
171.3331.7402.1102.5672.898
181.3301.7342.1012.5522.878
191.3281.7292.0932.5392.861
201.3251.7252.0862.5282.845
211.3231.7212.0802.5182.831
221.3211.7172.0742.5082.819
231.3191.7142.0692.5002.807
241.3181.7112.0642.4922.797
251.3161.7082.0602.4852.787
261.3151.7062.0562.4792.779
271.3141.7032.0522.4732.771
281.3131.7012.0482.4672.763
291.3111.6992.0452.4622.756
301.3101.6972.0422.4572.750
401.3031.6842.0212.4232.704
601.2961.6712.0002.3902.660
1201.2891.6581.9802.3582.617
1.2821.6451.9602.3262.576
degreesArea in Upper tail
of freedom0.100.050.0250.010.005
13.0786.31412.70631.82163.656
......
......
......
151.3411.7532.1312.6022.947
161.3371.7462.1202.5832.921
171.3331.7402.1102.5672.898
181.3301.7342.1012.5522.878
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Example: for a standard normal random variable z, what is the probability that z is between 0 and 2.0?2.0}Area of Probability =0.9772 0.5000 = 0.4772Area ofProbability = 0.5000{
Chart2
0.0013499672
0.0029798186
0.0062096799
0.0122244334
0.022750062
0.0400591136
0.0668072288
0.1056498391
0.1586552598
0.2266272797
0.33
0.42
0.45
0.42
0.33
0.2266272797
0.1586552598
0.1056498391
0.0668072288
0.0400591136
0.022750062
0.0122244334
0.0062096799
0.0029798186
0.0013499672
0
z
f(z)
Sheet17
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Normal Distribution Graphs
0.50.251.5
-30.00134996720.00000000100.01
-2.750.00297981860.00000001900.02
-2.50.00620967990.000000287100.03
-2.250.01222443340.000003400800.04
-20.0227500620.00003168600.06
-1.750.04005911360.000232673400.09
-1.50.06680722880.00134996720.0000000010.12
-1.250.10564983910.00620967990.00000028710.15
-10.15865525980.0227500620.0000316860.19
-0.750.22662727970.06680722880.00134996720.240.52.5
-0.50.330.15865525980.0227500620.29-30.00134996720.0010
-0.250.420.30853753260.15865525980.33-2.750.10.080
00.450.49999999980.54999999980.36-2.50.75210.60
-0.250.420.30853753260.15865525980.33-2.250.71050.550
-0.50.330.15865525980.0227500620.29-20.22510.30
-0.750.22662727970.06680722880.00134996720.24-1.750.10.20
-10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025
-1.250.10564983910.00620967990.00000028710.15-1.250.290.260.0000007178
-1.50.06680722880.00134996720.0000000010.12-10.430.330.0000792151
-1.750.04005911360.000232673400.09-0.750.290.250.0033749181
-20.0227500620.00003168600.06-0.50.050.230.0568751551
-2.250.01222443340.000003400800.04-0.250.080.330.3966381494
-2.50.00620967990.000000287100.0300.190.851.3749999995
-2.750.00297981860.00000001900.02-0.250.230.430.3966381494
-30.00134996720.00000000100.01-0.50.270.350.0568751551
-0.750.380.290.0033749181
-10.440.310.0000792151
-1.250.50.330.0000007178
-1.50.420.220.0000000025
-1.750.210.150
-20.090.050
-2.250.070.030
-2.50.030.020
-2.750.020.010
-30.010.0010
-40.000031686
-3.750.0000884446
-3.50.0002326734
-3.250.0005770865
-30.00134996720.00057708650.000031686
-2.750.00297981860.00134996720.0000884446
-2.50.00620967990.00297981860.0002326734
-2.250.01222443340.00620967990.0005770865
-20.0227500620.01222443340.0013499672
-1.750.04005911360.0227500620.0029798186
-1.50.06680722880.04005911360.0062096799
-1.250.10564983910.06680722880.0122244334
-10.15865525980.10564983910.022750062
-0.750.22662727970.15865525980.0400591136
-0.50.330.22662727970.0668072288
-0.250.420.330.1056498391
00.450.420.1586552598
-0.250.420.450.2266272797
-0.50.330.420.33
-0.750.22662727970.330.42
-10.15865525980.22662727970.45
-1.250.10564983910.15865525980.42
-1.50.06680722880.10564983910.33
-1.750.04005911360.06680722880.2266272797
-20.0227500620.04005911360.1586552598
-2.250.01222443340.0227500620.1056498391
-2.50.00620967990.01222443340.0668072288
-2.750.00297981860.00620967990.0400591136
-30.00134996720.00297981860.022750062
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Normal Distribution Graphs
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Normal Distribution Table
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m
x
f(x)
t-Distribution Table
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m
f(x)
Sheet3
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Sheet4
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Sheet7
0.00134996720.00000000100.01
0.00297981860.00000001900.02
0.00620967990.000000287100.03
0.01222443340.000003400800.04
0.0227500620.00003168600.06
0.04005911360.000232673400.09
0.06680722880.00134996720.0000000010.12
0.10564983910.00620967990.00000028710.15
0.15865525980.0227500620.0000316860.19
0.22662727970.06680722880.00134996720.24
0.330.15865525980.0227500620.29
0.420.30853753260.15865525980.33
0.450.49999999980.54999999980.36
0.420.30853753260.15865525980.33
0.330.15865525980.0227500620.29
0.22662727970.06680722880.00134996720.24
0.15865525980.0227500620.0000316860.19
0.10564983910.00620967990.00000028710.15
0.06680722880.00134996720.0000000010.12
0.04005911360.000232673400.09
0.0227500620.00003168600.06
0.01222443340.000003400800.04
0.00620967990.000000287100.03
0.00297981860.00000001900.02
0.00134996720.00000000100.01
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0
f(z,t)
z,t
Sheet8
0.001349967200.01
0.002979818600.02
0.006209679900.03
0.012224433400.04
0.02275006200.06
0.040059113600.09
0.06680722880.0000000010.12
0.10564983910.00000028710.15
0.15865525980.0000316860.19
0.22662727970.00134996720.24
0.330.0227500620.29
0.420.15865525980.33
0.450.54999999980.36
0.420.15865525980.33
0.330.0227500620.29
0.22662727970.00134996720.24
0.15865525980.0000316860.19
0.10564983910.00000028710.15
0.06680722880.0000000010.12
0.040059113600.09
0.02275006200.06
0.012224433400.04
0.006209679900.03
0.002979818600.02
0.001349967200.01
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0
f(z,t)
z,t
Sheet9
00.01
00.02
00.03
00.04
00.06
00.09
0.0000000010.12
0.00000028710.15
0.0000316860.19
0.00134996720.24
0.0227500620.29
0.15865525980.33
0.54999999980.36
0.15865525980.33
0.0227500620.29
0.00134996720.24
0.0000316860.19
0.00000028710.15
0.0000000010.12
00.09
00.06
00.04
00.03
00.02
00.01
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0
f(z,t)
z,t
Sheet10
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0
f(z,t)
z,t
Sheet11
m
x
f(x)
Sheet12
0
z
f(z)
Sheet13
m=5
x
f(x)
Sheet14
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m
m
m
x
f(x)
Sheet15
z0.000.010.020.030.040.050.060.070.080.09
0.0-0.00000.00400.00800.01200.01600.01990.02390.02790.03190.0359
0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753
0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141
0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517
0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879
0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224
0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549
0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852
0.80.28810.29100.29390.29670.29950.30230.30510.30780.31060.3133
0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389
1.00.34130.34380.34610.34850.35080.35310.35540.35770.35990.3621
1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830
1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015
1.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177
1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319
1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441
1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545
1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633
1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706
1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767
2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817
2.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857
2.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890
2.30.48930.48960.48980.49010.49040.49060.49090.49110.49130.4916
2.40.49180.49200.49220.49250.49270.49290.49310.49320.49340.4936
2.50.49380.49400.49410.49430.49450.49460.49480.49490.49510.4952
2.60.49530.49550.49560.49570.49590.49600.49610.49620.49630.4964
2.70.49650.49660.49670.49680.49690.49700.49710.49720.49730.4974
2.80.49740.49750.49760.49770.49770.49780.49790.49790.49800.4981
2.90.49810.49820.49820.49830.49840.49840.49850.49850.49860.4986
3.00.49870.49870.49870.49880.49880.49890.49890.49890.49900.4990
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Sheet16
degreesArea in Upper tail
of freedom0.100.050.0250.010.005
13.0786.31412.70631.82163.656
21.8862.9204.3036.9659.925
31.6382.3533.1824.5415.841
41.5332.1322.7763.7474.604
51.4762.0152.5713.3654.032
61.4401.9432.4473.1433.707
71.4151.8952.3652.9983.499
81.3971.8602.3062.8963.355
91.3831.8332.2622.8213.250
101.3721.8122.2282.7643.169
111.3631.7962.2012.7183.106
121.3561.7822.1792.6813.055
131.3501.7712.1602.6503.012
141.3451.7612.1452.6242.977
151.3411.7532.1312.6022.947
161.3371.7462.1202.5832.921
171.3331.7402.1102.5672.898
181.3301.7342.1012.5522.878
191.3281.7292.0932.5392.861
201.3251.7252.0862.5282.845
211.3231.7212.0802.5182.831
221.3211.7172.0742.5082.819
231.3191.7142.0692.5002.807
241.3181.7112.0642.4922.797
251.3161.7082.0602.4852.787
261.3151.7062.0562.4792.779
271.3141.7032.0522.4732.771
281.3131.7012.0482.4672.763
291.3111.6992.0452.4622.756
301.3101.6972.0422.4572.750
401.3031.6842.0212.4232.704
601.2961.6712.0002.3902.660
1201.2891.6581.9802.3582.617
1.2821.6451.9602.3262.576
degreesArea in Upper tail
of freedom0.100.050.0250.010.005
13.0786.31412.70631.82163.656
......
......
......
151.3411.7532.1312.6022.947
161.3371.7462.1202.5832.921
171.3331.7402.1102.5672.898
181.3301.7342.1012.5522.878
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2.0Area of Probability =1.0000 - 0.9772 = 0.0228What is the probability that z is at least 2.0?{
Chart2
0.0013499672
0.0029798186
0.0062096799
0.0122244334
0.022750062
0.0400591136
0.0668072288
0.1056498391
0.1586552598
0.2266272797
0.33
0.42
0.45
0.42
0.33
0.2266272797
0.1586552598
0.1056498391
0.0668072288
0.0400591136
0.022750062
0.0122244334
0.0062096799
0.0029798186
0.0013499672
0
z
f(z)
Sheet17
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Normal Distribution Graphs
0.50.251.5
-30.00134996720.00000000100.01
-2.750.00297981860.00000001900.02
-2.50.00620967990.000000287100.03
-2.250.01222443340.000003400800.04
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-0.750.22662727970.06680722880.00134996720.240.52.5
-0.50.330.15865525980.0227500620.29-30.00134996720.0010
-0.250.420.30853753260.15865525980.33-2.750.10.080
00.450.49999999980.54999999980.36-2.50.75210.60
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-10.15865525980.0227500620.0000316860.19-1.50.150.220.0000000025
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00.450.420.1586552598
-0.250.420.450.2266272797
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-0.750.22662727970.330.42
-10.15865525980.22662727970.45
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-20.0227500620.04005911360.1586552598
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-2.750.00297981860.00620967990.0400591136
-30.00134996720.00297981860.022750062
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Normal Distribution Graphs
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Normal Distribution Table
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m
x
f(x)
t-Distribution Table
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m
f(x)
Sheet3
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Sheet4
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Sheet5
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Sheet6
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Sheet7
0.00134996720.00000000100.01
0.00297981860.00000001900.02
0.00620967990.000000287100.03
0.01222443340.000003400800.04
0.0227500620.00003168600.06
0.04005911360.000232673400.09
0.06680722880.00134996720.0000000010.12
0.10564983910.00620967990.00000028710.15
0.15865525980.0227500620.0000316860.19
0.22662727970.06680722880.00134996720.24
0.330.15865525980.0227500620.29
0.420.30853753260.15865525980.33
0.450.49999999980.54999999980.36
0.420.30853753260.15865525980.33
0.330.15865525980.0227500620.29
0.22662727970.06680722880.00134996720.24
0.15865525980.0227500620.0000316860.19
0.10564983910.00620967990.00000028710.15
0.06680722880.00134996720.0000000010.12
0.04005911360.000232673400.09
0.0227500620.00003168600.06
0.01222443340.000003400800.04
0.00620967990.000000287100.03
0.00297981860.00000001900.02
0.00134996720.00000000100.01
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0
f(z,t)
z,t
Sheet8
0.001349967200.01
0.002979818600.02
0.006209679900.03
0.012224433400.04
0.02275006200.06
0.040059113600.09
0.06680722880.0000000010.12
0.10564983910.00000028710.15
0.15865525980.0000316860.19
0.22662727970.00134996720.24
0.330.0227500620.29
0.420.15865525980.33
0.450.54999999980.36
0.420.15865525980.33
0.330.0227500620.29
0.22662727970.00134996720.24
0.15865525980.0000316860.19
0.10564983910.00000028710.15
0.06680722880.0000000010.12
0.040059113600.09
0.02275006200.06
0.012224433400.04
0.006209679900.03
0.002979818600.02
0.001349967200.01
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0
f(z,t)
z,t
Sheet9
00.01
00.02
00.03
00.04
00.06
00.09
0.0000000010.12
0.00000028710.15
0.0000316860.19
0.00134996720.24
0.0227500620.29
0.15865525980.33
0.54999999980.36
0.15865525980.33
0.0227500620.29
0.00134996720.24
0.0000316860.19
0.00000028710.15
0.0000000010.12
00.09
00.06
00.04
00.03
00.02
00.01
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0
f(z,t)
z,t
Sheet10
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0
f(z,t)
z,t
Sheet11
m
x
f(x)
Sheet12
0
z
f(z)
Sheet13
m=5
x
f(x)
Sheet14
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m
m
m
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f(x)
Sheet15
z0.000.010.020.030.040.050.060.070.080.09
0.0-0.00000.00400.00800.01200.01600.01990.02390.02790.03190.0359
0.10.03980.04380.04780.05170.05570.05960.06360.06750.07140.0753
0.20.07930.08320.08710.09100.09480.09870.10260.10640.11030.1141
0.30.11790.12170.12550.12930.13310.13680.14060.14430.14800.1517
0.40.15540.15910.16280.16640.17000.17360.17720.18080.18440.1879
0.50.19150.19500.19850.20190.20540.20880.21230.21570.21900.2224
0.60.22570.22910.23240.23570.23890.24220.24540.24860.25170.2549
0.70.25800.26110.26420.26730.27040.27340.27640.27940.28230.2852
0.80.28810.29100.29390.29670.29950.30230.30510.30780.31060.3133
0.90.31590.31860.32120.32380.32640.32890.33150.33400.33650.3389
1.00.34130.34380.34610.34850.35080.35310.35540.35770.35990.3621
1.10.36430.36650.36860.37080.37290.37490.37700.37900.38100.3830
1.20.38490.38690.38880.39070.39250.39440.39620.39800.39970.4015
1.30.40320.40490.40660.40820.40990.41150.41310.41470.41620.4177
1.40.41920.42070.42220.42360.42510.42650.42790.42920.43060.4319
1.50.43320.43450.43570.43700.43820.43940.44060.44180.44290.4441
1.60.44520.44630.44740.44840.44950.45050.45150.45250.45350.4545
1.70.45540.45640.45730.45820.45910.45990.46080.46160.46250.4633
1.80.46410.46490.46560.46640.46710.46780.46860.46930.46990.4706
1.90.47130.47190.47260.47320.47380.47440.47500.47560.47610.4767
2.00.47720.47780.47830.47880.47930.47980.48030.48080.48120.4817
2.10.48210.48260.48300.48340.48380.48420.48460.48500.48540.4857
2.20.48610.48640.48680.48710.48750.48780.48810.48840.48870.4890
2.30.48930.48960.48