Statistical Tables

Normal Table

The function responsible for Standard Normal Table is the Normal Probability Function with mean 0 and standard deviation 1. The probability function for Standard Normal Distribution is,

\[ f(x)=\frac{1}{2\pi}\text{exp}\left[-\frac{1}{2}x^2\right] \]

The table is as follows,

Area under Standard Normal Distribution

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09
0 0.50 0.50 0.51 0.51 0.52 0.52 0.52 0.53 0.53 0.54
0.1 0.54 0.54 0.55 0.55 0.56 0.56 0.56 0.57 0.57 0.58
0.2 0.58 0.58 0.59 0.59 0.59 0.60 0.60 0.61 0.61 0.61
0.3 0.62 0.62 0.63 0.63 0.63 0.64 0.64 0.64 0.65 0.65
0.4 0.66 0.66 0.66 0.67 0.67 0.67 0.68 0.68 0.68 0.69
0.5 0.69 0.69 0.70 0.70 0.71 0.71 0.71 0.72 0.72 0.72
0.6 0.73 0.73 0.73 0.74 0.74 0.74 0.75 0.75 0.75 0.75
0.7 0.76 0.76 0.76 0.77 0.77 0.77 0.78 0.78 0.78 0.79
0.8 0.79 0.79 0.79 0.80 0.80 0.80 0.81 0.81 0.81 0.81
0.9 0.82 0.82 0.82 0.82 0.83 0.83 0.83 0.83 0.84 0.84
1 0.84 0.84 0.85 0.85 0.85 0.85 0.86 0.86 0.86 0.86
1.1 0.86 0.87 0.87 0.87 0.87 0.87 0.88 0.88 0.88 0.88
1.2 0.88 0.89 0.89 0.89 0.89 0.89 0.90 0.90 0.90 0.90
1.3 0.90 0.90 0.91 0.91 0.91 0.91 0.91 0.91 0.92 0.92
1.4 0.92 0.92 0.92 0.92 0.93 0.93 0.93 0.93 0.93 0.93
1.5 0.93 0.93 0.94 0.94 0.94 0.94 0.94 0.94 0.94 0.94
1.6 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95 0.95
1.7 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96 0.96
1.8 0.96 0.96 0.97 0.97 0.97 0.97 0.97 0.97 0.97 0.97
1.9 0.97 0.97 0.97 0.97 0.97 0.97 0.98 0.98 0.98 0.98
2 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98
2.1 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.98 0.99 0.99
2.2 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99
2.3 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99
2.4 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99
2.5 0.99 0.99 0.99 0.99 0.99 0.99 0.99 0.99 1.00 1.00
2.6 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
2.7 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
2.8 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
2.9 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
3 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00

Further, the area under the normal curve plays important role in various statistical applications. The standard Normal table consist of the area from the left end of the curve to the point \(z\). The table is generated by the function,

\[
\text{Area under the curve} = \int_{-\infty}^{z}{\frac{1}{2\pi}\text{exp}\left[-\frac{1}{2}x^2\right]}dx
\]

Following table contains the area under the normal curve;

Area under Standard Normal Distribution

Z0.000.010.020.030.040.050.060.070.080.09
0.00.500000.503990.507980.511970.515950.519940.523920.527900.531880.53586
0.10.539830.543800.547760.551720.555670.559620.563560.567490.571420.57535
0.20.579260.583170.587060.590950.594830.598710.602570.606420.610260.61409
0.30.617910.621720.625520.629300.633070.636830.640580.644310.648030.65173
0.40.655420.659100.662760.666400.670030.673640.677240.680820.684390.68793
0.50.691460.694970.698470.701940.705400.708840.712260.715660.719040.72240
0.60.725750.729070.732370.735650.738910.742150.745370.748570.751750.75490
0.70.758040.761150.764240.767300.770350.773370.776370.779350.782300.78524
0.80.788140.791030.793890.796730.799550.802340.805110.807850.810570.81327
0.90.815940.818590.821210.823810.826390.828940.831470.833980.836460.83891
1.00.841340.843750.846140.848490.850830.853140.855430.857690.859930.86214
1.10.864330.866500.868640.870760.872860.874930.876980.879000.881000.88298
1.20.884930.886860.888770.890650.892510.894350.896170.897960.899730.90147
1.30.903200.904900.906580.908240.909880.911490.913090.914660.916210.91774
1.40.919240.920730.922200.923640.925070.926470.927850.929220.930560.93189
1.50.933190.934480.935740.936990.938220.939430.940620.941790.942950.94408
1.60.945200.946300.947380.948450.949500.950530.951540.952540.953520.95449
1.70.955430.956370.957280.958180.959070.959940.960800.961640.962460.96327
1.80.964070.964850.965620.966380.967120.967840.968560.969260.969950.97062
1.90.971280.971930.972570.973200.973810.974410.975000.975580.976150.97670
2.00.977250.977780.978310.978820.979320.979820.980300.980770.981240.98169
2.10.982140.982570.983000.983410.983820.984220.984610.985000.985370.98574
2.20.986100.986450.986790.987130.987450.987780.988090.988400.988700.98899
2.30.989280.989560.989830.990100.990360.990610.990860.991110.991340.99158
2.40.991800.992020.992240.992450.992660.992860.993050.993240.993430.99361
2.50.993790.993960.994130.994300.994460.994610.994770.994920.995060.99520
2.60.995340.995470.995600.995730.995850.995980.996090.996210.996320.99643
2.70.996530.996640.996740.996830.996930.997020.997110.997200.997280.99736
2.80.997440.997520.997600.997670.997740.997810.997880.997950.998010.99807
2.90.998130.998190.998250.998310.998360.998410.998460.998510.998560.99861
3.00.998650.998690.998740.998780.998820.998860.998890.998930.998960.99900

t-table

A t-Table constitute of cumulative probability or Percentage points of t-distribution for different degree of freedom and level of significance. For instance, from the following table the t-value corresponding to \(7\) degree of freedom and \(0.05\) level of significance (for one tail) is \(1.895\), i.e. the probability that T is less than \(1.895\) is \(95\%\) for \(7\) degree of freedom. The probability density function of t-distribution is,

\[ f(t,\nu)=\frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\nu\pi}\Gamma(\frac{\nu}{2})}\left(1+\frac{x^2}{\nu}\right)^{-\frac{\nu+1}{2}}
\]

\( \nu \) (d.f.)0.400.250.100.050.0250.010.0050.00250.0010.0005
10.32491.00003.07776.313812.706231.820563.6567127.3213318.3088636.6192
20.28870.81651.88562.92004.30276.96469.924814.089022.327131.5991
30.27670.76491.63772.35343.18244.54075.84097.453310.214512.9240
40.27070.74071.53322.13182.77643.74694.60415.59767.17328.6103
50.26720.72671.47592.01502.57063.36494.03214.77335.89346.8688
60.26480.71761.43981.94322.44693.14273.70744.31685.20765.9588
70.26320.71111.41491.89462.36462.99803.49954.02934.78535.4079
80.26190.70641.39681.85952.30602.89653.35543.83254.50085.0413
90.26100.70271.38301.83312.26222.82143.24983.68974.29684.7809
100.26020.69981.37221.81252.22812.76383.16933.58144.14374.5869
110.25960.69741.36341.79592.20102.71813.10583.49664.02474.4370
120.25900.69551.35621.78232.17882.68103.05453.42843.92964.3178
130.25860.69381.35021.77092.16042.65033.01233.37253.85204.2208
140.25820.69241.34501.76132.14482.62452.97683.32573.78744.1405
150.25790.69121.34061.75312.13142.60252.94673.28603.73284.0728
160.25760.69011.33681.74592.11992.58352.92083.25203.68624.0150
170.25730.68921.33341.73962.10982.56692.89823.22243.64583.9651
180.25710.68841.33041.73412.10092.55242.87843.19663.61053.9216
190.25690.68761.32771.72912.09302.53952.86093.17373.57943.8834
200.25670.68701.32531.72472.08602.52802.84533.15343.55183.8495
210.25660.68641.32321.72072.07962.51762.83143.13523.52723.8193
220.25640.68581.32121.71712.07392.50832.81883.11883.50503.7921
230.25630.68531.31951.71392.06872.49992.80733.10403.48503.7676
240.25620.68481.31781.71092.06392.49222.79693.09053.46683.7454
250.25610.68441.31631.70812.05952.48512.78743.07823.45023.7251
260.25600.68401.31501.70562.05552.47862.77873.06693.43503.7066
270.25590.68371.31371.70332.05182.47272.77073.05653.42103.6896
280.25580.68341.31251.70112.04842.46712.76333.04693.40823.6739
290.25570.68301.31141.69912.04522.46202.75643.03803.39623.6594
300.25560.68281.31041.69732.04232.45732.75003.02983.38523.6460
400.25500.68071.30311.68392.02112.42332.70452.97123.30693.5510
600.25450.67861.29581.67062.00032.39012.66032.91463.23173.4602
1200.25390.67651.28861.65771.97992.35782.61742.85993.15953.3735
0.25330.67451.28161.64491.96002.32642.57582.80703.09023.2905

Chi-Square table

If \(Z_1, \ldots, Z_k \) are independent standard normal random variates, then the sum of their squares is,

\[ Q=\sum_{i=1}^k{Z_i} \]

is distributed as a chi-squared distribution with \(k \) degrees of freedom. This is usually denoted as

\[ Q\sim chi^2_k \]

The probability density function (pdf) of the chi-squared distribution is

\[ f(x,k)=\frac{x^{k/2-1}e^{-x/2}}{2^{k/2}\Gamma(k/2)} \]

where \(\Gamma(k/2)\) denotes the Gamma function.

Probability under Chi-Sq Distribution

\( \nu \) (d.f.)0.9950.9900.9750.9500.5000.0500.0250.0100.005
10.00000.00020.00100.00390.45493.84155.02396.63497.8794
20.01000.02010.05060.10261.38635.99157.37789.210310.5966
30.07170.11480.21580.35182.36607.81479.348411.344912.8382
40.20700.29710.48440.71073.35679.487711.143313.276714.8603
50.41170.55430.83121.14554.351511.070512.832515.086316.7496
60.67570.87211.23731.63545.348112.591614.449416.811918.5476
70.98931.23901.68992.16736.345814.067116.012818.475320.2777
81.34441.64652.17972.73267.344115.507317.534520.090221.9550
91.73492.08792.70043.32518.342816.919019.022821.666023.5894
102.15592.55823.24703.94039.341818.307020.483223.209325.1882
112.60323.05353.81574.574810.341019.675121.920024.725026.7568
123.07383.57064.40385.226011.340321.026123.336726.217028.2995
133.56504.10695.00885.891912.339822.362024.735627.688229.8195
144.07474.66045.62876.570613.339323.684826.118929.141231.3193
154.60095.22936.26217.260914.338924.995827.488430.577932.8013
165.14225.81226.90777.961615.338526.296228.845431.999934.2672
175.69726.40787.56428.671816.338227.587130.191033.408735.7185
186.26487.01498.23079.390517.337928.869331.526434.805337.1565
196.84407.63278.906510.117018.337730.143532.852336.190938.5823
207.43388.26049.590810.850819.337431.410434.169637.566239.9968
2510.519711.524013.119714.611424.336637.652540.646544.314146.9279
3013.786714.953516.790818.492729.336043.773046.979250.892253.6720
4020.706522.164324.433026.509339.335355.758559.341763.690766.7660
6035.534537.484940.481743.188059.334779.081983.297788.379491.9517
7043.275245.441748.757651.739369.334590.531295.0232100.4252104.2149
8051.171953.540157.153260.391579.3343101.8795106.6286112.3288116.3211
9059.196361.754165.646669.126089.3342113.1453118.1359124.1163128.2989
10067.327670.064974.221977.929599.3341124.3421129.5612135.8067140.1695

F-table

If \(Z_1 \) and \(Z_2 \) are independent chi square distribution with degree of freedom \(\nu_1 \) and \(\nu_2\) respectively, then,

\[ F=\frac{Z_1/\nu_1}{Z_2/\nu_2}\sim F(\nu_1, \nu_2) \]

is distributed as a F distribution with \(\nu_1\) and \(\nu_2\) degrees of freedom. The probability density function (pdf) of the F-distribution is

\[ f(x,\nu_1, \nu_2) = \frac{1}{B\left(\frac{\nu_1}{2},\frac{\nu_2}{2}\right)}\left(\frac{\nu_1}{\nu_2}\right)^{\frac{\nu_1}{2}}x^{\frac{\nu_1}{2}-1}\left(1+\frac{\nu_1}{\nu_2}\right)^{-\frac{\nu_1+\nu_2}{2}} \]

where \(B\left(\frac{\nu_1}{2},\frac{\nu_2}{2}\right)\) denotes a Beta function.

Probability under F-Distribution at 0.05 level of significance

\( \nu_1/\nu_2 \)12345678910204060\( \infty \)
218.5119.0019.1619.2519.3019.3319.3519.3719.3819.4019.4519.4719.4819.49
310.139.559.289.129.018.948.898.858.818.798.668.598.578.53
47.716.946.596.396.266.166.096.046.005.965.805.725.695.63
56.615.795.415.195.054.954.884.824.774.744.564.464.434.37
65.995.144.764.534.394.284.214.154.104.063.873.773.743.67
75.594.744.354.123.973.873.793.733.683.643.443.343.303.23
85.324.464.073.843.693.583.503.443.393.353.153.043.012.93
95.124.263.863.633.483.373.293.233.183.142.942.832.792.71
104.964.103.713.483.333.223.143.073.022.982.772.662.622.54
114.843.983.593.363.203.093.012.952.902.852.652.532.492.41
124.753.893.493.263.113.002.912.852.802.752.542.432.382.30
134.673.813.413.183.032.922.832.772.712.672.462.342.302.21
144.603.743.343.112.962.852.762.702.652.602.392.272.222.14
154.543.683.293.062.902.792.712.642.592.542.332.202.162.07
164.493.633.243.012.852.742.662.592.542.492.282.152.112.02
174.453.593.202.962.812.702.612.552.492.452.232.102.061.97
184.413.553.162.932.772.662.582.512.462.412.192.062.021.92
194.383.523.132.902.742.632.542.482.422.382.162.031.981.88
204.353.493.102.872.712.602.512.452.392.352.121.991.951.85
214.323.473.072.842.682.572.492.422.372.322.101.961.921.82
224.303.443.052.822.662.552.462.402.342.302.071.941.891.79
234.283.423.032.802.642.532.442.372.322.272.051.911.861.76
244.263.403.012.782.622.512.422.362.302.252.031.891.841.74
254.243.392.992.762.602.492.402.342.282.242.011.871.821.72
264.233.372.982.742.592.472.392.322.272.221.991.851.801.70
274.213.352.962.732.572.462.372.312.252.201.971.841.791.68
284.203.342.952.712.562.452.362.292.242.191.961.821.771.66
294.183.332.932.702.552.432.352.282.222.181.941.811.751.65
304.173.322.922.692.532.422.332.272.212.161.931.791.741.63
404.083.232.842.612.452.342.252.182.122.081.841.691.641.52
604.003.152.762.532.372.252.172.102.041.991.751.591.531.40
3.853.002.612.382.222.112.021.951.891.841.581.411.331.11

Probability under F-Distribution at 0.02 level of significance

\( \nu_1/\nu_2 \)12345678910204060\( \infty \)
238.5139.0039.1739.2539.3039.3339.3639.3739.3939.4039.4539.4739.4839.50
317.4416.0415.4415.1014.8814.7314.6214.5414.4714.4214.1714.0413.9913.91
412.2210.659.989.609.369.209.078.988.908.848.568.418.368.26
510.018.437.767.397.156.986.856.766.686.626.336.186.126.02
68.817.266.606.235.995.825.705.605.525.465.175.014.964.86
78.076.545.895.525.295.124.994.904.824.764.474.314.254.15
87.576.065.425.054.824.654.534.434.364.304.003.843.783.68
97.215.715.084.724.484.324.204.104.033.963.673.513.453.34
106.945.464.834.474.244.073.953.853.783.723.423.263.203.09
116.725.264.634.284.043.883.763.663.593.533.233.063.002.89
126.555.104.474.123.893.733.613.513.443.373.072.912.852.73
136.414.974.354.003.773.603.483.393.313.252.952.782.722.60
146.304.864.243.893.663.503.383.293.213.152.842.672.612.50
156.204.774.153.803.583.413.293.203.123.062.762.592.522.40
166.124.694.083.733.503.343.223.123.052.992.682.512.452.32
176.044.624.013.663.443.283.163.062.982.922.622.442.382.26
185.984.563.953.613.383.223.103.012.932.872.562.382.322.20
195.924.513.903.563.333.173.052.962.882.822.512.332.272.14
205.874.463.863.513.293.133.012.912.842.772.462.292.222.09
215.834.423.823.483.253.092.972.872.802.732.422.252.182.05
225.794.383.783.443.223.052.932.842.762.702.392.212.142.01
235.754.353.753.413.183.022.902.812.732.672.362.182.111.98
245.724.323.723.383.152.992.872.782.702.642.332.152.081.94
255.694.293.693.353.132.972.852.752.682.612.302.122.051.91
265.664.273.673.333.102.942.822.732.652.592.282.092.031.89
275.634.243.653.313.082.922.802.712.632.572.252.072.001.86
285.614.223.633.293.062.902.782.692.612.552.232.051.981.84
295.594.203.613.273.042.882.762.672.592.532.212.031.961.82
305.574.183.593.253.032.872.752.652.572.512.202.011.941.80
405.424.053.463.132.902.742.622.532.452.392.071.881.801.65
605.293.933.343.012.792.632.512.412.332.271.941.741.671.49
5.043.703.132.802.582.422.302.202.132.061.721.501.411.13

Probability under F-Distribution at 0.001 level of significance

\( \nu_1/\nu_2 \)12345678910204060\( \infty \)
2998.50999.00999.17999.25999.30999.33999.36999.37999.39999.40999.45999.47999.48999.50
3167.03148.50141.11137.10134.58132.85131.58130.62129.86129.25126.42124.96124.47123.53
474.1461.2556.1853.4451.7150.5349.6649.0048.4748.0546.1045.0944.7544.09
547.1837.1233.2031.0929.7528.8328.1627.6527.2426.9225.3924.6024.3323.82
635.5127.0023.7021.9220.8020.0319.4619.0318.6918.4117.1216.4416.2115.77
729.2521.6918.7717.2016.2115.5215.0214.6314.3314.0812.9312.3312.1211.72
825.4118.4915.8314.3913.4812.8612.4012.0511.7711.5410.489.929.739.36
922.8616.3913.9012.5611.7111.1310.7010.3710.119.898.908.378.197.84
1021.0414.9112.5511.2810.489.939.529.208.968.757.807.307.126.78
1119.6913.8111.5610.359.589.058.668.358.127.927.016.526.356.02
1218.6412.9710.809.638.898.388.007.717.487.296.405.935.765.44
1317.8212.3110.219.078.357.867.497.216.986.805.935.475.304.99
1417.1411.789.738.627.927.447.086.806.586.405.565.104.944.62
1516.5911.349.348.257.577.096.746.476.266.085.254.804.644.33
1616.1210.979.017.947.276.806.466.195.985.814.994.544.394.08
1715.7210.668.737.687.026.566.225.965.755.584.784.334.183.87
1815.3810.398.497.466.816.356.025.765.565.394.594.154.003.69
1915.0810.168.287.276.626.185.855.595.395.224.433.993.843.53
2014.829.958.107.106.466.025.695.445.245.084.293.863.703.40
2114.599.777.946.956.325.885.565.315.114.954.173.743.583.28
2214.389.617.806.816.195.765.445.194.994.834.063.633.483.17
2314.209.477.676.706.085.655.335.094.894.733.963.533.383.08
2414.039.347.556.595.985.555.234.994.804.643.873.453.292.99
2513.889.227.456.495.895.465.154.914.714.563.793.373.222.91
2613.749.127.366.415.805.385.074.834.644.483.723.303.152.84
2713.619.027.276.335.735.315.004.764.574.413.663.233.082.78
2813.508.937.196.255.665.244.934.694.504.353.603.183.022.72
2913.398.857.126.195.595.184.874.644.454.293.543.122.972.66
3013.298.777.056.125.535.124.824.584.394.243.493.072.922.61
4012.618.256.595.705.134.734.444.214.023.873.142.732.572.25
6011.977.776.175.314.764.374.093.863.693.542.832.412.251.92
10.896.965.464.654.143.783.513.303.132.992.301.871.691.22

Leave a Reply