Chi-Square Table

The chi-square distribution with d degrees of freedom gives distribution of the sum of the squares of d normal random variables. Its probability density function is



The chi-square distribution is used in hypothesis testing to determine the likelihood of the null hypothesis--that the observations are consistent with a theoretically assumed distribution. The test statistic is χ² is



where Oi is the number of observations in group i, and Ei is the expected number in group i. The degrees of freedom is always 1 less than the number of observations. In practical applications, researchers often reject the null hypothesis if the probability is less than a pre-determined threshold, such as 0.05 or 0.01.

Chi-Square Table

You can use a chi-square table to find the approximate probability of obtaining a certain χ² given the number of degrees of freedom. (You can also use this Chi-Square Test Calculator.) The table below lists the χ² values for the most commonly used probability thresholds.

D.F.Probability (Critical Value)
0.9950.9750.200.100.050.0250.020.010.0050.0020.001
10.00003930.0009821.6422.7063.8415.0245.4126.6357.8799.55010.828
2 0.01000.05063.2194.6055.9917.3787.8249.21010.59712.42913.816
30.07170.2164.6426.2517.8159.3489.83711.34512.83814.79616.266
40.2070.4845.9897.7799.48811.14311.66813.27714.86016.92418.467
50.4120.8317.2899.23611.07012.83313.38815.08616.75018.90720.515
60.6761.2378.55810.64512.59214.44915.03316.81218.54820.79122.458
70.9891.6909.80312.01714.06716.01316.62218.47520.27822.60124.322
81.3442.18011.03013.36215.50717.53518.16820.09021.95524.35226.124
91.7352.70012.24214.68416.91919.02319.67921.66623.58926.05627.877
102.1563.24713.44215.98718.30720.48321.16123.20925.18827.72229.588
112.6033.81614.63117.27519.67521.92022.61824.72526.75729.35431.264
123.0744.40415.81218.54921.02623.33724.05426.21728.30030.95732.909
133.5655.00916.98519.81222.36224.73625.47227.68829.81932.53534.528
144.0755.62918.15121.06423.68526.11926.87329.14131.31934.09136.123
154.6016.26219.31122.30724.99627.48828.25930.57832.80135.62837.697
165.1426.90820.46523.54226.29628.84529.63332.00034.26737.14639.252
175.6977.56421.61524.76927.58730.19130.99533.40935.71838.64840.790
186.2658.23122.76025.98928.86931.52632.34634.80537.15640.13642.312
196.8448.90723.90027.20430.14432.85233.68736.19138.58241.61043.820
207.4349.59125.03828.41231.41034.17035.02037.56639.99743.07245.315
218.03410.28326.17129.61532.67135.47936.34338.93241.40144.52246.797
228.64310.98227.30130.81333.92436.78137.65940.28942.79645.96248.268
239.26011.68928.42932.00735.17238.07638.96841.63844.18147.39149.728
249.88612.40129.55333.19636.41539.36440.27042.98045.55948.81251.179
2510.52013.12030.67534.38237.65240.64641.56644.31446.92850.22352.620
2611.16013.84431.79535.56338.88541.92342.85645.64248.29051.62754.052
2711.80814.57332.91236.74140.11343.19544.14046.96349.64553.02355.476
2812.46115.30834.02737.91641.33744.46145.41948.27850.99354.41156.892
2913.12116.04735.13939.08742.55745.72246.69349.58852.33655.79258.301
3013.78716.79136.25040.25643.77346.97947.96250.89253.67257.16759.703
3114.45817.53937.35941.42244.98548.23249.22652.19155.00358.53661.098
3215.13418.29138.46642.58546.19449.48050.48753.48656.32859.89962.487
3315.81519.04739.57243.74547.40050.72551.74354.77657.64861.25663.870
3416.50119.80640.67644.90348.60251.96652.99556.06158.96462.60865.247
3517.19220.56941.77846.05949.80253.20354.24457.34260.27563.95566.619
3617.88721.33642.87947.21250.99854.43755.48958.61961.58165.29667.985
3718.58622.10643.97848.36352.19255.66856.73059.89362.88366.63369.346
3819.28922.87845.07649.51353.38456.89657.96961.16264.18167.96670.703
3919.99623.65446.17350.66054.57258.12059.20462.42865.47669.29472.055
4020.70724.43347.26951.80555.75859.34260.43663.69166.76670.61873.402
4121.42125.21548.36352.94956.94260.56161.66564.95068.05371.93874.745
4222.13825.99949.45654.09058.12461.77762.89266.20669.33673.25476.084
4322.85926.78550.54855.23059.30462.99064.11667.45970.61674.56677.419
4423.58427.57551.63956.36960.48164.20165.33768.71071.89375.87478.750
4524.31128.36652.72957.50561.65665.41066.55569.95773.16677.17980.077
4625.04129.16053.81858.64162.83066.61767.77171.20174.43778.48181.400
4725.77529.95654.90659.77464.00167.82168.98572.44375.70479.78082.720
4826.51130.75555.99360.90765.17169.02370.19773.68376.96981.07584.037
4927.24931.55557.07962.03866.33970.22271.40674.91978.23182.36785.351
5027.99132.35758.16463.16767.50571.42072.61376.15479.49083.65786.661
5128.73533.16259.24864.29568.66972.61673.81877.38680.74784.94387.968
5229.48133.96860.33265.42269.83273.81075.02178.61682.00186.22789.272
5330.23034.77661.41466.54870.99375.00276.22379.84383.25387.50790.573
5430.98135.58662.49667.67372.15376.19277.42281.06984.50288.78691.872
5531.73536.39863.57768.79673.31177.38078.61982.29285.74990.06193.168
5632.49037.21264.65869.91974.46878.56779.81583.51386.99491.33594.461
5733.24838.02765.73771.04075.62479.75281.00984.73388.23692.60595.751
5834.00838.84466.81672.16076.77880.93682.20185.95089.47793.87497.039
5934.77039.66267.89473.27977.93182.11783.39187.16690.71595.14098.324
6035.53440.48268.97274.39779.08283.29884.58088.37991.95296.40499.607
6136.30141.30370.04975.51480.23284.47685.76789.59193.18697.665100.888
6237.06842.12671.12576.63081.38185.65486.95390.80294.41998.925102.166
6337.83842.95072.20177.74582.52986.83088.13792.01095.649100.182103.442
6438.61043.77673.27678.86083.67588.00489.32093.21796.878101.437104.716
6539.38344.60374.35179.97384.82189.17790.50194.42298.105102.691105.988
6640.15845.43175.42481.08585.96590.34991.68195.62699.330103.942107.258
6740.93546.26176.49882.19787.10891.51992.86096.828100.554105.192108.526
6841.71347.09277.57183.30888.25092.68994.03798.028101.776106.440109.791
6942.49447.92478.64384.41889.39193.85695.21399.228102.996107.685111.055
7043.27548.75879.71585.52790.53195.02396.388100.425104.215108.929112.317
7144.05849.59280.78686.63591.67096.18997.561101.621105.432110.172113.577
7244.84350.42881.85787.74392.80897.35398.733102.816106.648111.412114.835
7345.62951.26582.92788.85093.94598.51699.904104.010107.862112.651116.092
7446.41752.10383.99789.95695.08199.678101.074105.202109.074113.889117.346
7547.20652.94285.06691.06196.217100.839102.243106.393110.286115.125118.599
7647.99753.78286.13592.16697.351101.999103.410107.583111.495116.359119.850
7748.78854.62387.20393.27098.484103.158104.576108.771112.704117.591121.100
7849.58255.46688.27194.37499.617104.316105.742109.958113.911118.823122.348
7950.37656.30989.33895.476100.749105.473106.906111.144115.117120.052123.594
8051.17257.15390.40596.578101.879106.629108.069112.329116.321121.280124.839
8151.96957.99891.47297.680103.010107.783109.232113.512117.524122.507126.083
8252.76758.84592.53898.780104.139108.937110.393114.695118.726123.733127.324
8353.56759.69293.60499.880105.267110.090111.553115.876119.927124.957128.565
8454.36860.54094.669100.980106.395111.242112.712117.057121.126126.179129.804
8555.17061.38995.734102.079107.522112.393113.871118.236122.325127.401131.041
8655.97362.23996.799103.177108.648113.544115.028119.414123.522128.621132.277
8756.77763.08997.863104.275109.773114.693116.184120.591124.718129.840133.512
8857.58263.94198.927105.372110.898115.841117.340121.767125.913131.057134.745
8958.38964.79399.991106.469112.022116.989118.495122.942127.106132.273135.978
9059.19665.647101.054107.565113.145118.136119.648124.116128.299133.489137.208
9160.00566.501102.117108.661114.268119.282120.801125.289129.491134.702138.438
9260.81567.356103.179109.756115.390120.427121.954126.462130.681135.915139.666
9361.62568.211104.241110.850116.511121.571123.105127.633131.871137.127140.893
9462.43769.068105.303111.944117.632122.715124.255128.803133.059138.337142.119
9563.25069.925106.364113.038118.752123.858125.405129.973134.247139.546143.344
9664.06370.783107.425114.131119.871125.000126.554131.141135.433140.755144.567
9764.87871.642108.486115.223120.990126.141127.702132.309136.619141.962145.789
9865.69472.501109.547116.315122.108127.282128.849133.476137.803143.168147.010
9966.51073.361110.607117.407123.225128.422129.996134.642138.987144.373148.230
10067.32874.222111.667118.498124.342129.561131.142135.807140.169145.577149.449
10168.14675.083112.726119.589125.458130.700132.287136.971141.351146.780150.667
10268.96575.946113.786120.679126.574131.838133.431138.134142.532147.982151.884
10369.78576.809114.845121.769127.689132.975134.575139.297143.712149.183153.099
10470.60677.672115.903122.858128.804134.111135.718140.459144.891150.383154.314
10571.42878.536116.962123.947129.918135.247136.860141.620146.070151.582155.528
10672.25179.401118.020125.035131.031136.382138.002142.780147.247152.780156.740
10773.07580.267119.078126.123132.144137.517139.143143.940148.424153.977157.952
10873.89981.133120.135127.211133.257138.651140.283145.099149.599155.173159.162
10974.72482.000121.192128.298134.369139.784141.423146.257150.774156.369160.372
11075.55082.867122.250129.385135.480140.917142.562147.414151.948157.563161.581
11176.37783.735123.306130.472136.591142.049143.700148.571153.122158.757162.788
11277.20484.604124.363131.558137.701143.180144.838149.727154.294159.950163.995
11378.03385.473125.419132.643138.811144.311145.975150.882155.466161.141165.201
11478.86286.342126.475133.729139.921145.441147.111152.037156.637162.332166.406
11579.69287.213127.531134.813141.030146.571148.247153.191157.808163.523167.610
11680.52288.084128.587135.898142.138147.700149.383154.344158.977164.712168.813
11781.35388.955129.642136.982143.246148.829150.517155.496160.146165.900170.016
11882.18589.827130.697138.066144.354149.957151.652156.648161.314167.088171.217
11983.01890.700131.752139.149145.461151.084152.785157.800162.481168.275172.418
12083.85291.573132.806140.233146.567152.211153.918158.950163.648169.461173.617
12184.68692.446133.861141.315147.674153.338155.051160.100164.814170.647174.816
12285.52093.320134.915142.398148.779154.464156.183161.250165.980171.831176.014
12386.35694.195135.969143.480149.885155.589157.314162.398167.144173.015177.212
12487.19295.070137.022144.562150.989156.714158.445163.546168.308174.198178.408
12588.02995.946138.076145.643152.094157.839159.575164.694169.471175.380179.604
12688.86696.822139.129146.724153.198158.962160.705165.841170.634176.562180.799
12789.70497.698140.182147.805154.302160.086161.834166.987171.796177.743181.993
12890.54398.576141.235148.885155.405161.209162.963168.133172.957178.923183.186
12991.38299.453142.288149.965156.508162.331164.091169.278174.118180.103184.379
13092.222100.331143.340151.045157.610163.453165.219170.423175.278181.282185.571
13193.063101.210144.392152.125158.712164.575166.346171.567176.438182.460186.762
13293.904102.089145.444153.204159.814165.696167.473172.711177.597183.637187.953
13394.746102.968146.496154.283160.915166.816168.600173.854178.755184.814189.142
13495.588103.848147.548155.361162.016167.936169.725174.996179.913185.990190.331
13596.431104.729148.599156.440163.116169.056170.851176.138181.070187.165191.520
13697.275105.609149.651157.518164.216170.175171.976177.280182.226188.340192.707
13798.119106.491150.702158.595165.316171.294173.100178.421183.382189.514193.894
13898.964107.372151.753159.673166.415172.412174.224179.561184.538190.688195.080
13999.809108.254152.803160.750167.514173.530175.348180.701185.693191.861196.266
140100.655109.137153.854161.827168.613174.648176.471181.840186.847193.033197.451
141101.501110.020154.904162.904169.711175.765177.594182.979188.001194.205198.635
142102.348110.903155.954163.980170.809176.882178.716184.118189.154195.376199.819
143103.196111.787157.004165.056171.907177.998179.838185.256190.306196.546201.002
144104.044112.671158.054166.132173.004179.114180.959186.393191.458197.716202.184
145104.892113.556159.104167.207174.101180.229182.080187.530192.610198.885203.366
146105.741114.441160.153168.283175.198181.344183.200188.666193.761200.054204.547
147106.591115.326161.202169.358176.294182.459184.321189.802194.912201.222205.727
148107.441116.212162.251170.432177.390183.573185.440190.938196.062202.390206.907
149108.291117.098163.300171.507178.485184.687186.560192.073197.211203.557208.086
150109.142117.985164.349172.581179.581185.800187.678193.208198.360204.723209.265
151109.994118.871165.398173.655180.676186.914188.797194.342199.509205.889210.443
152110.846119.759166.446174.729181.770188.026189.915195.476200.657207.054211.620
153111.698120.646167.495175.803182.865189.139191.033196.609201.804208.219212.797
154112.551121.534168.543176.876183.959190.251192.150197.742202.951209.383213.973
155113.405122.423169.591177.949185.052191.362193.267198.874204.098210.547215.149
156114.259123.312170.639179.022186.146192.474194.384200.006205.244211.710216.324
157115.113124.201171.686180.094187.239193.584195.500201.138206.390212.873217.499
158115.968125.090172.734181.167188.332194.695196.616202.269207.535214.035218.673
159116.823125.980173.781182.239189.424195.805197.731203.400208.680215.197219.846
160117.679126.870174.828183.311190.516196.915198.846204.530209.824216.358221.019
161118.536127.761175.875184.382191.608198.025199.961205.660210.968217.518222.191
162119.392128.651176.922185.454192.700199.134201.076206.790212.111218.678223.363
163120.249129.543177.969186.525193.791200.243202.190207.919213.254219.838224.535
164121.107130.434179.016187.596194.883201.351203.303209.047214.396220.997225.705
165121.965131.326180.062188.667195.973202.459204.417210.176215.539222.156226.876
166122.823132.218181.109189.737197.064203.567205.530211.304216.680223.314228.045
167123.682133.111182.155190.808198.154204.675206.642212.431217.821224.472229.215
168124.541134.003183.201191.878199.244205.782207.755213.558218.962225.629230.383
169125.401134.897184.247192.948200.334206.889208.867214.685220.102226.786231.552
170126.261135.790185.293194.017201.423207.995209.978215.812221.242227.942232.719
171127.122136.684186.338195.087202.513209.102211.090216.938222.382229.098233.887
172127.983137.578187.384196.156203.602210.208212.201218.063223.521230.253235.053
173128.844138.472188.429197.225204.690211.313213.311219.189224.660231.408236.220
174129.706139.367189.475198.294205.779212.419214.422220.314225.798232.563237.385
175130.568140.262190.520199.363206.867213.524215.532221.438226.936233.717238.551
176131.430141.157191.565200.432207.955214.628216.641222.563228.074234.870239.716
177132.293142.053192.610201.500209.042215.733217.751223.687229.211236.023240.880
178133.157142.949193.654202.568210.130216.837218.860224.810230.347237.176242.044
179134.020143.845194.699203.636211.217217.941219.969225.933231.484238.328243.207
180134.884144.741195.743204.704212.304219.044221.077227.056232.620239.480244.370
181135.749145.638196.788205.771213.391220.148222.185228.179233.755240.632245.533
182136.614146.535197.832206.839214.477221.251223.293229.301234.891241.783246.695
183137.479147.432198.876207.906215.563222.353224.401230.423236.026242.933247.857
184138.344148.330199.920208.973216.649223.456225.508231.544237.160244.084249.018
185139.210149.228200.964210.040217.735224.558226.615232.665238.294245.234250.179
186140.077150.126202.008211.106218.820225.660227.722233.786239.428246.383251.339
187140.943151.024203.052212.173219.906226.761228.828234.907240.561247.532252.499
188141.810151.923204.095213.239220.991227.863229.935236.027241.694248.681253.659
189142.678152.822205.139214.305222.076228.964231.040237.147242.827249.829254.818
190143.545153.721206.182215.371223.160230.064232.146238.266243.959250.977255.976
191144.413154.621207.225216.437224.245231.165233.251239.386245.091252.124257.135
192145.282155.521208.268217.502225.329232.265234.356240.505246.223253.271258.292
193146.150156.421209.311218.568226.413233.365235.461241.623247.354254.418259.450
194147.020157.321210.354219.633227.496234.465236.566242.742248.485255.564260.607
195147.889158.221211.397220.698228.580235.564237.670243.860249.616256.710261.763
196148.759159.122212.439221.763229.663236.664238.774244.977250.746257.855262.920
197149.629160.023213.482222.828230.746237.763239.877246.095251.876259.001264.075
198150.499160.925214.524223.892231.829238.861240.981247.212253.006260.145265.231
199151.370161.826215.567224.957232.912239.960242.084248.329254.135261.290266.386
200152.241162.728216.609226.021233.994241.058243.187249.445255.264262.434267.541
201153.112163.630217.651227.085235.077242.156244.290250.561256.393263.578268.695
202153.984164.532218.693228.149236.159243.254245.392251.677257.521264.721269.849
203154.856165.435219.735229.213237.240244.351246.494252.793258.649265.864271.002
204155.728166.338220.777230.276238.322245.448247.596253.908259.777267.007272.155
205156.601167.241221.818231.340239.403246.545248.698255.023260.904268.149273.308
206157.474168.144222.860232.403240.485247.642249.799256.138262.031269.291274.460
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