Chi-squared Χ2 test of goodness-of-fit

applied to categorical data to evaluate how likely it is that differences between the actual observed data and its expected/theoretical values arose by chance.
It tests a null hypothesis that the frequency distribution of certain events observed in a sample is consistent with a particular theoretical distribution. The events must be mutually exclusive and have total probability 1.
     
Resembles a normalized sum of squared deviations between observed and theoretical frequencies. Asymptotically approaches a Χ2 distribution.

Observed values Oi:
Expected values Ei: (AKA Theoretical values)

Select a desired level of confidence (significance level, p-value or alpha level) for the result of the test:

0.90 0.95 0.975 0.99 0.999

p: Theoretical distribution's number of parameters; reduction in df. (usually 1. 3 for Normal. 2 for Poisson)

ΣOi=N=
degrees of freedom df= the number of categories reduced by the number of parameters of the fitted distribution, i.e. n-p degrees of freedom, where n is the number of categories, p the number of parameters.
Χ2 statistic=       critical value=      p_value:
Accept or reject the null hypothesis that the observed distribution is the same as the theoretical/expected distribution based on whether the test statistic > the critical value. If test statistic > critical value, then null hypothesis (H0 that there is no difference between the distributions) can be rejected, and the alternative hypothesis (H1 that there is a difference between the distributions) can be accepted, both with the selected level of confidence. 

Upper-tail critical values of chi-square distribution.

df     Probability less than the critical value
        0.90 	0.95 	0.975 	0.99 	0.999

1 	2.706 	3.841 	5.024 	6.635 	10.828
2 	4.605 	5.991 	7.378 	9.210 	13.816
3 	6.251 	7.815 	9.348 	11.345 	16.266
4 	7.779 	9.488 	11.143 	13.277 	18.467
5 	9.236 	11.070 	12.833 	15.086 	20.515
6 	10.645 	12.592 	14.449 	16.812 	22.458
7 	12.017 	14.067 	16.013 	18.475 	24.322
8 	13.362 	15.507 	17.535 	20.090 	26.125
9 	14.684 	16.919 	19.023 	21.666 	27.877
10 	15.987 	18.307 	20.483 	23.209 	29.588
11 	17.275 	19.675 	21.920 	24.725 	31.264
12 	18.549 	21.026 	23.337 	26.217 	32.910
13 	19.812 	22.362 	24.736 	27.688 	34.528
14 	21.064 	23.685 	26.119 	29.141 	36.123
15 	22.307 	24.996 	27.488 	30.578 	37.697
16 	23.542 	26.296 	28.845 	32.000 	39.252
17 	24.769 	27.587 	30.191 	33.409 	40.790
18 	25.989 	28.869 	31.526 	34.805 	42.312
19 	27.204 	30.144 	32.852 	36.191 	43.820
20 	28.412 	31.410 	34.170 	37.566 	45.315
21 	29.615 	32.671 	35.479 	38.932 	46.797
22 	30.813 	33.924 	36.781 	40.289 	48.268
23 	32.007 	35.172 	38.076 	41.638 	49.728
24 	33.196 	36.415 	39.364 	42.980 	51.179
25 	34.382 	37.652 	40.646 	44.314 	52.620
26 	35.563 	38.885 	41.923 	45.642 	54.052
27 	36.741 	40.113 	43.195 	46.963 	55.476
28 	37.916 	41.337 	44.461 	48.278 	56.892
29 	39.087 	42.557 	45.722 	49.588 	58.301
30 	40.256 	43.773 	46.979 	50.892 	59.703
31 	41.422 	44.985 	48.232 	52.191 	61.098
32 	42.585 	46.194 	49.480 	53.486 	62.487
33 	43.745 	47.400 	50.725 	54.776 	63.870
34 	44.903 	48.602 	51.966 	56.061 	65.247
35 	46.059 	49.802 	53.203 	57.342 	66.619
36 	47.212 	50.998 	54.437 	58.619 	67.985
37 	48.363 	52.192 	55.668 	59.893 	69.347
38 	49.513 	53.384 	56.896 	61.162 	70.703
39 	50.660 	54.572 	58.120 	62.428 	72.055
40 	51.805 	55.758 	59.342 	63.691 	73.402
41 	52.949 	56.942 	60.561 	64.950 	74.745
42 	54.090 	58.124 	61.777 	66.206 	76.084
43 	55.230 	59.304 	62.990 	67.459 	77.419
44 	56.369 	60.481 	64.201 	68.710 	78.750
45 	57.505 	61.656 	65.410 	69.957 	80.077
46 	58.641 	62.830 	66.617 	71.201 	81.400
47 	59.774 	64.001 	67.821 	72.443 	82.720
48 	60.907 	65.171 	69.023 	73.683 	84.037
49 	62.038 	66.339 	70.222 	74.919 	85.351
50 	63.167 	67.505 	71.420 	76.154 	86.661
51 	64.295 	68.669 	72.616 	77.386 	87.968
52 	65.422 	69.832 	73.810 	78.616 	89.272
53 	66.548 	70.993 	75.002 	79.843 	90.573
54 	67.673 	72.153 	76.192 	81.069 	91.872
55 	68.796 	73.311 	77.380 	82.292 	93.168
56 	69.919 	74.468 	78.567 	83.513 	94.461
57 	71.040 	75.624 	79.752 	84.733 	95.751
58 	72.160 	76.778 	80.936 	85.950 	97.039
59 	73.279 	77.931 	82.117 	87.166 	98.324
60 	74.397 	79.082 	83.298 	88.379 	99.607
61 	75.514 	80.232 	84.476 	89.591 	100.888
62 	76.630 	81.381 	85.654 	90.802 	102.166
63 	77.745 	82.529 	86.830 	92.010 	103.442
64 	78.860 	83.675 	88.004 	93.217 	104.716
65 	79.973 	84.821 	89.177 	94.422 	105.988
66 	81.085 	85.965 	90.349 	95.626 	107.258
67 	82.197 	87.108 	91.519 	96.828 	108.526
68 	83.308 	88.250 	92.689 	98.028 	109.791
69 	84.418 	89.391 	93.856 	99.228 	111.055
70 	85.527 	90.531 	95.023 	100.425 	112.317
71 	86.635 	91.670 	96.189 	101.621 	113.577
72 	87.743 	92.808 	97.353 	102.816 	114.835
73 	88.850 	93.945 	98.516 	104.010 	116.092
74 	89.956 	95.081 	99.678 	105.202 	117.346
75 	91.061 	96.217 	100.839 	106.393 	118.599
76 	92.166 	97.351 	101.999 	107.583 	119.850
77 	93.270 	98.484 	103.158 	108.771 	121.100
78 	94.374 	99.617 	104.316 	109.958 	122.348
79 	95.476 	100.749 	105.473 	111.144 	123.594
80 	96.578 	101.879 	106.629 	112.329 	124.839
81 	97.680 	103.010 	107.783 	113.512 	126.083
82 	98.780 	104.139 	108.937 	114.695 	127.324
83 	99.880 	105.267 	110.090 	115.876 	128.565
84 	100.980 	106.395 	111.242 	117.057 	129.804
85 	102.079 	107.522 	112.393 	118.236 	131.041
86 	103.177 	108.648 	113.544 	119.414 	132.277
87 	104.275 	109.773 	114.693 	120.591 	133.512
88 	105.372 	110.898 	115.841 	121.767 	134.746
89 	106.469 	112.022 	116.989 	122.942 	135.978
90 	107.565 	113.145 	118.136 	124.116 	137.208
91 	108.661 	114.268 	119.282 	125.289 	138.438
92 	109.756 	115.390 	120.427 	126.462 	139.666
93 	110.850 	116.511 	121.571 	127.633 	140.893
94 	111.944 	117.632 	122.715 	128.803 	142.119
95 	113.038 	118.752 	123.858 	129.973 	143.344
96 	114.131 	119.871 	125.000 	131.141 	144.567
97 	115.223 	120.990 	126.141 	132.309 	145.789
98 	116.315 	122.108 	127.282 	133.476 	147.010
99 	117.407 	123.225 	128.422 	134.642 	148.230
100 	118.498 	124.342 	129.561 	135.807 	149.449

PDFs of chi-squared functions for first few values of k:

PDFs of chi-squared functions for various values of k:

Γ gamma function

k Γ(k/2) =/≈
1 Γ(1/2) 1.7724
2 Γ(1) 1
3 Γ(3/2) .8862
4 Γ(2)=1! 1
5 Γ(5/2) 1.3293
6 Γ(3)=2! 2
7 Γ(7/2) 3.3233
8 Γ(4)=3! 6
9 Γ(9/2)
10 Γ(5)=4! 24
20 Γ(10)9! 362880