This is probably the most basic and widely used of the non-parametric statistical tests. Developed in the 1930s by Andrei Nikolaevich Kolmogorov and Nikolai Vasilyevich Smirnov, test allows the comparison of a frequency distribution to some other known (continuous one-dimensional) distribution, such as a Gaussian normal distribution.

Some people use the K-S test as an alternative to the student T-test. Unfortunately, this use violates basic constraints of the K-S test (i.e. that one distribution must be known beforehand), and is meaningless. (However, this is OK if you're comparing a sample to an entire population).

The test works by comparing cumulative frequency values for the data points against the cumulative frequencies expected for the distribution in question.

Although it's called a 'goodness of fit' test, it's really a 'badness of fit' test, as all it can tell you is whether the data deviate significantly from the distribution. Because of this, we have to formulate our hypotheses in a counterintuituve way: The null hypothesis H_{0} is that the data do not significantly deviate from the distribution.

Before you can perform the test, you must have an "empirical cumulative distribution function" (ECDF) describing the expected cumulative frequencies. After deciding on a confidence interval you can then order the data, and calculate the actual cumulative frequencies for the points. You then determine a statistic

**D** = max |ECDF(i) - F(i)|

from these values. If this **D** value is less than a critical value for the number of points and the confidence interval you chose,
you cannot reject your null hypothesis, and must proceed as if the data fit the distribution.

Obviously, if you *want* the data to fit the distribution, it's tempting to jump to the conclusion that the data *do* fit the distribution. Unfortunately, this latter fact isn't determined by the test. A rather unsatisfying conclusion, and the only way to achieve a higher comfort level is to use more data points (which lowers the critical values).

For an example, let's test a sample of random numbers. I generated 41 random numbers (in the Data column of the table below) to represent sample data points. We'll pick the confidence level of 0.05 used commonly in the social sciences. The critical value for D is thus 0.3004.

To test the sample distribution against a normal distribution, we first calculate a Z score for each sample point. Then generate the expected cumulative frequency for each Z from a formula which estimates this value for the normal distribution^{1}. F(Z) is the actual cumulative frequency encountered (the point index divided by 41). Finally, calculate the difference between ECDF(Z) and F(Z) for each Z:

Point Data Z ECDF(z) F(z) |F(z)-ECDF(z)|
----- ------- -------- ------ ------ --------
1 0.7807 -1.5248 0.0608 0.0244 0.0364
2 2.5164 -1.4670 0.0690 0.0488 0.0202
3 7.9011 -1.2875 0.0981 0.0732 0.0249
4 8.0303 -1.2832 0.0989 0.0976 0.0013
5 10.6726 -1.1951 0.1155 0.1220 0.0064
6 12.8190 -1.1235 0.1303 0.1463 0.0161
7 14.6225 -1.0634 0.1436 0.1707 0.0271
8 17.1340 -0.9797 0.1635 0.1951 0.0316
9 20.6383 -0.8629 0.1941 0.2195 0.0254
10 21.2010 -0.8441 0.1993 0.2439 0.0446
11 21.7543 -0.8257 0.2045 0.2683 0.0638
12 23.5697 -0.7652 0.2221 0.2927 0.0706
13 25.2114 -0.7104 0.2387 0.3171 0.0783
14 29.7567 -0.5589 0.2881 0.3415 0.0533
15 30.1014 -0.5474 0.2921 0.3659 0.0738
16 33.3612 -0.4388 0.3304 0.3902 0.0598
17 33.5060 -0.4340 0.3322 0.4146 0.0825
18 35.2678 -0.3752 0.3538 0.4390 0.0853
19 38.5917 -0.2644 0.3957 0.4634 0.0677
20 39.3931 -0.2377 0.4061 0.4878 0.0818
21 39.5253 -0.2333 0.4078 0.5122 0.1044
22 41.9588 -0.1522 0.4395 0.5366 0.0971
23 44.3276 -0.0732 0.4708 0.5610 0.0902
24 45.0259 -0.0500 0.4801 0.5854 0.1053
25 48.1024 0.0526 0.5210 0.6098 0.0888
26 49.3623 0.0946 0.5377 0.6341 0.0965
27 49.7351 0.1070 0.5426 0.6585 0.1159
28 50.6010 0.1359 0.5540 0.6829 **0.1289**
29 62.2574 0.5244 0.7000 0.7073 0.0073
30 68.9057 0.7460 0.7722 0.7317 0.0405
31 76.8052 1.0094 0.8436 0.7561 0.0875
32 78.6105 1.0695 0.8576 0.7805 0.0771
33 79.5384 1.1005 0.8644 0.8049 0.0596
34 84.1996 1.2559 0.8954 0.8293 0.0661
35 87.4173 1.3631 0.9136 0.8537 0.0599
36 91.1650 1.4880 0.9316 0.8780 0.0536
37 94.2047 1.5894 0.9440 0.9024 0.0416
38 95.7160 1.6397 0.9495 0.9268 0.0226
39 96.4987 1.6658 0.9521 0.9512 0.0009
40 96.8248 1.6767 0.9532 0.9756 0.0224
41 99.8979 1.7791 0.9624 1.0000 0.0376

The maximum value in the last column occurs at point 28, and so our D statistic is 0.1289. This is well below our critical value. We cannot reject our null hypothesis, and must proceed as if the points were normally distributed.

The test can also be performed by picking Z values at regular intervals, and generating F(Z) by counting the number of points whose Z values are less than each critical Z:

Z ECDF(Z) pts<z F(z) |F(z)-ECDF(z)|
---- ------- ----- ------ ------
-2.0 0.0065 0 0.0000 0.0065
-1.8 0.0276 0 0.0000 0.0276
-1.6 0.0509 0 0.0000 0.0509
-1.4 0.0792 2 0.0488 0.0304
-1.2 0.1145 4 0.0976 0.0170
-1.0 0.1585 7 0.1707 0.0122
-0.8 0.2119 11 0.2683 0.0564
-0.6 0.2743 13 0.3171 0.0428
-0.4 0.3446 17 0.4146 0.0700
-0.2 0.4207 21 0.5122 0.0915
0.0 0.5000 24 0.5854 0.0854
0.2 0.5793 28 0.6829 0.1037
0.4 0.6554 28 0.6829 0.0275
0.6 0.7257 29 0.7073 0.0184
0.8 0.7881 30 0.7317 0.0564
1.0 0.8413 30 0.7317 **0.1096**
1.2 0.8849 33 0.8049 0.0801
1.4 0.9192 35 0.8537 0.0656
1.6 0.9452 37 0.9024 0.0428
1.8 0.9641 41 1.0000 0.0359
2.0 0.9772 41 1.0000 0.0228

(US) National Institute of Standards and Technology, Engineering Statistics Handbook.
1.3.5.16. Kolmogorov-Smirnov Goodness-of-Fit Test
*http://www.itl.nist.gov/div898/handbook/eda/section3/eda35g.htm*

^{1}US Department of Commerce, *Handbook of Mathematical Functions*, June 1964
26.2 (especially 26.2.18), Normal or Gaussian Probability Function, pp. 931-933

The original references appear to be

Kolmogorov, A. N. (1933) "On the empirical determination of a distribution function," (Italian) *Giornale dellâ€™Instituto Italiano degli Attuari*, 4, 83-91.

Smirnov, N. V. (1939), "On the estimation of the discrepancy between empirical curves of distribution for two independent samples." (Russian) *Bulletin of Moscow University*, 2, 3-16.