ks.test {stats}R Documentation

Kolmogorov-Smirnov Tests


Performs one or two sample Kolmogorov-Smirnov tests.


ks.test(x, y, ..., alternative = c("two.sided", "less", "greater"),
        exact = NULL)


x a numeric vector of data values.
y either a numeric vector of data values, or a character string naming a distribution function.
... parameters of the distribution specified (as a character string) by y.
alternative indicates the alternative hypothesis and must be one of "two.sided" (default), "less", or "greater". You can specify just the initial letter of the value, but the argument name must be give in full. See Details for the meanings of the possible values.
exact NULL or a logical indicating whether an exact p-value should be computed. See Details for the meaning of NULL. Not used for the one-sided two-sample case.


If y is numeric, a two-sample test of the null hypothesis that x and y were drawn from the same continuous distribution is performed.

Alternatively, y can be a character string naming a continuous distribution function. In this case, a one-sample test is carried out of the null that the distribution function which generated x is distribution y with parameters specified by ....

The presence of ties generates a warning, since continuous distributions do not generate them.

The possible values "two.sided", "less" and "greater" of alternative specify the null hypothesis that the true distribution function of x is equal to, not less than or not greater than the hypothesized distribution function (one-sample case) or the distribution function of y (two-sample case), respectively. This is a comparison of cumulative distribution functions, and the test statistic is the maximum difference in value, with the statistic in the "greater" alternative being D^+ = max_u [ F_x(u) - F_y(u) ]. Thus in the two-sample case alternative="greater" includes distributions for which x is stochastically smaller than y (the CDF of x lies above and hence to the left of that for y), in contrast to t.test or wilcox.test.

Exact p-values are not available for the one-sided two-sample case, or in the case of ties. If exact = NULL (the default), an exact p-value is computed if the sample size if less than 100 in the one-sample case, and if the product of the sample sizes is less than 10000 in the two-sample case. Otherwise, asymptotic distributions are used whose approximations may be inaccurate in small samples. In the one-sample two-sided case, exact p-values are obtained as described in Marsaglia, Tsang & Wang (2003). The formula of Birnbaum & Tingey (1951) is used for the one-sample one-sided case.

If a single-sample test is used, the parameters specified in ... must be pre-specified and not estimated from the data. There is some more refined distribution theory for the KS test with estimated parameters (see Durbin, 1973), but that is not implemented in ks.test.


A list with class "htest" containing the following components:

statistic the value of the test statistic.
p.value the p-value of the test.
alternative a character string describing the alternative hypothesis.
method a character string indicating what type of test was performed.
data.name a character string giving the name(s) of the data.


Z. W. Birnbaum & Fred H. Tingey (1951), One-sided confidence contours for probability distribution functions. The Annals of Mathematical Statistics, 22/4, 592–596.

William J. Conover (1971), Practical Nonparametric Statistics. New York: John Wiley & Sons. Pages 295–301 (one-sample “Kolmogorov” test), 309–314 (two-sample “Smirnov” test).

Durbin, J. (1973) Distribution theory for tests based on the sample distribution function. SIAM.

George Marsaglia, Wai Wan Tsang & Jingbo Wang (2003), Evaluating Kolmogorov's distribution. Journal of Statistical Software, 8/18. http://www.jstatsoft.org/v08/i18/.

See Also

shapiro.test which performs the Shapiro-Wilk test for normality.


x <- rnorm(50)
y <- runif(30)
# Do x and y come from the same distribution?
ks.test(x, y)
# Does x come from a shifted gamma distribution with shape 3 and rate 2?
ks.test(x+2, "pgamma", 3, 2) # two-sided, exact
ks.test(x+2, "pgamma", 3, 2, exact = FALSE)
ks.test(x+2, "pgamma", 3, 2, alternative = "gr")

# test if x is stochastically larger than x2
x2 <- rnorm(50, -1)
plot(ecdf(x), xlim=range(c(x, x2)))
plot(ecdf(x2), add=TRUE, lty="dashed")
t.test(x, x2, alternative="g")
wilcox.test(x, x2, alternative="g")
ks.test(x, x2, alternative="l")

[Package stats version 2.5.0 Index]