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.
exact NULL or a logical indicating whether an exact p-value should be computed. See Details for the meaning of NULL. Only used in the two-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.

Exact p-values are only available for the two-sided two-sample test with no ties. In that case, if exact = NULL (the default) an exact p-value is computed if the product of the sample sizes is less than 10000. Otherwise, asymptotic distributions are used whose approximations may be inaccurate in small samples.

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.


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.

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 scale 2?
ks.test(x+2, "pgamma", 3, 2) # two-sided
ks.test(x+2, "pgamma", 3, 2, alternative = "gr")

[Package stats version 2.1.0 Index]