Non Parametric Data and Tests (Distribution Free Tests)

What is a Non Parametric Test?

A non parametric test (sometimes called a distribution free test) does not assume anything about the underlying distribution (for example, that the data comes from a normal distribution). That’s compared to parametric test, which makes assumptions about a population’s parameters (for example, the mean or standard deviation); When the word “non parametric” is used in stats, it doesn’t quite mean that you know nothing about the population. It usually means that you know the population data does not have a normal distribution.

For example, one assumption for the one way ANOVA is that the data comes from a normal distribution. If your data isn’t normally distributed, you can’t run an ANOVA, but you can run the nonparametric alternative—the Kruskal-Wallis test.

If at all possible, you should us parametric tests, as they tend to be more accurate. Parametric tests have greater statistical power, which means they are likely to find a true significant effect. Use nonparametric tests only if you have to (i.e. you know that assumptions like normality are being violated). Nonparametric tests can perform well with non-normal continuous data if you have a sufficiently large sample size (generally 15-20 items in each group).

When to use it

Non parametric tests are used when your data isn’t normal. Therefore the key is to figure out if you have normally distributed data. For example, you could look at the distribution of your data. If your data is approximately normal, then you can use parametric statistical tests.
Q. If you don’t have a graph, how do you figure out if your data is normally distributed?
A. Check the skewness and Kurtosis of the distribution using software like Excel (See: Skewness in Excel 2013 and Kurtosis in Excel 2013).
A normal distribution has no skew. Basically, it’s a centered and symmetrical in shape. Kurtosis refers to how much of the data is in the tails and the center. The skewness and kurtosis for a normal distribution is about 1.

non parametric
Negative kurtosis (left) and positive kurtosis (right)

If your distribution is not normal (in other words, the skewness and kurtosis deviate a lot from 1.0), you should use a non parametric test like chi-square test. Otherwise you run the risk that your results will be meaningless.

Data Types

Does your data allow for a parametric test, or do you have to use a non parametric test like chi-square? The rule of thumb is:

nonparametric tests
A skewed distribution is one reason to run a nonparametric test.
Other reasons to run nonparametric tests:

  • One or more assumptions of a parametric test have been violated.
  • Your sample size is too small to run a parametric test.
  • Your data has outliers that cannot be removed.
  • You want to test for the median rather than the mean (you might want to do this if you have a very skewed distribution).

Types of Nonparametric Tests

When the word “parametric” is used in stats, it usually means tests like ANOVA or a t test. Those tests both assume that the population data has a normal distribution. Non parametric do not assume that the data is normally distributed. The only non parametric test you are likely to come across in elementary stats is the chi-square test. However, there are several others. For example: the Kruskal Willis test is the non parametric alternative to the One way ANOVA and the Mann Whitney is the non parametric alternative to the two sample t test.

The main nonparametric tests are:

The following table lists the nonparametric tests and their parametric alternatives.

Advantages and Disadvantages

Compared to parametric tests, nonparametric tests have several advantages, including:

However, they do have their disadvantages. The most notable ones are:

  • Less powerful than parametric tests if assumptions haven’t been violated.
  • More labor-intensive to calculate by hand (for computer calculations, this isn’t an issue).
  • Critical value tables for many tests aren’t included in many computer software packages. This is compared to tables for parametric tests (like the z-table or t-table) which usually are included.


Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences, Wiley.
Lindstrom, D. (2010). Schaum’s Easy Outline of Statistics, Second Edition (Schaum’s Easy Outlines) 2nd Edition. McGraw-Hill Education

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