###### Testing Independence based on Continge ...

Parametric tests are statistical tests in which we make assumptions regarding population distribution. Such tests involve the estimation of the key parameters of a distribution. For example, we may wish to estimate the mean or compare population proportions.

When carrying out statistical tests, assuming the parameter follows a specific distribution, the distribution chosen directly affects the formulation of the test statistic. For instance, if we assume that a parameter follows a normal distribution, we have to compute the z-statistic.

When conducting parametric testing, it may be necessary to approximate the normal distribution for non-normal distributions. This approximation is possible because of the central limit theorem, which asserts that as the sample size increases, most non-normal distributions “tend to normalize.”

Parametric tests are generally considered to be stronger compared to non-parametric ones.

Non-parametric tests – also called distribution-free tests by some researchers – are tests that do not make any assumption regarding the distribution of the parameter under study. Researchers use non-parametric testing when there are concerns about some quantities other than the parameter of the distribution.

You can conduct a parametric test using non-normal data courtesy of the central limit theorem. There are situations, however, where the mean may not provide a fair measure of the central tendency, i.e., when there are large outliers.

Consider a situation where we want to establish the center of a rather skewed distribution such as that of the income of the residents of a given city. While majority of the residents could be categorized as the middle class, the presence of just a few billionaires in a sample can greatly increase the mean income. Such a mean, therefore, may not provide a very reliable or realistic measure of income.

Instead, it may be more appropriate to use the median. Compared to the mean, the median can give a better representation of the center of the income distribution. This is due to the fact that 50% of the residents will be above median and the remaining 50%, below it.

In a summary, “outliers” affect the mean when dealing with skewed data. The median, on the other hand, sticks closer to the center of the distribution.

The absence of outright normal distribution of data might leave an analyst with insufficient information or too small a sample size to justify normal approximation. In the end, the analyst might not even successfully carry out distribution tests such as the goodness-of-fit test. In such a scenario, a non-parametric test may be appropriate.

Although non-parametric tests are usually easier to conduct than parametric ones, they do not have as much statistical power. Nonetheless, they provide an efficient tool for analyzing ordinal, ranked, or very skewed data.