# Nonparametric Statistics

A method that makes statistical inference without regard to any underlying distribution

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## What is Nonparametric Statistics?

Nonparametric statistics is a method that makes statistical inferences without regard to any underlying distribution. The method fits a normal distribution under no assumptions. Habitually, the approach uses data that is often ordinal because it relies on rankings rather than numbers.

Nonparametric statistics can be contrasted with parametric statistics. The latter approach makes explicit assumptions about the distribution of observed data and estimates the parameters of the distribution using the same data.

### Summary

• Nonparametric statistics is a method that disregards any underlying distribution when making statistical inferences.
• Nonparametric statistical methods aim to discover the unknown underlying distribution of the observed data and make a statistical inference in the absence of the underlying distribution.
• Researchers are advised to consider weaknesses, strengths, and potential pitfalls of nonparametric statistics.

### Understanding Nonparametric Statistics

Consider the data with unknown parameters µ (mean) and σ2 (variance). While parametric statistics assume that the data were drawn from a normal distribution, a nonparametric statistic does not assume that the data is normally distributed or quantitative. In that regard, nonparametric statistics would estimate the shape of the distribution itself instead of estimating the individual µ and σ2.

On the other hand, parametric statistics would employ sample mean and sample standard deviation to estimate the values of µ and σ2, respectively. The model structure of nonparametric statistics is deduced from the observed data instead of a specified priori. The term nonparametric itself implies that the number and nature of parameters are flexible and not that they entirely lack parameters.

### Types of Nonparametric Statistics

There are two main types of nonparametric statistical methods. The first method seeks to discover the unknown underlying distribution of the observed data, while the second method attempts to make a statistical inference regarding the underlying distribution.

Kernel methods and histograms are commonly used to estimate the values of the parameters in the first approach. In contrast, the latter method involves testing hypotheses without the actual data values but rather based on the rank ordering of the data.

The nonparametric statistics tests tend to be easier to apply than parametric statistics, given the lack of assumption about the population parameters. Standard mathematical procedures for hypotheses testing make no assumptions about the probability distributions – including distribution t-tests, sign tests, and single-population inferences.

For example, when testing for the hypothesis that “there is a difference in medians,” the two random variables, X and Y, define two continuous distributions between where the hypothesis is performed, and paired samples are drawn. In addition to having general applicability, the test also lacks the statistical power of other tests, given that it works under a few assumptions.

### Examples of Nonparametric Statistics

Let us assume that a researcher is interested in estimating the number of babies born with jaundice in the state of California. An analysis of the data set may be performed by taking a sample of 5,000 babies. An estimate of the entire population of babies bearing jaundice born the following year is the derived measurement.

For a second case, consider two groups of different researchers. They are interested in knowing whether blanket marketing or commercial marketing is associated with how fast a company gains brand positioning. Assuming that the sample size is chosen randomly, its distribution regarding how fast a company realizes a brand positioning can be assumed to be normal. Nevertheless, an experiment that measures the company’s strategic goals to address market dynamics (which also determines brand positioning) cannot be assumed to take on a normal distribution.

The main idea behind the phenomenon is that randomly selected data may contain factors such as market dynamics. At the other extreme, if factors such as market segment and competition come into play, the company’s strategic objectives are not likely to impact the sample size. Such an approach is effective when the data lacks a clear numerical interpretation.

For example, tests on whether customers prefer a particular product because of its nutritional value may include ranking its metrics as strongly agree, agree, indifferent, disagree, and strongly disagree. In such a scenario, a nonparametric method comes in handy.

### Key Takeaways

Using nonparametric statistics approaches in research calls for due diligence on its weaknesses, strengths, and potential pitfalls. For data distribution with excess kurtosis or skewness, rank-based nonparametric tests turn out to be more potent than parametric tests.

Even so, not all instances where, if parametric assumptions are not met, we adopt nonparametric statistics as the substitute methods because of the comparatively low degree of confidence obtained from the earlier statistics.

Nonparametric statistics are appreciated because they can be applied with ease. The data becomes more applicable to various tests since the parameters are not mandatory. More importantly, the statistics can be used in the absence of vital information, such as the mean, standard deviation, or sample size. The features make nonparametric statistics have a broader scope of application compared to parametric statistics.