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# Non Parametric Statistics

Dive into the world of Non Parametric Statistics, a pivotal area within the field of Engineering. This comprehensive guide will unravel the intricate meaning of Non-Parametric Statistics, elaborating on definitions and key concepts. You will gain insightful knowledge on varied test methods, discerning the differentiation between parametric and non-parametric approaches. You'll also find an in-depth exploration of their individual properties and understand their impact on data analysis. Furthermore, practical applications and real-world examples of Non-Parametric Statistics illuminate its uses, supplemented with useful formulas. Equip yourself with these resilient statistical tools and enhance your understanding of this integral aspect of engineering analysis.

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## Understanding Non Parametric Statistics

Non Parametric Statistics, a vital topic in engineering, provides robust and versatile statistical methods that make fewer assumptions about the data being analysed. Notably, these techniques are less reliant on normal distribution assumptions about the population and are more tolerant of outliers in the data. As a result, non parametric statistics holds ideal implications in real-world applications where data often does not strictly adhere to normality assumption.

### Non-Parametric Statistics Meaning

Non-parametric statistics, often known as distribution-free statistics, offer a way of analysing data without requiring the stringent conditions traditionally imposed by parametric statistics.

Non-parametric statistics refers to statistical methods that do not assume a specific distribution to the data. They are often used when the data is not normally distributed and can't easily fit into a particular statistical model.

It is especially valuable in situations where the sample size is small, when data have outliers, or when stringent assumptions about data distribution are unrealistic.

#### Definitions and Key Concepts

To familiarize with non-parametric statistics more efficiently, understanding some key concepts and definitions is crucial. They include:
• $$\textbf{Population:}$$ The complete set of observations or data points that are being analysed.
• $$\textbf{Sample:}$$ A subset of data points selected from the population.
• $$\textbf{Distribution:}$$ The way the data points are spread across the value range.
• $$\textbf{Outlier:}$$ A data point that is significantly different from the other data points in a dataset.
The most noteworthy concept in non-parametric statistics is the element of ranking the data. Non-parametric tests fundamentally convert data into ranks and then analyse the ranks rather than actual data values.
 Concept Definition Rank A numerical position of a data point in a dataset when data points are arranged in ascending or descending order.
For instance, in a dataset of test scores, ranks imply the positions of individual scores when arranged in order from highest to lowest or vice versa.

For example, a dataset {50, 60, 70, 80} will have ranks {1, 2, 3, 4} respectively when arranged in ascending order.

In-depth, ranking is powerful in non-parametric statistics because it enables the analyst to reduce complex and varied data into uniform ranks. Hence, data analysis becomes much simpler and intuitive, accommodating a broader variety of data types and making the statistical tests more robust to outliers or skewed data.

## The Different Test Methods in Non Parametric Statistics

Non-parametric statistical tests provide a range of methods for performing analyses when data do not meet the assumptions of parametric tests. These tests help to overcome challenges related to distribution, sample size, outliers, and skewed data. Let's delve into the different methods, including one-sample tests and two-sample tests.

### One Sample NonParametric Test

The one-sample non-parametric test allows for the analysis of a single population in situations where parametric assumptions may not be satisfied. It's typically used to test whether the median of a distribution is equal to a hypothesised value.

The one-sample non-parametric test is a statistical method used to determine if a sample comes from a particular population with a specific median.

The most common one-sample non-parametric tests are the Sign Test and the Wilcoxon Signed-Rank Test.
• $$\textbf{Sign Test:}$$ This test evaluates the median of a distribution. It only considers the 'signs' of the differences between the actual and hypothesised median, disregarding their magnitudes.
• $$\textbf{Wilcoxon Signed-Rank Test:}$$ This test is similar to the Sign Test but takes into account the magnitude of the differences, providing a more exact result.

#### Steps to Conduct a One Sample Non-Parametric Test

The process of conducting a one-sample non-parametric test involves a series of distinct steps. Let's discuss conducting the Wilcoxon Signed-Rank Test regarding the hypothesised median.
1. Formulate the null hypothesis ($$H_0$$) and the alternative hypothesis ($$H_A$$). Typically, $$H_0$$ is that the population median is equal to the hypothesised value, whereas $$H_A$$ is that the population median is not equal to the hypothesised value.
2. Rank the absolute differences between the data values and the hypothethised median, disregarding the signs.
3. Apply these ranks back to their corresponding data values with their original sign (+/-).
4. Sum up the positive ranks and the negative ranks separately.
5. Calculate the test statistic, which is the smaller of the two summed rank values.
6. Compare your test statistic to the critical value from the Wilcoxon test table. If the test statistic is less than the critical value, then reject $$H_0$$.

### Non-Parametric Statistical Tests for Two Samples

Two-sample non-parametric statistical tests allow you to compare two independent data sets. These tests are useful when you want to determine whether there's a significant difference between two groups.

Two-sample non-parametric tests are tests that compare the medians of two independent samples to determine whether they come from the same population or different populations.

Two common two-sample non-parametric tests are the Mann-Whitney U Test and the Wilcoxon Rank-Sum Test.
• $$\textbf{Mann-Whitney U Test:}$$ This test compares two independent samples to see if they originate from the same distribution.
• $$\textbf{Wilcoxon Rank-Sum Test:}$$ This test, just like the Mann-Whitney U test, compares two independent samples. The ranking process, however, is different.

#### Comparing Two Sets of Data Using Non-Parametric Statistics

The process to compare two sets of data using non-parametric statistics involves several steps. Here's an overview using the Mann-Whitney U Test.
1. Firstly, state the null hypothesis ($$H_0$$) that the samples come from the same population and the alternate hypothesis ($$H_A$$) that the samples come from different populations.
2. Combine the data from both samples, then rank the data from smallest to largest.
3. Sum the ranks for the values from each sample separately.
4. The Mann-Whitney U statistic is the smaller of the two sums of ranks.
5. Check the critical values from the Mann-Whitney U distribution to judge the null hypothesis.
These steps offer a robust method of comparing two datasets non-parametrically to draw meaningful conclusions while negating the limitations of parametric methods.

## Explaining the Difference Between Parametric and Non Parametric Statistics

Both parametric and non-parametric statistics present pivotal methodologies in data analyses and interpretations. Paramount in various practical scenarios, these statistical techniques hold their unique advantages and potentially some limitations. It's their individual characteristics that call for careful selection based on the data type and research questions at hand.

### Advantages and Limitations of Both Approaches

To make a more significant comparative understanding of parametric and non-parametric statistics, it's beneficial to examine the advantages and limitations of these two approaches.

Parametric statistics are statistical techniques that assume data has been drawn from a specific population where the population parameters (mean, standard deviation) are well-defined and known. On the other hand, non-parametric statistics make fewer assumptions about the population parameters, hence its popular label as 'distribution-free' statistics.

• $$\textbf{Parametric Statistics}$$ Parametric statistical methods offer the following advantages:
• More efficient if their assumptions hold - provide more statistically significant results for the same amount of data.
• Offer wider options of tests and models, enhancing the capacity to model and understand complex relationships in the data.
• Allow for more detailed and informative inferences since they also estimate the parameters of the population distribution.
However, they encounter limitations:
• They require a significant amount of strict assumptions about the nature of the underlying data. The data should be numerical and often assumed to follow a Normal distribution.
• In an instance where the data has outliers, or it's skewed, using parametric tests could lead to misleading results.
• $$\textbf{Non-Parametric Statistics}$$ Non-parametric statistical methods have the following advantages:
• They have less stringent requirements about the underlying data and can be used with ordinal, interval or ratio data.
• Highly robust to outliers since they don't make assumptions about the population parameters and are based on ranks.
However, non-parametric methods also have limitations:
• They may require more data to achieve the same level of statistical power as parametric methods.
• While they can tell you if there is a significant effect or relationship, they don't provide as detailed information on the size or nature of the effect as their parametric counterparts.
Noticeably, the major distinguishing feature between the two is that parametric statistics assume the data is of a particular type (e.g., normally distributed), whereas non-parametric statistics do not rely on such assumptions. Nonetheless, both methodologies can offer valuable insights into your data if chosen wisely and implemented correctly. You need to consider the nature of your data, its distribution, and the specific research question being addressed in determining whether parametric or non-parametric techniques are the most appropriate.

## A Closer Look at Non-Parametric Statistics Properties

Non-parametric statistics, often lauded for their versatility, hold an edge in certain analysis scenarios. Especially when dealing with skewed data or categorical information, non-parametric methods prove dynamic. Let's explore the distinct properties that characterise these statistics and how they significantly influence data analysis.

### Common Properties of Non-Parametric Statistics

Non-parametric statistics, often referred to as distribution-free methods, stand out for their inherent properties. These properties account for their stringent applicability across various data types and analytical situations.

Non-parametric statistics are characterised by their lack of reliance on specific population parameters. This means they operate without the conventional constraints of normality and homogeneity of variance.

The observed common properties of non-parametric statistics include the following:

• $$\textbf{No Distributional Assumptions:}$$ These statistics do not rely on assumptions of the data conforming to specific distributions. Hence, they can efficiently handle data that do not satisfy the normality assumption of many parametric tests.
• $$\textbf{Robustness:}$$ Non-parametric methods are fairly robust against the presence of outliers or extreme values which could distort the results of parametric tests.
• $$\textbf{Ordinal Use:}$$ These methods can be utilised with data that are measured on ordinal scales, enriching their application scope.
• $$\textbf{Flexible Data Types:}$$ They are capable of analysing different types of data, ranging from ordinal to nominal, and even numerical data, increasing their versatility.

Robustness, in statistical terms, refers to a method's ability to produce consistent and reliable results even when the underlying assumptions are not strictly met. This property is a key advantage of non-parametric statistics, making them favourable in scenarios where outliers and skewed data cannot be avoided.

The absence of distributional assumptions alongside the robustness of these tests represents a significant advantage in dealing with real world, complex data structures. The ability to comfortably handle different data types ensures non-parametric techniques can be flexibly applied across diverse data landscapes.

#### How these Properties Influence Data Analysis

The properties of non-parametric statistics allow these methods to influence data analysis by providing valid results even when the data do not meet assumptions of parametric tests, by handling different levels of measurement and by being less sensitive to outliers.

The influence of the inherent properties of non-parametric statistical methods on data analysis become visible through three primary facets:
• $$\textbf{Practicality:}$$ Non-parametric tests become practical alternatives when parametric test assumptions can’t be fulfilled by the given data – when the population is not normally distributed, or population parameters are unknown.
• $$\textbf{Ranks as Data:}$$ In non-parametric tests, the data is often converted into ranks, enhancing the interpretability of the analysis results. This is particularly helpful when dealing with ordinal data, where magnitude differences are not necessarily homogeneous or meaningful. All attributions are treated equally, circumventing the adverse effect of outlier values.
• $$\textbf{Power:}$$ Even though non-parametric tests are considered to be less powerful than parametric ones when assumptions of the latter are met, they can equal or exceed the power of parametric tests when data are heavily skewed or contain outliers.

For instance, if you're examining customer satisfaction of a product, using a Likert scale that ranges from 'Very Unsatisfied' to 'Very Satisfied', a non-parametric test would be more appropriate than a parametric test. Such data clearly does not follow a normal distribution and does not possess equal intervals. Engaging a non-parametric test would cater better to these features of the data – neither ignoring the ordinal nature nor being affected by the absence of homogeneous intervals.

Overall, the properties of non-parametric statistics not only grant them more extensive applicability but also make them competent alternatives in several real-world analytical scenarios. Consider the properties and influences of non-parametric methods when you are working with complicated data structures or when key assumptions of parametric methods are violated.

## Practical Applications and Examples of Non-Parametric Statistics

Making practical sense of non-parametric statistics involves uncovering situations where these distribution-free methods apply. Their versatility is often showcased in real-life applications where assumptions of parametric tests are violated, and where data comes in different forms, scales, and distributions.

### Non-Parametric Statistics Applications in Real Life

In real life, data seldom fits into the strict stipulations required by parametric tests. Hence, non-parametric statistics find wide application in various research fields and industries, likely due to their relaxed requirements and unquestionable robustness.

Non-parametric statistics play a crucial role in circumstances where the fundamental assumptions of parametric models, such as normally distributed data or known population parameters, aren’t met or data is ordinal, ranking, or non-numerical.

• $$\textbf{Market Research:}$$ Flowing from customer surveys, data could be ordinal or non-numerical. Non-parametric tests, such as the Chi-square goodness of fit test, are applied to evaluate the success of campaigns or customer preferences.
• $$\textbf{Medical and Health Sciences:}$$ In clinical trials, non-parametric tests are utilised in comparing the ranks of two or more independent samples. The Mann-Whitney U test or Wilcoxon signed-rank test, for example, provide exceptional utility.
• $$\textbf{Environmental Science:}$$ When studying phenomena like pollution levels or climate change impacts, outliers are common. Non-parametric tests are often preferred due to their ability to marshall skewed data or outliers.
• $$\textbf{Social Sciences:}$$ Non-parametric methods can help measure non-quantifiable factors such as attitudes, perceptions, and beliefs. Here, the ordinal data benefits immensely from these distribution-free tests.
Importantly, the application of non-parametric tests is not constrained to when data fails to meet parametric assumptions. They're also deployed when data is ordinal or categorical by nature, or when data is prone to contain outliers.

#### Non-Parametric Statistics Formula

For non-parametric statistics, the data is frequently ranked, and test statistics used for comparing two samples is based on these ranks. This process is clearly evident in two widely-used non-parametric tests: the Mann-Whitney U test and the Wilcoxon signed-rank test.

The Mann-Whitney U test is employed to check whether two independent samples hail from populations with similar distributions. The Wilcoxon signed-rank test, conversely, compares two related samples to determine if their differences are symmetric around zero.

The Mann-Whitney U test statistic is given by: $U = n_1n_2 + \frac{n_1(n_1+1)}{2} - R_1$ where $$n_1$$ and $$n_2$$ are the sample sizes and $$R_1$$ is the sum of the ranks in the first sample. The Wilcoxon signed-rank test statistic, W, is calculated as the smaller of the two sums of positive and negative ranks, denoted as $$W^{+}$$ and $$W^{-}$$ respectively. $W = min(W^{+}, W^{-})$ Understanding the basis of these formulas not only gives you valuable insight into how non-parametric statistics functions, but it also improves your ability to interpret the results and drive valid conclusions.

#### Non-Parametric Statistics Examples

Putting the concept of non-parametric statistics into context requires concrete examples. These examples spotlight the practical use and utility of these tools in interpreting and analysing data from diverse sources.

Here's an instance of the use of non-parametric statistics in a marketing context. Let's say you have acquired data from a customer satisfaction survey with three categorical responses: 'Satisfied', 'Neutral', and 'Unsatisfied'. The goal here is to ascertain whether the distribution of customer attitudes deviates significantly from an expected even distribution. Here, the Chi-Square Test of Goodness-of-Fit, a classic non-parametric test, can come into play. The result enables you to make informed decisions about potential adjustments to improve customer satisfaction.

In the medical field, let's assume you're interested in comparing the effectiveness of two treatments. To this end, you collect data on patient recovery times for both treatments. Since recovery time isn't necessarily normally distributed, a Mann-Whitney test could be utilised to test for a significant difference between the two treatments. The output of the test allows for informed decisions regarding the preferred treatment option.

Embracing non-parametric statistical methods in your analytical toolkit provides a robust response to diverse and complex data scenarios. By understanding its various applications, mastering the underlying formulas, and learning from practical examples, you can ensure statistically valid interpretations and conclusions, regardless of data characteristics.

## Non Parametric Statistics - Key takeaways

• Non-parametric statistical tests are used when data does not meet the assumptions of parametric tests, including distribution, sample size, outliers, and skewness.
• The one-sample non-parametric test is used for situations where parametric assumptions may not be satisfied and to test if the median of a distribution matches a hypothesized value.
• Two common one-sample non-parametric tests are the Sign Test (which evaluates the median of a distribution) and the Wilcoxon Signed-Rank Test (which takes into account the magnitude of the differences).
• Two-sample non-parametric statistical tests like the Mann-Whitney U Test and the Wilcoxon Rank-Sum Test, are used to compare two independent data sets.
• The major difference between parametric and non-parametric statistics is that parametric statistics assume the data is of a particular type (e.g., normally distributed), whereas non-parametric statistics do not rely on such assumptions.

#### Flashcards in Non Parametric Statistics 15

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What is the difference between parametric and non-parametric statistics? Please write in UK English.
Parametric statistics make assumptions about population parameters and rely on the distribution of data, like normal distribution. Non parametric statistics, on the other hand, don't make such assumptions and can be used with data not fitting specific distribution patterns.
What are non-parametric tests in statistics?
Non-parametric tests in statistics are methods used for analysis when data doesn’t fit normal distribution or when data's nature doesn't require any assumptions about the parameters. These tests are less powerful but more flexible, making them useful with non-quantitative data or skewed distributions.
When should Non-Parametric Statistics be used?
Non-parametric statistics are used when the data fails to meet the assumptions required for parametric statistics, such as lack of normal distribution or homogeneity of variance. They can also be used with ordinal data, ranking data, or when the data set is small or has outliers.
What is Non-Parametric Rank Statistics? Write in UK English.
Non-parametric rank statistics is a statistical method used when data doesn't follow a specific distribution. Instead of focusing on parameters like mean or variance, it ranks data points and analyses patterns within these ranks, making it useful for data that doesn't meet standard assumption criteria.
Why should we use non-parametric statistical tests? Write in UK English.
Non-parametric statistical tests are used when the data is not normally distributed or when the sample size is small. They make fewer assumptions about the data's distribution and are more flexible, which makes them useful for non-numerical or ordinal data.

## Test your knowledge with multiple choice flashcards

What are the steps to conduct a one sample non-parametric test using the Wilcoxon Signed-Rank Test?

How does the property of robustness in non-parametric statistics influence data analysis?

Why is the concept of ranking important in non-parametric statistics?

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