Rank

 Rank means the order of data. Level by statistics. In arithmetic, the standard is the conversion of data in which numeric or ordinal values ​​are replaced by the rate at which the data is sorted. For example, data on the numbers 3.4, 5.1, 2.6, and 7.3 is recognizable, the levels of these data items will be 2, 3, 1, and 4 respectively.

In statistics, "rank" refers to the position of an element or data point in a dataset when it is sorted in ascending or descending order. The rank provides information about the relative position or order of each data point within the dataset. Ranks are often used in various statistical analyses and calculations, especially in non-parametric statistics and rank-based methods.

Here are some key points about rank in statistics:

Ascending and Descending Rank: Depending on the context, you can calculate ranks in ascending order (from the smallest value to the largest) or descending order (from the largest value to the smallest). Ascending rank is more common, but both types have their uses.

Tied Values: In real-world data, it's common to have tied values, which are data points that have the same value. When calculating ranks, there are different methods for dealing with tied values, including assigning them the same rank or using an average rank.

Ranking Methods: Various methods exist for assigning ranks, and the choice of method can impact the results of statistical analysis. Some common methods include:

Standard Competition Ranking (SCR): Assigns the same rank to tied values and then skips the next rank(s) accordingly.

Modified Competition Ranking (MCR): Similar to SCR but assigns an average rank to tied values.

Dense Ranking: Assigns consecutive integers to tied values, with no gaps in the ranks.

Fractional Ranking: Assign each tied value an average of the ranks they would occupy without ties.

Use Cases: Rank is frequently used in non-parametric statistical tests, such as the Mann-Whitney U test, Wilcoxon signed-rank test, and Spearman's rank correlation coefficient. These tests use rank-based comparisons rather than the actual values of data points, making them robust against outliers and less reliant on assumptions about data distribution.

Percentile Rank: Percentile rank is a concept related to rank. It represents the percentage of data points in a dataset that have values less than or equal to a particular data point. For example, if your score on a test ranks at the 90th percentile, it means you scored better than 90% of the test-takers.

Ranking in Data Visualization: Rank can be used in data visualization to create various types of charts and plots, such as ranked bar charts or dot plots, to visually compare the relative positions of data points.

Applications: Rank is used in a wide range of fields, including finance (e.g., ranking stocks by performance), sports analytics (e.g., ranking teams or players), and academic assessment (e.g., ranking students based on test scores).

In summary, rank in statistics provides a way to order and compare data points within a dataset, and it plays a crucial role in various statistical analyses, especially in situations where assumptions about data distribution are not met or when dealing with tied values. It is a versatile tool for making comparisons and drawing conclusions from data.

The Wilcoxon Signed-Rank Test

Suppose that we are willing to assume that the population of interest is continuous and symmetric. Our interest focuses on the median, if mean = median ( symmetric distributions). A disadvantage of the sign test in this situation is that it considers only the signs of the deviations X(i)- median (0) and not their magnitudes. The Wilcoxon signed-rank test was designed to overcome that disadvantage.

We are interested in testing H(0):mu=mu(0) against the usual alternative. Assume that X1, X2,...Xn is a random sample from a continuous and symmetric distribution with mean(and median) mu.Compute the differences Xi-mu(0),i=1,2,....n. Rank the absolute differences |Xi-mu(0)|,i=1,2...n in ascending order, and then give the ranks the signs of their corresponding differences.

Let R+ be the sum of the positive ranks R- be the absolute value of the sum of negative ranks, and let R=min(R+, R-).

Ties in the Wilcoxon Signed Rank Test

Because the underlying population is continuous, ties are theoretically impossible, although they will sometimes occur in practice. If several observations have the same absolute magnitude, they are assigned the average of the ranks that they would receive if they differed slightly from one another.

Paired Observations

The Wilcoxon signed-rank test is applied to paired data.Let(X1j, X2j),j=1,2,3.....n be a collection of paired observations from continuous distributions that differ only with respect to their mean(It is not necessary that the distribution of X1  and X2 be symmetric.).This assures that the distribution of the differences Dj=X1j-X2j is continuous and symmetric.

The Wilcoxon Rank-Sum Test

Suppose that we have two independent continuous populations X1 and X2 with means mu1 and mu2. The distributions of X1 and X2 have the same shape and spread and differ only(possibly) in their means. The Wilcoxon rank-sum test can be used to test the hypothesis H0: mu1=mu2. Sometimes this procedure is called the Mann-Whitney test although the Mann-Whitney test statistic is usually expressed in a different form.

Rank Transformation

The procedure used in the previous of replacing the observations by their ranks is called rank transformation. It is a very powerful and widely useful technique. If we were to apply the ordinary F-test to the ranks rather than to the original data, we would obtain the test statistic.

F0=[K/(a-1)]/(N-1-K)/(N-a)

Note that as the Kruskal-Wallis statistic K increases or decreases, F0 also increases or decreases, so the Kruskal-Wallis test is nearly equivalent to applying the usual analysis of variance to the ranks.

The rank transformation has wide applicability in experimental design problems for which no nonparametric alternative to the analysis of variance exists.

More:

F-test

T-test

Z-test

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