Explain parametric and non-parametric statistics and differentiate between them.

1. Introduction to Parametric and Non-parametric Statistics

Parametric and non-parametric statistics are two approaches used in statistical analysis to make inferences about population parameters based on sample data. These approaches differ in their assumptions, measurement scales, and statistical techniques.

2. Parametric Statistics

2.1 Assumptions:
Parametric statistics assume that the data are drawn from a specific probability distribution, typically the normal distribution. This distributional assumption allows parametric methods to make precise estimates of population parameters, such as means and variances.

2.2 Measurement Scales:
Parametric statistics are suitable for continuous or interval-ratio data, where the measurement scale allows for meaningful calculations of means, variances, and other descriptive statistics. Parametric methods rely on numerical values and assume equal intervals between values.

2.3 Examples:
Common parametric tests include t-tests, analysis of variance (ANOVA), regression analysis, and correlation analysis. These tests require assumptions about the distribution of the data and the parameters being estimated.

2.4 Strengths:
Parametric statistics offer greater statistical power and efficiency when the underlying assumptions are met. They provide precise estimates of population parameters and are often preferred when sample sizes are large and data are normally distributed.

3. Non-parametric Statistics

3.1 Assumptions:
Non-parametric statistics make fewer assumptions about the underlying probability distribution of the data. Instead of assuming a specific distribution, non-parametric methods rely on the ranks or orderings of the data, making them more robust to deviations from normality.

3.2 Measurement Scales:
Non-parametric statistics are suitable for ordinal, nominal, and sometimes interval-ratio data. These methods do not require numerical values and can be applied to data that do not meet the assumptions of parametric tests.

3.3 Examples:
Common non-parametric tests include the Wilcoxon rank-sum test, Kruskal-Wallis test, Mann-Whitney U test, and Spearman's rank correlation coefficient. These tests use ranks or categorical data and do not assume a specific probability distribution.

3.4 Strengths:
Non-parametric statistics are more flexible and robust in the presence of non-normal data distributions, outliers, or small sample sizes. They provide valid statistical inference even when parametric assumptions are violated and are suitable for analyzing data with skewed distributions or outliers.

4. Differences Between Parametric and Non-parametric Statistics

4.1 Assumptions:
Parametric statistics require assumptions about the underlying probability distribution, while non-parametric statistics make fewer distributional assumptions and are more robust to deviations from normality.

4.2 Measurement Scales:
Parametric statistics are suitable for continuous or interval-ratio data with numerical values, while non-parametric statistics can be applied to ordinal, nominal, or sometimes interval-ratio data without numerical values.

4.3 Statistical Techniques:
Parametric statistics use mathematical models based on specific probability distributions to estimate population parameters, while non-parametric statistics use rank-based or categorical data and do not rely on distributional assumptions.

4.4 Flexibility and Robustness:
Non-parametric statistics are more flexible and robust in analyzing data with non-normal distributions, outliers, or small sample sizes. Parametric statistics may be more powerful and efficient when assumptions are met but are sensitive to violations of those assumptions.

5. Conclusion

Parametric and non-parametric statistics offer different approaches to statistical analysis, each with its own strengths and limitations. Researchers must carefully consider the nature of their data, the assumptions of the statistical tests, and the research questions being addressed when choosing between parametric and non-parametric methods. By understanding the differences between these approaches, researchers can select the most appropriate statistical techniques to draw valid and reliable conclusions from their data.