Key Statistical Techniques: Assumptions and Applications

Introduction

Statistics is defined as a methodology of gathering data, analyzing it, understanding it, and forming appropriate conclusions from the analysis. Not many subjects are as broad or have as many important applications as statistics (Wooldridge & Jeffrey, 2013). Mathematicians, scientists, and researchers in general rely on statistics to interpret the information they collect or encounter in their specific fields of work. In general, nearly everything that covers the collection, interpretation, manipulation, and presentation of data is considered statistics (Armitage & Berry, 2011). This paper discusses various statistical instruments, their assumptions, and their uses.

Pearson's Correlation Coefficient

Pearson's correlation coefficient, r, is a statistical instrument that measures the strength and direction of the relationship between two variables. It is more accurately known as the Pearson product-moment correlation coefficient (Lange, Little & Taylor, 2010). Simply put, it is a statistical instrument designed to weigh the relationship between two related variables. For example, it can be used to understand the link between the time one spends revising for a test and his or her eventual performance. It could also be used to examine whether there is a link between duration of unemployment and depression. One can also think of the coefficient as the line of best fit drawn through the data collected for two variables.

Assumptions

Assumption 1: When using the coefficient, the assumption is made that the data is normally distributed. This is because one cannot evaluate the statistical significance of the correlation between two variables without bivariate normality. However, this assumption is not easy to evaluate, so a more practical method is often used instead. This method entails evaluating the normality of each variable individually, which can be done using SPSS Statistics to conduct the Shapiro-Wilk test of normality.

Assumption 2: For Pearson's correlation coefficient to be accurate, the data from both variables must be measured at the ratio or interval level — that is, they must be continuous (Lange, Little & Taylor, 2010). Examples of this kind of data include weight measured in pounds or kilograms, test scores measured between 0 and 100, intelligence measured using an IQ score, and study time measured in minutes or hours.

Range

Pearson's coefficient ranges from −1 to +1.

Example of Use

Pearson's coefficient can be applied to many different types of investigations and analyses. For instance, Chinese scientists have used it to investigate genetic divergence between different varieties of rice (Armitage & Berry, 2011). The objective of that study was to determine the evolutionary propensity of the different rice varieties. When the correlation was measured, it revealed a positive correlation of between 0.78 and 0.89. The researchers concluded that the evolutionary potential was very high.

ANOVA

Overview

ANOVA (Analysis of Variance) is a statistical instrument that measures whether the results of a test, experiment, or survey are significant — that is, whether they are meaningful. Simply put, the ANOVA test helps a researcher decide whether to accept or reject the null hypothesis (Armitage & Berry, 2011). It tests different groups of data to determine whether they are similar or different. There are several situations in which the ANOVA test is useful (Wooldridge & Jeffrey, 2013). For example, when different types of therapies — such as biofeedback, medication, and counseling — are tried with a group of psychiatric patients, the test can determine which therapy is most effective. Similarly, when a manufacturer uses two different processes to make the same type of product, the test can reveal which process yields the better result (Lange, Little & Taylor, 2010). Additionally, if university students from different campuses sit the same examination, examiners can use the ANOVA test to determine which campus performed best.

Assumptions

• The groups being tested should have equal sample sizes.
• The data should be normally distributed or near-normally distributed.
• Population variances should be equal (homogeneity of variance).
• Samples should be independent of one another.

Range and Uses

The ANOVA test examines whether the results of a particular study or test are statistically significant. When conducting the test, researchers typically aim to answer two questions: (1) Is the variance between the groups significant? (2) If so, how large is that variance?

Many tests compare group standard deviations, medians, and means (Armitage & Berry, 2011). For comparing group means, several procedures are available, including the Tukey HSD procedure. This procedure can reveal group differences by plotting the means and identifying cases where there is no overlap.

Example of Use

The ANOVA test is used to measure whether there is a significant difference between groups in response to some form of stimulus (Wooldridge & Jeffrey, 2013). This has many important applications. For example, when a manufacturer uses two distinct processes to make a lightbulb, ANOVA can determine which process is superior. Similarly, when students from different countries sit the same examination, ANOVA can indicate which group outperformed the other.

Conclusion

Statistics is a broad discipline whose main goal is to investigate sample populations and extrapolate to the wider population using that information (Lange, Little & Taylor, 2010). There are many different statistical instruments available that can be used to investigate and interpret data in the most appropriate way. The instruments discussed in this paper include Pearson's correlation coefficient, ANOVA, the T-test, and the OLS regression model. Each model investigates the extent of variance, significance, or certainty in data collected from sample populations. The degree of significance indicates how much confidence can be placed in the conclusions or interpretations drawn (Wooldridge & Jeffrey, 2013). However, for the conclusions produced by these instruments to be valid, certain assumptions must be satisfied. This paper has outlined some of the most important assumptions for each method. All of the instruments discussed have significant real-world applications across science, medicine, education, business, and econometrics.

References

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