Paradigm

 

Any discussion on the History and Philosophy of Science necessarily involves the work of Thomas Samuel Kuhn (1922-1996), who authored, among other works, the book "The Structure of Scientific Revolutions."

In this book, Kuhn explains that scientific practice alternates between periods of "normal science" – based on a paradigm – and periods of "scientific revolution," when a paradigm shift occurs. Let's discuss the meaning of these terms.

Normal Science means conducting research based on established scientific achievements recognized by a specific community of scientists. Since scientists are committed to the same way of doing science, they share knowledge and ways of thinking. Thus, researchers learn how to solve problems through courses, laboratories, books, and manuals.

Puzzles and Exemplars
The search for solutions to problems, which Kuhn called "puzzles," is carried out using techniques learned from exemplars. These are the theories that dictate research methods and provide guidelines for scientific work.

Paradigm
Scientific research is guided not only by theories but by something broader: the paradigm. But what is a paradigm?

It is a set of practices that defines the behavior of scientists during a specific period.

In a given scientific field, you have a paradigm when you know:

·  The established truths.

·  What can be observed and examined.

·  What kinds of questions can be asked and researched to obtain answers on the subject.

·  How these questions should be structured.

·  How the results of scientific research should be interpreted.

Kuhn conceptualized paradigm in 1970 as:

“a complete set of beliefs, values, techniques, and everything else that is shared by the members of a given community.”

Later, Kuhn explained that:

“paradigms [are] actual solutions to puzzles that, used as models or examples, can be treated as if they were explicit rules and serve as the basis for solving the remaining puzzles of normal science.”

However, the word "paradigm" took on a life of its own. In fact, Kuhn acknowledged that the concept escaped his initial intentions.
In the Brazilian Portuguese translation, the concept of paradigm is inadequately expressed. It states: “paradigms are universally recognized scientific realizations that, for a time, provide model problems and solutions for a community of practitioners of a science.”

Paradigm Shift
Changing a paradigm is not easy. It means "acquiring" new values (an effort) and erasing old ones (a greater effort).

Changing a paradigm is not about changing techniques, texts, or equipment, as some think – but rather acquiring a "new worldview," which can be shared by an entire community of scientists.

Ultimately, a paradigm's success depends on the space it creates for new discoveries. If a scientific achievement can solve puzzles that previously had no satisfactory solution and is sufficiently original to attract a group of good scientists to the point of making them abandon the paradigm they knew, then you are facing a "revolution."

Scientific Revolution occurs when a paradigm shift happens. Following this change, science evolves normally for some time within the new paradigm. But the force of a paradigm is powerful.

Most of the time, science exhibits adherence to the paradigm. The puzzles proposed for scientists to solve are confined within it. This would explain why scientific revolutions are rare.

Anomalous Cases
Science enters a crisis when confidence is lost in the paradigm's ability to resolve discrepant cases – the so-called "anomalous cases." This then paves the way for a scientific revolution and the construction of a new paradigm.

An Example of a Paradigm Shift

When Christian Barnard replaced one man's heart with another's, he showed the world that a person could live with another's heart. He displayed not just a result – but shattered a paradigm ("one must die when the heart dies") and, in its place, another emerged: "organs can be transplanted."

When a new paradigm emerges, the structure of the entire scientific community is affected. The acceptance of a new paradigm – at least for a time – is not due solely to logical resources or evidence, experimental or otherwise. The truth is that scientists adhering to different paradigms have different views of the same phenomenon (while one sees the Sun revolving around the Earth, the other sees the Earth revolving around the Sun).

Sometimes, it becomes impossible to justify a scientist's or a group of scientists' preference for a particular paradigm. Those defending the new paradigm can campaign and seek new adherents through conversion or simply wait for the most resistant to die out. But there will always be a period of accommodation. On the other hand, some recognize a change immediately.

A Recognition of Paradigm Shift

The dentist Horace Wells (1815-1848) was undoubtedly the first to use anesthesia in surgical procedures. At the time, the importance of his proposal was not recognized. But, to demonstrate the anesthetic effect of sulfuric ether, a Harvard Medical student asked the Professor of Surgery to anesthetize a patient scheduled for a leg amputation at Massachusetts General Hospital. This was done in 1846. The patient showed no signs of pain during the operation, and Professor John Warren Collins (1778-1856) was moved to tears. He immediately recognized the change in the course of surgical history.

But when this happens – a paradigm shift – many things also change: how a scientist sees the world; the criteria for selecting important problems; research techniques; how phenomena are interpreted; the criteria for evaluating theories.

Scientific activity is critical. Being critical implies admitting the probability of error. Therefore, since it is possible we are wrong, we must seek evidence for our judgments about facts. Furthermore, we must know that what is considered evidence today may not be evidence tomorrow. After all, we are confined within a specific time and place, to say the least.

References

1.                   Kuhn, T. S. The Structure of Scientific Revolutions. 3rd ed. The University Chicago Press, 1996.

2.                   Kuhn, T. S. The Structure of Scientific Revolutions. 2nd ed., University of Chicago Press, Chicago & London, 1970, p.175.

3.                   KATZ, J. Experimentation with human beings. New York: Russel Sage Foundation. 1973.

4.                   VIEIRA, S. e HOSSNE, W. S. Experimentação com seres humanos. São Paulo: Moderna, 1986.

A Practical Rule for Residual Degrees of Freedom in ANOVA

 

Imagine you are conducting an experiment in an area with a fertility gradient. The land is on a slope and is therefore more fertile at the bottom than at the top. You want to compare four treatments, which we will call A, B, C, and D, and you decide to arrange them in five blocks. Each block can accommodate four plots. The experimental design could be the one shown in Figure 1.

            Figure 1: Layout of a randomized complete block design

Table 1 presents the Analysis of Variance (ANOVA) for this experiment.

             Table 1: Analysis of Variance (ANOVA)

This design is appropriate because the variation within each block has been minimized (by grouping similar fertility levels together), and the variation between blocks has been maximized. But what can be said about the number of residual degrees of freedom?

The most repeated criticism in experimental work is that the sample size is too small. Sometimes it is also argued that the number of residual degrees of freedom should be greater than 10 or 12. But why?

Remember that you want to compare four treatments. Therefore, the degrees of freedom for treatments are necessarily 3. If you increase the sample size, by how much does the residual degrees of freedom increase? Look at Table 2, which shows the increase in residual degrees of freedom as the sample size—more specifically, the number of blocks—increases.

   Table 2: Residual Degrees of Freedom for 4 Treatments and a Varying Number of Blocks

Now, observe Table 3 below. It provides some critical values of F for 3 degrees of freedom in the numerator (because you are comparing four treatments) and various degrees of freedom in the denominator (the residual). Notice that the critical F-values stabilize after the denominator has about 12 degrees of freedom. Therefore, increasing the number of blocks beyond this point does not help much in achieving statistical significance.

           Table 3: Critical F-values at the 5% significance level for 3 numerator df and various denominator df.

This becomes clearer by looking at Figure 2. The F-value is what determines significance. So, your ability to detect differences between the means of the four treatments improves if you organize five blocks instead of four (the critical F decreases from 3.86 to 3.49). However, it does not improve as much if you use six blocks instead of five (the critical F only decreases from 3.49 to 3.29).

        Figure 2: A graph plotting the data from Table 3, showing the critical F-value rapidly decreasing and then leveling off as the residual df increases.

This is the origin of the practical rule: aim for at least 12 residual degrees of freedom in the ANOVA. But note well: this is for 4 treatments. In agricultural sciences, it is common to compare 4 or even more treatments. Therefore, this rule is quite reasonable.

Summary

Here is a well-established and practical rule of thumb.

 

·        The power of an ANOVA F-test to detect differences between treatments depends on the critical F-value.

·        This critical F-value drops quickly as the residual degrees of freedom (df) increase from a low number but stabilizes after around 10-12 df.

·        Therefore, beyond a certain point (e.g., 12 residual df), adding more replicates (blocks) provides diminishing returns for the cost and effort involved.

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Beyond the Average: 5 Types of Means

       

         Introduction

When we talk about the "average," we're usually thinking of just one number. But in statistics, there isn't just one way to find the center of your data—there are several. Each type of mean provides a unique perspective, and choosing the right one can reveal a more accurate story hidden within the numbers.

Here are five essential means, from the everyday arithmetic mean to the robust trimmed mean.

 

1.     Arithmetic mean

 

The arithmetic mean of a set of data is the sum of all data divided by the number of data in the set. For example, a student obtained grades of 7.0, 3.0, 5.5, 6.5, and 8.0 in mathematics. He passed because the average grade is:

                                              ·        Mean = (7.0 + 3.0 + 5.5 + 6.5 + 8.0)/5 = 6.0 

 

The arithmetic mean of a sample is represented by x (read as x-bar or x-slash). The sample size is indicated by n. So, the formula for calculating the arithmetic mean of a sample is:

 

·        x̄ = (1/n) Σ xᵢ = (x₁ + x₂ + ... + xₙ)/n

       2- Weighted average 

The weighted average is the sum of the products of the data (x) by their respective weights (p), divided by the sum of the weights.

To understand how weighted averages are calculated, imagine that a student took three tests in a certain subject in which the material is cumulative, that is:

• in the first test, questions were asked about the material taught up to the date of that first test; 
• in the second test, questions were asked about the material taught from the beginning of the course up to the date of that second test.
• in the third test, questions were asked about the material taught from the beginning of the course up to the end of the course.

It is reasonable that the grade for the first test should have less weight (in other words, count for less in the final grade) than the second; it is also reasonable that the grade for the second test should have less weight than the third. Consider the following weights were  proposed: 1, 2, 3.

Imagine that the student obtained grades of 4, 7, and 6, which had weights of 1, 2, and 3, respectively. The student's weighted average is

                                           ·        x̄ = (1×4 + 2×7 + 3×6)/(1 + 2 + 3) = 36/6 = 6.0

Notice that to obtain the weighted average of a student's grades, each grade by was multiplied by its respective weight; products were added; weights were added and was applied the formula:

·        Formula: x̄ = Σ(xᵢpᵢ)/Σpᵢ

 3 Geometric mean

 

The geometric mean is given by the n^th root of the product of n data points.

The geometric mean is difficult to calculate and, perhaps because of this characteristic, is rarely used.

Here is an example. Let's calculate the geometric mean of the following data: 2, 3, 5, and 10.

 

                                       ·        G = ⁴√(2×3×5×10) = ⁴√300 = 4.16

To perform this calculation, use a calculator or apply logarithms. Since

      Therefore

        So, given n values of variable X,the geometric mean is

 

                                                           
 

The Greek letter ∏ (pronounced pi) is used as a mathematical symbol to indicate that all observed values of X must be multiplied. In mathematics, this letter is read as product.

4. Harmonic mean

The harmonic mean of n data points is the inverse of the arithmetic mean of the inverse of these values.

As an example, consider two numbers, 2 and 4. To calculate the harmonic mean, indicated here by H, invert the numbers, determine the arithmetic mean of these inverses, and invert the arithmetic mean to find the harmonic mean:

 

To calculate the harmonic mean, apply the formula:

5. Trimmed mean

 

Trimmed mean is a way to calculate an average by first removing a small percentage of the highest and lowest values. After taking out these extreme values, the average is calculated using the usual method.

Let’s say, as an example, a figure skating competition produces the following scores:

6.0, 8.1, 8.3, 9.1, 9.9.

 

The mean for the scores would equal:

 

    

To trim the mean by a total of 40%, we remove the lowest 20% and the highest 20% of values, eliminating the scores of 6.0 and 9.9.

Next, we calculate the mean based on the calculation:                                                      

Conclusion

Although the arithmetic mean is the most familiar, it is not always the most appropriate measure.
Choosing the right type of mean depends on the nature of the data and the purpose of the analysis.
Understanding the differences among arithmetic, weighted, geometric, harmonic, and trimmed means ensures that statistical summaries accurately reflect what the data reveal.

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