# Section 2.4: Percentages & Rates

The term percent means per one hundred.  You can think of a percentage as a special fraction that always has 100 in the denominator.  For example, if males make up 49% of the population, then 49 out of every 100 people will be male.  Any fraction can be converted to a percentage.  This is very useful because it facilitates comparisons of things that would otherwise be hard to compare because they have different frequencies.

For example, a criminologist wanting to compare homicides in rural areas versus urban areas would not be able to use the actual number of homicides because there are far fewer people in rural areas, so homicides would be far more infrequent in those rural areas.  By using percentages (or rates), the criminologist has a better method of comparing Monticello, Arkansas (population 10,000) with New York City.

To delve deeper, percentages are often instrumental in fields like psychology. For instance, a researcher might be interested in comparing rates of anxiety among college students during exam periods at different universities. Even if University A has 2,000 students and University B has 20,000 students, using raw numbers would skew the results, given the disparity in the total student population. However, by expressing the data in percentages, the researcher can more accurately compare the prevalence of anxiety in both institutions, irrespective of their size.

Similarly, in sociology, percentages are invaluable when studying phenomena like divorce rates across different cultures or societies. Let’s say a sociologist is keen to contrast the divorce rates in a more traditionally oriented society versus a more modern one. By using percentages, the sociologist can sidestep issues related to sheer population size and instead focus on the proportion of the population that experiences divorce, allowing for a more apples-to-apples comparison.

In the domain of public health, which often intersects with sociology and psychology, percentages also play a pivotal role. When examining issues like substance abuse or vaccine uptake rates among different demographic groups, raw numbers can be misleading due to varying group sizes. But by converting these numbers into percentages, public health officials can gain clearer insights into which groups may be at higher risk or in need of targeted interventions. In sum, percentages provide clarity and a level playing field, facilitating more accurate comparisons across diverse entities in the social and behavioral sciences.

## Computing a Percentage

To compute a percentage, simply divide the part by the whole and multiply the result by 100.  For example, if 7 out of 113 convicted house burglars report carrying a firearm while committing their crime, the 6.2% reported carrying firearms in the commission of burglaries (7/113 = .062 x 100 = 6.2%).

## Proportions

A proportion is a part of one.  We compute a proportion following the same procedure we use to compute a percent, but we do not multiply the decimal number by 100.  That is, a proportion is a fractional part of a whole where the whole is always equal to one.  For example, a proportion of .50 is the same as saying 50% or the fraction ½.  Proportions are rarely ever reported in the popular press because the average person without the benefit of statistical training does not understand them.  Percentages are preferred.  Social scientists reporting research results in professional journals are far more likely to report findings as proportions.

Within the realm of social and behavioral sciences, the use of proportions offers a refined way to communicate findings, especially among peers with statistical expertise. Consider a psychologist conducting a study on the effectiveness of a particular therapeutic intervention. When analyzing the results, the psychologist might find that .65 of the participants reported significant improvement in their symptoms. While this could also be reported as 65%, expressing it as a proportion immediately conveys the magnitude of the effect without the additional layer of percentage interpretation, making it efficient for other researchers to quickly grasp.

Anthropologists often employ proportions when examining patterns of behavior or cultural traits within a community or group. Suppose an anthropologist is studying the prevalence of a specific ritual in a community. If they find that .2 of the observed events contained elements of that ritual, it directly signifies that one-fifth of the events had that trait. Proportions here help in maintaining a level of precision, especially when diving deep into further sub-segments of data or when proportions are not intuitively easy to convert into percentages, such as .137 or .082.

Similarly, in the field of political science, proportions are crucial when analyzing voting behavior or political affiliations. For instance, if in a specific region, .48 of the population align with a particular political ideology, it can suggest a near-even split, providing insights into potential election outcomes or policy support levels. By expressing this as a proportion, the analyst is adhering to a standard that is commonly accepted and understood in academic and research publications. In essence, while percentages might be more accessible for general audiences, proportions offer a concise and precise metric for scholarly discourse in the social and behavioral sciences.

### Why Proportions are Sometimes Preferred

there are several mathematical and statistical reasons why proportions might be preferred over percentages in certain contexts, especially within research and academic settings:

1. Simplicity in Calculations: Working directly with proportions can simplify calculations, especially when dealing with statistical formulas. For example, when computing the expected frequencies in a chi-squared test, it’s more straightforward to use proportions rather than converting percentages back into raw counts.
2. Flexibility in Data Transformation: Proportions are more amenable to certain mathematical transformations. In logistic regression, a common technique in statistical analysis to predict the probability of a binary outcome, the log odds are derived from proportions, not percentages.
3. Avoiding Misinterpretations: Percentages can sometimes lead to misconceptions. For instance, an increase from 1% to 2% and an increase from 50% to 100% are both technically “100% increases,” but their implications and magnitudes are vastly different. Working with proportions can help maintain a clear focus on the actual values rather than the potentially misleading relative changes.
4. Probability Theory Alignment: Proportions align directly with probability values. In statistics, probabilities always lie between 0 and 1, identical to the range of proportions. By working with proportions, statisticians can immediately recognize and interpret values in the context of probability theory.
5. Enhanced Precision: When dealing with very small or very large percentages, rounding can lead to loss of information. For instance, a percentage like 0.03% would be 0.0003 as a proportion. In contexts where that extra precision is critical, such as in epidemiological studies dealing with rare events, using proportions can be more appropriate.
6. Multiplicative Models: When building multiplicative models, proportions are naturally suited. For instance, in scenarios where events are independent, probabilities (or proportions) multiply together. If events A and B are independent, then the probability of both A and B occurring is the product of their individual probabilities.
7. Avoiding Percent Bias: Percentages can sometimes create a bias in interpretation, especially when dealing with growth or changes. For example, going from 2% to 4% might be seen as “doubling,” which can sound more significant than stating a proportion increased from 0.02 to 0.04.

In summary, while percentages have their merits and are generally more accessible to a wider audience, proportions offer a level of mathematical rigor, precision, and clarity that is often required in research and statistical analyses. This makes them especially favored in academic and professional settings where the nuances of data interpretation are paramount.

## Summary

Percentages are fractions with a denominator of 100, facilitating comparisons across different frequencies. For instance, criminologists use percentages to compare crime rates in areas with varying populations, ensuring fair comparisons. In the social and behavioral sciences, percentages are also employed for their clarity in diverse scenarios, such as comparing anxiety rates between universities or divorce rates across cultures.

Proportions, which represent parts of a whole equating to one, aren’t popularly understood and thus are less frequent in mainstream media. However, in professional academic journals, they’re preferred for their precision and clarity. Proportions provide straightforward insights into data, like psychologists identifying therapy effectiveness or anthropologists examining cultural traits. They’re particularly useful when figures aren’t easily translatable into percentages.

Mathematically, proportions have advantages over percentages. They simplify computations, align with probability theory, and are versatile in data transformations. Proportions avoid potential percentage misinterpretations, ensure enhanced precision, and are apt for multiplicative models. Overall, while percentages are more broadly understood, proportions provide the precision, rigor, and clarity crucial in academic and professional research contexts.

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`Last Modified:  09/25/2023`

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