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Variance and standard deviation of a sample : Summarizing quantitative data More on standard deviation : Summarizing quantitative data Box and whisker plots : Summarizing quantitative data Other measures of spread : Summarizing quantitative data. Modeling data distributions.

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Percentiles : Modeling data distributions Z-scores : Modeling data distributions Effects of linear transformations : Modeling data distributions. Density curves : Modeling data distributions Normal distributions and the empirical rule : Modeling data distributions Normal distribution calculations : Modeling data distributions More on normal distributions : Modeling data distributions. Exploring bivariate numerical data. Introduction to scatterplots : Exploring bivariate numerical data Correlation coefficients : Exploring bivariate numerical data Introduction to trend lines : Exploring bivariate numerical data.

Least-squares regression equations : Exploring bivariate numerical data Assessing the fit in least-squares regression : Exploring bivariate numerical data More on regression : Exploring bivariate numerical data. Study design. Statistical questions : Study design Sampling and observational studies : Study design Sampling methods : Study design. Types of studies experimental vs.

Statistics: History, Interpretation, and Application |

Basic theoretical probability : Probability Probability using sample spaces : Probability Basic set operations : Probability Experimental probability : Probability. Randomness, probability, and simulation : Probability Addition rule : Probability Multiplication rule for independent events : Probability Multiplication rule for dependent events : Probability Conditional probability and independence : Probability. Counting, permutations, and combinations.

Counting principle and factorial : Counting, permutations, and combinations Permutations : Counting, permutations, and combinations Combinations : Counting, permutations, and combinations. Combinatorics and probability : Counting, permutations, and combinations. Random variables. Discrete random variables : Random variables Continuous random variables : Random variables Transforming random variables : Random variables Combining random variables : Random variables. Binomial random variables : Random variables Binomial mean and standard deviation formulas : Random variables Geometric random variables : Random variables More on expected value : Random variables Poisson distribution : Random variables.

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Sampling distributions. What is a sampling distribution? Confidence intervals. Introduction to confidence intervals : Confidence intervals Estimating a population proportion : Confidence intervals Estimating a population mean : Confidence intervals. Hopefully, students who learn the applicability of math to biology in this way will be better prepared to succeed in graduate studies and will contribute to the development of new ways in which the fields of math and biology can enhance knowledge Miller and Walston, The time—now—and place—Puerto Rico—chosen to implement efforts to improve mathematical skills among biology students is just right for several reasons.

Despite these efforts, teachers are still deficient in quantitative skills and limited in their ability to show their students the connections between these disciplines Quintero, Future math and biology teachers learn their science content in our courses together with science majors. Therefore, the learning activities we provide in the classroom have the potential to influence students further down on the educational ladder. In addition, the curricular reform in our biology department calls for student involvement in undergraduate research.

Development of quantitative skills with relevance to biological problems, as modeled herein, will help these students to see the application of mathematics to bioinformatics, biotechnology, rates of change, and data analysis. As a result, several consortia between industry and UPR have been developed to promote student training in biotechnology and development of scientific process skills Potera, Undergraduate teaching initiatives that promote interdisciplinarity between math and science are key to achieving this goal.

The student populations in the genetics and zoology courses are similar, as they have already completed a year of introductory biology; however, while statistics is a prerequisite for genetics, not all students in zoology have taken it. Genetics is a 1. Students usually take the lab the semester after they take the lecture course. Most of the activities that emphasized development of quantitative skills in genetics were conducted in the first 6 wk of the semester. These activities involved solving genetic crosses; analyzing data describing the distribution of quantitative traits, and allelic, genotypic, and phenotypic frequencies; and genetic mapping of eukaryotic chromosomes.

Typically, students allocated 30—40 min of the class period to solving and discussing problem sets in cooperative groups. The Blackboard academic online platform was used to distribute supplementary modules, practice exercises, and weekly quizzes that enforced student preparation. The peer-tutoring walk-in sessions were an important component of this course, because they provided another opportunity for students to get help on the assigned problem sets. These peer mentors were volunteer undergraduate students who had passed the genetics course in previous semesters with an A or B. The genetics tutors held weekly meetings with the professor M.

As part of their training, tutors also took a weekly quiz through Blackboard that included content-related questions, as well as math—genetics exercises.

For this, we translated and slightly modified a survey by Metz designed to measure knowledge of statistics among incoming students in general biology. The modified test instrument included two different surveys: a item pretest and a item posttest that shared only five identical questions.

The other seven questions measured knowledge of the same concepts, but encouraged higher-level thinking skills by requiring application of the quantitative concepts discussed in class to solve genetics problems. Content validity was achieved by the critical examination of the test by three other genetics professors and two biology professors engaged in science education and math—biology integration.

The pretest survey was administered on the first day of class, whereas the posttest was administered during the last week of the semester. Students were asked to voluntarily complete both surveys. Of 33 students registered for this session, only 16 students took both the pretest and the posttest. All the class and assessment materials, including the pretests and posttests, are available upon request in Spanish and English see Supplemental Material 1 and 2 or contact the corresponding author.

B Change in levels of achievement throughout the genetics course as measured by the pretest and the posttest. Low achievement is equivalent to a score of 0—5 points; medium, 6—8 points; and high, 9—12 points out of a maximum of 12 available points for both tests. A total of 40 students were enrolled in the zoology course. During the semester, students undertook three projects requiring the application of some kind of computational skill to answer questions about animals.

The first project aimed at developing simple bioinformatics skills by requiring consultation, analysis, and synthesis of data available in the International Union for Conservation of Nature website. The second project required the application of statistical methods to determine morphological patterns in bats. The third project involved the application of precalculus and calculus concepts to develop a model of sustainable yield for two valuable resources in an Amazonian forest. The projects were presented in class as biological problems, and the needed statistical or mathematical skills were discussed and explained in the context of the question.

Students worked in pairs, and had 2 wk to turn in a five-page written report, which included appropriate tables and figures. Because we focus on the integration of statistics to biology in this paper, we limit our discussion to the second project, which dealt with bat morphology and statistics.

Students were given an Excel worksheet with data on the wingspan cm , weight g , and sex male vs. The data were based on field observations Rodriguez-Duran, but were artificially generated to ensure that they would meet all the assumptions of parametric statistics. In class, we discussed the ecology and reproductive biology of this bat, mentioning relevant details, such as the fact that Caribbean fruit bats reproduce twice a year and female fruit bats give birth to a single offspring, which they carry around, and lactate for 3—4 mo Gannon et al.

Before analyzing the data, students were required to formulate a biological hypothesis for each question and to state the null and alternative hypotheses for each statistical test considered. Finally, students were expected to predict which sex was capable of carrying more weight during flight, and discuss what selective forces may have favored the evolution of the characteristics of body size and morphology observed in these bats.

Biological and statistical questions addressed by the students through the bat activity in zoology class. By looking at student performance on each of the 12 items, we were able to identify specific skills where students showed improvement in the posttest.

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These questions included statistical concepts of basic probability, probability applied to genetics, and data interpretation, which were discussed during the first 6 wk of the course relevant to the topics of Mendelian genetics, population genetics, and quantitative genetics. The one item that showed no change, but was well achieved even in the pretest, asked students to predict the gametes involved in a cross between individuals of known genotype. This suggests that students understood this basic Mendelian concept, perhaps from studying it in their general biology course.

This question was at the highest level of Bloom's taxonomy of learning domains Bloom and Krathwohl, , because it required comparison, evaluation, and formulation of a conclusion regarding the phenotypic, genotypic, and allelic frequencies of two populations. All of these questions involved analysis of quantitative concepts in a biological context.

Students improved mostly in skills that required interpretation of graphs and correlation coefficients. However, the concept of probability did not show considerable improvement, perhaps because the questions were applied to difficult genetics concepts that required higher levels of analysis than the ones presented in the pretest. Questions dealing with interpretation of p value presented a decrease in correct responses in the posttest. The items testing knowledge and understanding of the p value items 4 and 7 had different elements and distractors in the posttest that augmented the difficulty and the level of analysis required to answer the question.

The decrease in scores observed for these items shows students had a very basic knowledge of the concept of p value before the genetics course, and our efforts to improve their understanding of it to a point where they could synthesize resulting data and create a conclusion were not enough. The results show that learning objectives B, C, F, G, and J were not achieved by students, regardless of class instruction and the opportunity to work on a relevant biological problem requiring the application of statistical methods.

Getting Started with Statistics Concepts

Misconceptions were noted in the inability to formulate hypotheses and understanding of the significance of the probability that statistical tests associate to these hypotheses. Although the concept of correlation between variables with sex was understood, as indicated by students being able to ascertain that bat weight and wingspan were associated with sex from calculating high Pearson product-moment correlation coefficients r , students were unable to distinguish the added value of a regression analysis.

This paper presents the results of two independent projects aimed at integrating statistical concepts into undergraduate biology courses—genetics and zoology—in two very different ways. In general, we found an increase of understanding of and ability to apply statistical concepts of correlation and association between variables to biology.

In addition, students expressed their understanding of the relevance of statistics as a tool to analyze biological data and understand its significance. However, we identified other concepts for which students had more difficulty demonstrating knowledge gain despite in-class instruction and educational activities. Interestingly, although the instructional approach and the assessment instruments used were different, these two courses yielded similar results.

We will discuss the implications of these results in light of the challenges we face in order to further integrate math and biology, and we provide some suggestions to meet the need for interdisciplinary teaching in these fields. The lack of understanding of probability distributions, interpretation of the p value associated with statistics, and when to reject a null hypothesis was evident in both courses. Mathematical concepts associated with linear relationships between two variables were also weak.

Specific to the students in the zoology course was the inability to extrapolate the biological meaning of strong positive or negative slopes versus flat slopes describing the relationship between two variables, a concept that can be traced back to high school geometry and algebra. In both courses, we could see that students were able to distinguish the meaning of high or low correlation coefficients, but it was then difficult for them to synthesize a conclusion explaining the relationship between the variables involved, and to predict potential biological causation for the event described by the data.