Math concepts and applications for biology

Master basic arithmetic and the order of operations, from brackets to exponents, root extraction, multiplication, division, subtraction, and addition, with biology examples on a 0 to 1 scale.

Learn data normalization and transformation to compare plant growth across conditions. Calculate growth inhibition from exposed versus unexposed plants using initial and final lengths and their growth ratio.

Explore proportional reasoning in biology by distinguishing direct and inverse proportionality and applying compound proportion and the rule of three to multivariable problems, illustrated through area and bird population examples.

Explore mathematical functions that describe relationships between variables, model processes like bacterial growth, and evaluate how inputs map to outputs with a sample function f(x) = x^2 - 3x + 3.

Visualizes a function on a Cartesian graph with f(x)=x+3, and shows how the linear coefficient changes the line's vertical position while the angular coefficient changes its slope.

Explore probability basics, including the multiplication rule for two events happening at the same time and the addition rule for either-or outcomes, with biology-inspired examples from genetics and population genetics.

Learn how to compute genotype and phenotype probabilities in a simple fish example, using A and a alleles to predict yellow or blue color and zygote outcomes with and/or rules.

Explore three challenges on selection coefficient in natural selection, calculating parasite growth inhibition from drug efficacy, and using proportions to identify the cheapest protein source per dollar.

Model plant height with a function: starting at 30 cm, growth is 1.1 times the original height each day, predicting height after x days.

Identify the selection coefficient as one minus the chance of reproduction, linking higher selection to lower fitness. The example 0.8 corresponds to a 20% chance of reproduction.

Compare protein sources by grams per dollar using proportions, calculate total protein per package and price, and identify product three as the cheapest source.

Calculate parasite growth inhibition from an experiment by comparing treated and untreated cells, using the formula 1 minus treated over untreated times 100 to obtain a percentage.

Explore descriptive statistics fundamentals, including data collection, sampling, analysis, interpretation, and variable types and graphs, with biology-focused examples in epidemiology, bioinformatics, experiment design, ecology, and biodiversity.

Learn how to calculate mean, mode, and median as central tendencies, including mean calculation, mode as the most frequent value, and median as the ordered middle value.

Explore how standard deviation measures how much data deviates from the average, and how variance, its squared form, differs in units and sensitivity to outliers.

Explore the four main variable types—nominal, ordinal, discrete, and continuous—and how they function as categorical or numerical variables with biology-based examples.

Explore how graphs reveal patterns in biological data, enabling comparisons across groups. Learn histograms for distributions of continuous data with bins and bar charts for category comparisons.

Explore line charts and scatter plots to analyze trends over time and correlations between continuous variables in biology, with examples like CO2, viral load, and plant height.

Explore box plots, pie charts, and heat maps in biology, showing data distribution, parts of a whole, and gene expression through medians, quartiles, outliers, and color intensity.

Identify appropriate statistical tests for three-group comparisons in drug and fertilizer studies and for smoking-related gray hair differences, and define dependent and independent variables.

Introduce statistical inference and explain how samples support population conclusions, including hypothesis testing with null and alternative hypotheses, confidence intervals, normal distribution, central limit theorem, and significance levels.

Explore the t test for comparing two groups in biology, including one-sample and paired t tests, interpret the t value, and distinguish null and alternative hypotheses.

Apply the chi-square test to compare observed and expected counts for categorical data, using degrees of freedom and a significance level to decide whether to reject the null hypothesis.

Explore how to apply the chi-square test to Hardy-Weinberg equilibrium in population genetics, using allele frequencies and observed versus expected genotypes to assess evolutionary stability.

Compare three or more groups with anova. Show how post tests identify which groups differ and how f values indicate significance in a parasite growth inhibition context.

Identify appropriate statistical methods for comparing three treatment groups in drug and fertilizer trials and for assessing gray hair differences between smokers and non-smokers, noting dependent and independent variables.

Explore calculus concepts, including differential and integral calculus, to analyze rates of change and sums over intervals in biology, from population dynamics to enzyme kinetics and drug clearance.

Explore limits and infinitesimals by examining f(x)=1/x and f(x)=x; as x grows, f(x) approaches zero or infinity, illustrating the limit notation for infinity.

Explore the intuitive idea of differential calculus and how derivatives describe the rate of change, with distance over time yielding speed and applications to protein synthesis and population growth.

Explore integral calculus through the area under a deforestation curve, showing how smaller time intervals yield a closer approximation to real area, and relate distance to the integral of speed.

Explore evolutionary algorithms that train computer programs through mutation, crossing over, and selection to solve combinatorial optimization. Understand genes and chromosomes driving variation toward optimal solutions, as in antenna design.