Introductory Statistics Practice, Set Where Biology Uses It
by Professor Stephen T. Abedon Steve Abedon, Ph.D., has taught well over 100 individual classes and ~10 individual courses (including introductory biology, microbiology, physiology, biochemistry, etc.) over his 30+ years as a college professor. He has over 150 publications, including multiple monographs, and is ranked in the top 1 or 2 percent among researchers of all disciplines worldwide. His research focuses on bacteriophage biology and phage therapy. (abedon.1@osu.edu)
phage.org | phage-therapy.org | biologyaspoetry.org | abedon.phage.org | google scholar | publications
Jump to: 🗺️ Topic Map | 📊 Describing Data | 🎲 Probability | 🔔 The Normal Curve | 🎯 Sampling and Standard Error | ↔️ Confidence Intervals | ⚖️ Hypothesis Tests | 👥 Comparing Two Groups | 🧬 Counts and Chi-Square | 📈 Correlation and Regression | 📝 Word Problems | 📋 Your Data | 🧮 More Calculators
biologyaspoetry.org/laboratories/statistics/ · Abedon’s Books
How can I improve this page? contact: statistics@biologyaspoetry.org
Every skill on this page is listed below by tab, with a mark where a biology version of its problems is available. Pick a skill and press Practice, or go straight to a tab.
| Skill | Biology version |
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The page follows the usual one-semester introductory statistics course, the kind taken by students in biology, nursing, and the other sciences, and the statistics a first biology laboratory course asks for. Analysis of variance, nonparametric tests, multiple regression, and the design of experiments are named on the Comparing Two Groups tab, so that you know when you need them, but calculating them is left to your textbook and to statistical software. The order of topics differs from school to school and from instructor to instructor, so your own syllabus is the final word.
Something missing? If your course covers a topic that isn’t here, if a problem doesn’t match the way your course does it, or if an answer you believe is right is marked wrong, say so at statistics@biologyaspoetry.org. The page grows from what students ask for.
The page remembers the tab you were on and, on each tab, the skill, the problem, any answer you had typed, and how you have been doing. It keeps this in your web browser on this device only; nothing is sent anywhere. Using another browser or device starts fresh.
The answer box reads numbers the way you would type them in most online homework systems. A few conventions help:
A few conventions differ from textbook to textbook, and this page states the one it uses in each problem. Standard deviations are sample standard deviations, dividing by n − 1, unless a problem says the numbers are a whole population. Quartiles are the medians of the lower and upper halves of the data, leaving the median itself out of both halves when the count is odd, the method of most introductory textbooks and of the TI-83 and TI-84 calculators; spreadsheets and some software use other methods and can give slightly different quartiles. The two-sample t statistic is the unpooled one, which does not assume that the two groups have equal variances. Worked by hand, it is usually given the conservative degrees of freedom, the smaller of n1 − 1 and n2 − 1; software, and the Your Data tab, use a more exact formula (Welch’s), which gives a few more degrees of freedom and a slightly smaller p-value.
Your answers to the practice problems are kept as you go, grouped by skill. In most browsers they are still here if you leave and come back on the same device, though a private window, a different device, or cleared browser data will start you fresh. When you are done, download a summary to hand in or keep. It lists each problem you checked, the answers you gave, and whether the first one was right, and it notes when a hint or steps of the worked solution were showing before you answered. A guided problem counts as right the first time only if every step was. These answers are kept apart from your saved place: starting over above does not clear them, and clearing them does not change your place or the running count on each practice card.
Before any test, look at the numbers and summarize them: a center, a spread, and anything unusual. The mean and standard deviation use every value, which makes them the basis of most of what follows, but a single extreme value can pull them a long way. The median and the interquartile range are resistant to such outliers, which is why they are reported for skewed data such as incomes, survival times, and bacterial counts. This tab covers the mean, median, and mode, the standard deviation and variance, quartiles and the 1.5 × IQR rule for outliers, and the mean of data given as a frequency table.
In biology, variation is not a nuisance to be averaged away. Measure the beaks of a hundred finches and no two are the same, and that variation among individuals is the raw material of natural selection. A standard deviation is a measure of it: roughly, how far a typical individual is from the mean.
Probability is the language that statistics is written in. This tab covers the basic rules (the complement, “or” for events that cannot happen together, and “and” for independent events), conditional probability read from a two-way table, and the binomial formula for the number of successes in a fixed number of independent trials.
Genetics is probability. Each child of two carriers of a recessive allele has a 1/4 chance of inheriting two copies, independently of brothers and sisters, so the chance that exactly one of three children is affected is a binomial probability. Conditional probability is how a medical test is judged: the chance that a test is positive given disease is not the same as the chance of disease given a positive test, and confusing the two is one of the most common errors in reading medical results.
Many measurements, when collected in quantity, pile up in a symmetric bell shape, the normal distribution, which is described completely by its mean and its standard deviation. A z-score says how many standard deviations a value is from the mean, and turns any normal distribution into the standard one, so that one table serves for all of them. This tab covers z-scores, the 68–95–99.7 rule, the area under the curve to the left of, to the right of, or between values, and working backward from a percentile to a value.
Heights, birth weights, blood pressures, and the beak depths of finches are roughly normal; counts of rare events and many concentrations are not, and are often skewed to the right. Reference ranges for clinical tests are often the middle 95% of healthy people, which is why about one healthy person in twenty falls outside the range on any one test. For every problem here, sketch the curve and shade the area you want before you calculate: the sketch catches most wrong-side errors.
A sample mean is itself a random quantity: take another sample and you get another mean. The standard error, the standard deviation divided by the square root of the sample size, says how much sample means vary from sample to sample. Means vary less than individuals do, and the larger the sample, the less they vary. The central limit theorem adds that sample means are close to normally distributed even when the individuals are not, provided the sample is large enough. This tab covers the standard error and probabilities for a sample mean.
The standard deviation and the standard error answer different questions. The standard deviation describes how much individual mice differ; the standard error describes how precisely the mean of your mice estimates the mean of all such mice. Error bars in biology papers are sometimes one and sometimes the other, and a figure legend should always say which.
An estimate without a measure of its uncertainty is only half an answer. A confidence interval is the estimate plus or minus a margin of error, where the margin is a critical value (from the t table for a mean, the z table for a proportion) times the standard error. This tab covers intervals for a mean and for a proportion, the sample size needed for a given margin of error, and what a 95% confidence level does and does not mean.
The interpretation trips up nearly everyone at first. The 95% describes the method: of all the intervals made this way from different samples, about 95% would contain the true value. Any one interval either contains it or does not. And wider is not worse: a wide interval honestly reports a small or variable sample.
A hypothesis test asks whether a result could easily be due to chance alone. The null hypothesis says that it could: no difference, no effect. The test statistic measures how far the data are from what the null hypothesis predicts, in standard errors, and the p-value is the probability of a result at least that far out if the null hypothesis were true. A small p-value, below the significance level α chosen in advance, leads to rejecting the null hypothesis. This tab covers writing hypotheses, the one-sample t statistic, p-values from z, decisions and conclusions in words, Type I and Type II errors, and the one-proportion z test.
Two warnings that matter in biology. Failing to reject the null hypothesis is not evidence that it is true; a small experiment can easily miss a real effect. And statistically significant does not mean large or important: with enough data, a difference too small to matter biologically can still be significant.
Most experiments compare: treated with control, mutant with wild type, before with after. When the two groups are separate individuals, the two-sample t test compares their means. When the measurements come in pairs, the same patients before and after a drug, or the left and right leaves of the same plants, the paired t test works on the differences within each pair, which removes the variation from one pair to the next and usually makes the test far more sensitive. This tab covers both t statistics and choosing the right test for a study.
Choosing the test comes first, and it depends on the kind of data and the design, not on the numbers. More than two groups calls for analysis of variance (ANOVA) rather than a series of t tests; counts in categories call for chi-square; and two measured variables on the same individuals call for correlation and regression.
When the data are counts in categories rather than measurements, the chi-square statistic compares the counts observed with the counts expected: for each category, (observed − expected)2 ÷ expected, added up. The goodness-of-fit test compares one set of counts with a ratio set in advance; the test of independence asks whether two categorical variables, recorded on the same individuals, are related. This tab covers both, including expected counts, degrees of freedom, and the decision.
The goodness-of-fit test is the genetics student’s test: does a cross fit 3:1, or 9:3:3:1, or 1:2:1? A good fit does not prove the genetic hypothesis, but a poor one says something else is going on, such as linked genes, reduced survival of one genotype, or a mistake in scoring. Use the counts themselves, never percentages, and check that every expected count is at least 5.
Two measurements made on each individual, such as body mass and metabolic rate, or temperature and the chirping rate of crickets, can be related. The correlation coefficient r measures how closely the points fall along a straight line, from −1 to +1; the regression line is that straight line, used to predict one variable from the other; and r2 is the fraction of the variation in the response that the line accounts for. This tab covers r, the regression line from summary statistics, predictions and residuals, and r2.
A standard curve, the line that turns a spectrophotometer reading into a protein or DNA concentration, is a regression line, and so is every “line of best fit” in a laboratory report. Two cautions apply to all of them: a correlation, however strong, does not by itself show that one variable causes the other, and a prediction beyond the range of the data, an extrapolation, can be far off.
A whole test, from the question to the conclusion, is where the pieces on the other tabs come together, and where they most often go wrong. The problems on this tab are guided. Each step asks you a question before it shows you anything: which hypotheses, which test, what the statistic is, how it compares with the critical value, and what the result means for the biology. A wrong choice gets an explanation of why it is wrong, not just a red mark.
If the harder part is working out what a problem is asking in the first place, the companion page Overcoming Biology Anxiety is about exactly that step.
Type or paste your own numbers to get the usual summaries and tests. Numbers can be separated by commas, spaces, or new lines, so a column copied from a spreadsheet works. Nothing you type here leaves your browser.
A calculator answers the question you give it, whether or not it is the right question. Look at your data first, choose the test from the design of the study (the Comparing Two Groups tab has practice at that), and report what the result means for the biology, not just the p-value. p-values are two-sided unless a box says otherwise.