Table of Contents
Introduction to Biostatistics for USMLE
Biostatistics gives you the language and tools to interpret medical research, assess diagnostic tests, and answer many data based questions on the USMLE. At this level, your goal is not to become a statistician, but to recognize common measures, understand what they mean clinically, and avoid typical traps in exam questions.
In this chapter you will focus on core ideas that repeatedly appear in questions about clinical studies, screening tests, and evidence based medicine. Concepts that belong to separate topics like epidemiologic study designs or ethics of research will be covered elsewhere, so here they are only touched when needed for context.
Populations, Samples, and Types of Data
In biostatistics, we almost never measure every person in a population. Instead, we collect a sample and then use it to make inferences about the larger population.
A population is the entire group you are interested in, for example all adults with hypertension in the United States. A sample is the subset actually studied. The key idea is that good samples are selected in a way that they represent the population fairly. Problems in sampling strategies are mostly discussed in epidemiology, but you must remember that any statistical result is only as trustworthy as the sample it came from.
Medical data come in several types. For USMLE, it helps to distinguish these because they determine which statistical tests are appropriate.
A useful classification is:
| Type of data | Description | Example |
|---|---|---|
| Nominal | Categories without order | Blood type A, B, AB, O; sex; presence of disease |
| Ordinal | Categories with a natural order | Cancer stage I, II, III, IV; pain scale mild to severe |
| Discrete numerical | Countable numbers | Number of asthma attacks per year |
| Continuous | Any value in a range | Blood pressure, serum sodium, height |
On exams, clinical measurements like blood pressure and lab values are usually treated as continuous variables, even though they are recorded with finite precision. Categorical outcomes, such as alive or dead or improved or not improved, are nominal.
Measures of Central Tendency
To summarize a group of numbers, you often describe a "typical" value. Three common measures of central tendency are mean, median, and mode. They sound similar but behave differently in the presence of outliers.
The mean is the arithmetic average. If there are $n$ values $x_1, x_2, \dots, x_n$, then
$$\text{mean} = \bar{x} = \frac{x_1 + x_2 + \dots + x_n}{n}.$$
The median is the middle value when numbers are ordered. If there is an odd number of observations, the median is the exact middle. If there is an even number, it is the average of the two middle values. The mode is the most frequently occurring value.
Clinical data are often skewed by a few extreme values. Income, hospital length of stay, or very high lab values can pull the mean away from the bulk of the data. In positively skewed distributions, a few very large values stretch the right tail. In that case the mean is greater than the median, which is greater than the mode. In negatively skewed distributions, extreme low values pull the left tail, and the mean is less than the median, which is less than the mode.
USMLE questions frequently test which measure is most appropriate to describe skewed data. In skewed distributions, the median is generally preferred because it is less influenced by outliers. When the distribution is symmetric, such as an ideal normal distribution, the mean, median, and mode are approximately equal.
Measures of Variability and the Normal Distribution
Describing only the center of the data is not enough. You also need to quantify how spread out the values are. Several measures exist, but in medical statistics the two most important are variance and standard deviation.
The variance of a sample is roughly the average squared distance of each observation from the mean. If the sample mean is $\bar{x}$ then the sample variance $s^2$ is
$$s^2 = \frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}.$$
The denominator $n - 1$ rather than $n$ is used for an unbiased estimate of the population variance. The standard deviation (SD) is the square root of the variance,
$$s = \sqrt{s^2}.$$
Standard deviation is easier to interpret because it is in the same units as the original measurement, while variance is in squared units.
Many biological measurements follow, or approximately follow, a normal distribution. A normal distribution is the classic bell shaped curve that is symmetric around its mean. For a perfectly normal distribution, there are precise relationships between standard deviation and the proportion of observations.
In a normal distribution:
About $68\%$ of observations lie within $\pm 1$ SD of the mean.
About $95\%$ lie within $\pm 2$ SD of the mean.
About $99.7\%$ lie within $\pm 3$ SD of the mean.
This is often called the 68–95–99.7 rule. On the exam, you may be asked what percentage of patients fall within a certain range, or you may need to recognize whether an observation is very unusual. Values more than 2 SD away from the mean are relatively uncommon and may be considered abnormal in many clinical contexts. For instance, many lab "normal ranges" are constructed as mean ± 2 SD, which captures roughly 95 percent of healthy individuals.
Standard Error and Confidence Intervals
So far the focus has been on describing data in a sample. However, medical research is interested in learning about the population from which the sample is drawn. Because a sample is only an approximation, any estimate like a sample mean has uncertainty. That uncertainty is quantified by the standard error.
The standard error of the mean (SEM) measures how much the sample mean would vary across repeated samples from the same population. It is calculated as
$$\text{SEM} = \frac{s}{\sqrt{n}},$$
where $s$ is the sample standard deviation and $n$ is the sample size.
Standard error of the mean: $\displaystyle \text{SEM} = \frac{s}{\sqrt{n}}$.
As sample size $n$ increases, SEM decreases.
This is a central idea for interpreting studies. Larger samples do not reduce the variability among individual patients, but they give a more precise estimate of the true population mean.
To express both an estimate and its uncertainty, researchers report confidence intervals. A confidence interval (CI) gives a range of plausible values for the true population parameter, such as the mean or a difference between means. The most common on USMLE is the 95 percent confidence interval.
For a sample mean when the population is large and the distribution is approximately normal, a 95 percent confidence interval is often approximated as
$$\bar{x} \pm 1.96 \times \text{SEM}.$$
On exams, this is often simplified to $\bar{x} \pm 2 \times \text{SEM}$. The exact 1.96 multiplier comes from the normal distribution.
Interpretation is subtle and often tested. A 95 percent confidence interval does not mean that there is a 95 percent probability that the true mean lies in that specific interval. Instead, if you repeated the study many times and calculated a 95 percent confidence interval for each study, about 95 percent of those intervals would contain the true population mean.
In hypothesis testing, confidence intervals also relate to statistical significance. For example, when comparing a difference between two means, if the 95 percent confidence interval for the difference does not include 0, that difference is statistically significant at the 0.05 level. When examining a ratio measure, such as a relative risk, if the 95 percent confidence interval does not include 1, the result is statistically significant at the 0.05 level. These points are strongly tested in USMLE style questions about study results.
Hypotheses, Type I Error, and Type II Error
Biostatistics formalizes the idea of testing whether an observed effect could be due to chance. This is done with hypothesis testing. Two hypotheses are defined.
The null hypothesis $H_0$ usually states that there is no effect or no difference. For example, "The new drug has the same mean blood pressure effect as placebo." The alternative hypothesis $H_a$ states that there is some effect or difference. For example, "The new drug changes mean blood pressure compared with placebo."
A statistical test uses data from the sample to decide whether to reject the null hypothesis. However, this decision is made with uncertainty, and mistakes are possible. Two types of errors are important.
A type I error occurs when you reject a true null hypothesis. In simple terms, you conclude there is a difference or effect when in reality there is none. This is a "false positive" result. The probability of making a type I error is called alpha ($\alpha$), which is usually set at 0.05 in medical research. That corresponds to a 5 percent chance of incorrectly rejecting a true null hypothesis.
A type II error occurs when you fail to reject a false null hypothesis. In other words, you conclude there is no difference when a true difference actually exists. This is a "false negative" result. The probability of making a type II error is called beta ($\beta$). The power of a study is defined as $1 - \beta$. Power is the probability that the study will correctly reject a false null hypothesis and detect a true effect.
Type I error: reject $H_0$ when $H_0$ is true. Probability $= \alpha$.
Type II error: fail to reject $H_0$ when $H_0$ is false. Probability $= \beta$.
Power of a study: $\text{Power} = 1 - \beta$.
On the exam, you must be able to match definitions to type I and type II errors, understand how alpha relates to p values, and understand what affects power. Power increases with larger sample size, larger effect size, lower variability in the measurements, and higher alpha. A crucial implication is that small studies have low power and may miss clinically important differences.
p Values and Statistical Significance
The p value is a central quantity in biostatistics and a common source of confusion. It is the probability, assuming the null hypothesis is true, of obtaining a result as extreme as or more extreme than the one observed. If the p value is small, such a result would be unlikely just by chance, so it provides evidence against the null hypothesis.
Researchers choose a significance level alpha ($\alpha$), commonly 0.05. If the calculated p value is less than alpha, we say the result is statistically significant and reject the null hypothesis.
If $p < \alpha$ (often $0.05$), reject the null hypothesis. The result is "statistically significant."
If $p \ge \alpha$, fail to reject the null hypothesis. You cannot conclude there is a difference.
Several misunderstandings are frequently tested. A p value does not tell you the probability that the null hypothesis is true or false. It does not measure the size or clinical importance of the effect. A tiny p value can occur with a very small, clinically unimportant difference if the sample size is large. Likewise, a meaningful clinical effect may fail to reach statistical significance if the study is underpowered.
USMLE questions may give p values alongside confidence intervals. You should be able to recognize their consistency. If a 95 percent confidence interval for a difference excludes 0, the p value for a two sided test at alpha equal to 0.05 is less than 0.05. If the interval includes 0, the p value is greater than or equal to 0.05.
Common Statistical Tests
Specific statistical tests are chosen according to the type of data and the study design. On the exam, you are not usually asked to perform the calculations, but you must recognize which test is appropriate for a given scenario and what its result means.
A useful high level view is to connect tests with the number of groups compared and the type of variable measured.
| Purpose | Data type of outcome | Typical test |
|---|---|---|
| Compare mean of one group to a known value | Continuous | One sample $t$ test |
| Compare means of two independent groups | Continuous | Two sample (unpaired) $t$ test |
| Compare means before and after treatment in same subjects | Continuous | Paired $t$ test |
| Compare means of three or more groups | Continuous | Analysis of variance (ANOVA) |
| Compare proportions between groups | Categorical | Chi square test (or Fisher exact) |
The $t$ test family is used when you compare means. The unpaired $t$ test compares the means of two independent groups, such as treated versus control patients. The paired $t$ test compares two measurements from the same subjects, for example blood pressure before and after a drug in the same group of patients. ANOVA extends this to three or more groups and assesses whether there is any difference in means across groups. If ANOVA detects a difference, follow up methods identify which specific groups differ, but details of those approaches are generally beyond USMLE expectations.
The chi square test (written $\chi^2$) assesses the association between categorical variables. Typical questions involve constructs like a 2 by 2 table of improved or not improved vs treated or not treated. When cell counts are very small, Fisher exact test is used instead of chi square, but the underlying idea is the same.
USMLE questions may briefly mention nonparametric tests such as the Mann–Whitney U test or Wilcoxon signed rank test. These are used when data are not normally distributed or are ordinal, but you do not need to know their formulas. You mainly need to recognize that they are alternatives to parametric tests like the $t$ test or paired $t$ test when assumptions of normality are not satisfied.
Correlation and Regression
Correlation describes the strength and direction of a linear relationship between two quantitative variables. The most common measure is the Pearson correlation coefficient $r$, which ranges from $-1$ to $+1$.
Values near $+1$ indicate a strong positive linear relationship, where higher values of one variable tend to be associated with higher values of the other. Values near $-1$ indicate a strong negative linear relationship, where higher values of one variable are associated with lower values of the other. Values near 0 indicate little or no linear relationship.
On the exam, several points about correlation are key. First, correlation does not imply causation. A high correlation between two variables does not prove that one causes the other; both might be affected by a third factor. Second, correlation assesses linear relationships; variables can have strong nonlinear relationships with low Pearson correlation.
Regression goes beyond correlation by modeling how one variable, the dependent or outcome variable, changes as a function of one or more independent or predictor variables. In simple linear regression, the relationship is modeled by a line
$$\hat{y} = \beta_0 + \beta_1 x,$$
where $\hat{y}$ is the predicted value of the outcome, $x$ is the predictor variable, $\beta_0$ is the intercept, and $\beta_1$ is the slope. The slope tells you how much the outcome changes on average for each one unit change in the predictor.
Multiple regression includes more than one independent variable and allows adjustment for potential confounders. Logistic regression is commonly used when the outcome is binary, like disease present or absent. It models the log odds of the outcome and yields odds ratios. While detailed formulas are beyond the scope of USMLE preparation at this level, you must recognize that logistic regression is appropriate for binary outcomes and that it provides adjusted estimates for the effect of predictors.
Interpreting Diagnostic Test Statistics
Diagnostic tests and screening tools are evaluated with specific biostatistical measures, and these measures appear very frequently on USMLE. Although epidemiology covers study designs for evaluating tests, here the focus is on the actual statistical quantities and how to calculate and interpret them.
Any diagnostic test can be summarized in a 2 by 2 table that cross classifies test result (positive or negative) with true disease status (present or absent).
| Disease present | Disease absent | |
|---|---|---|
| Test positive | True positive | False positive |
| Test negative | False negative | True negative |
From this table you derive several key measures.
Sensitivity is the proportion of people with the disease who test positive. It answers the question: "If a patient has the disease, how likely is the test to be positive?" If TP is the number of true positives and FN is the number of false negatives, then
$$\text{Sensitivity} = \frac{\text{TP}}{\text{TP} + \text{FN}}.$$
Specificity is the proportion of people without the disease who test negative. It answers: "If a patient does not have the disease, how likely is the test to be negative?" If TN is true negatives and FP is false positives, then
$$\text{Specificity} = \frac{\text{TN}}{\text{TN} + \text{FP}}.$$
Positive predictive value (PPV) is the probability that a patient who tests positive actually has the disease. It is defined as
$$\text{PPV} = \frac{\text{TP}}{\text{TP} + \text{FP}}.$$
Negative predictive value (NPV) is the probability that a patient who tests negative truly does not have the disease,
$$\text{NPV} = \frac{\text{TN}}{\text{TN} + \text{FN}}.$$
Sensitivity and specificity are properties of the test. They do not change with disease prevalence.
PPV and NPV depend strongly on disease prevalence in the tested population.
USMLE questions often ask how PPV and NPV change when disease prevalence changes. When prevalence increases, PPV increases and NPV decreases. When prevalence decreases, PPV decreases and NPV increases. Sensitivity and specificity remain constant because they are intrinsic test characteristics.
It is also useful to know heuristics. A highly sensitive test with a negative result helps you rule out a disease. Similarly, a highly specific test with a positive result helps you rule in a disease. The exact mnemonics and applications belong in clinical testing strategy, but the definitions are biostatistical.
Beyond these basic measures, likelihood ratios compress information about sensitivity and specificity into single numbers that can update disease probability. They are less commonly tested but useful.
The positive likelihood ratio (LR+) is
$$\text{LR}^+ = \frac{\text{Sensitivity}}{1 - \text{Specificity}},$$
and the negative likelihood ratio (LR−) is
$$\text{LR}^- = \frac{1 - \text{Sensitivity}}{\text{Specificity}}.$$
LR+ greater than 10 and LR− less than 0.1 are generally considered strong. Interpretation and the use of pretest and posttest probabilities draw on clinical reasoning and epidemiology, but recognizing these formulas and their qualitative meaning may be useful.
Measures of Association: Risk, Odds, and Ratios
Biostatistics often examines associations between an exposure and an outcome. You will encounter several related measures, especially in questions about clinical trials and observational studies.
Risk is the probability of an outcome in a defined group over a specified time. For example, among 100 smokers followed for 10 years, if 20 develop lung cancer, the risk is 20 out of 100 or 0.20 over ten years.
Relative risk (RR) compares the risk in an exposed group with the risk in an unexposed group. If $R_e$ is risk in the exposed group and $R_u$ is risk in the unexposed group, then
$$\text{RR} = \frac{R_e}{R_u}.$$
Relative risk equal to 1 means no association. Relative risk greater than 1 suggests that exposure is associated with increased risk. Relative risk less than 1 suggests that exposure is protective.
Odds are related but not identical to risk. Odds are the ratio of the probability that an event occurs to the probability that it does not occur. If the risk of an event is $p$, the odds are $\dfrac{p}{1 - p}$. In a group of 100 patients where 20 have a disease and 80 do not, the risk is 0.20 and the odds are $20 / 80 = 0.25$.
Odds ratio (OR) compares the odds of disease in exposed and unexposed groups. It is widely used in case control studies and in logistic regression. Like relative risk, an odds ratio of 1 indicates no association, greater than 1 indicates increased odds, and less than 1 indicates decreased odds.
USMLE questions may ask you to interpret confidence intervals for relative risk or odds ratio. The key rule is that if the 95 percent confidence interval for RR or OR includes 1, there is no statistically significant association at alpha equal to 0.05. If it does not include 1, the association is statistically significant.
Reliability and Validity of Measurements
In clinical research and testing, you often assess not only how accurate a measure is, but also how consistent it is. Two related concepts summarize this: reliability and validity.
Reliability refers to the consistency or reproducibility of a measurement. A test is reliable if it yields similar results when repeated under similar conditions. Types of reliability include test retest reliability, where the same test is given again to the same subjects; interrater reliability, where different observers measure the same subjects; and internal consistency, where different items in a test correlate with each other. Histology slide grading by pathologists is a common scenario where interrater reliability matters.
Validity refers to whether a test measures what it is supposed to measure. A valid blood pressure monitor gives readings that closely reflect the true blood pressure. A cognitive test that predicts job performance has higher predictive validity than one that does not.
A crucial distinction is that a test can be reliable without being valid. For example, a miscalibrated scale may give the same incorrect weight every time. Its reliability is high but its validity is low. For a test to be useful clinically, it should be both reliable and valid.
These concepts also appear in the context of questionnaires and rating scales in psychiatry and psychology, where internal consistency and construct validity are evaluated statistically. While details of psychometrics are beyond the scope here, USMLE questions commonly examine whether you can distinguish reliability from validity conceptually.
Summary
Biostatistics provides the technical framework that supports evidence based medicine and clinical decision making. For USMLE preparation, the focus is on understanding and correctly interpreting standard measures such as mean and standard deviation, confidence intervals, p values, diagnostic test characteristics, and basic measures of association. You do not have to perform complex calculations, but you must be comfortable with the meanings of key formulas, recognize appropriate statistical tests for simple study descriptions, and avoid common interpretive errors. These skills allow you to read tables and figures in questions quickly, identify which results are statistically and clinically meaningful, and apply quantitative evidence to patient scenarios.