This refresher introduces the questions behind common statistical terms. Allow about 35–45 minutes, including the examples. You do not need statistical software. All numbers below are invented for teaching.
Probability: begin with the denominator
A probability describes how likely an event is under a specified model, on a scale from 0 to 1. A proportion describes what happened in an observed sample. If 12 of 100 samples pass a laboratory check, the observed proportion is 0.12; it is an estimate of an underlying probability, not a guarantee for the next batch.
Always ask: “Among which samples?” The probability of finding a variant among people with a condition differs from the probability of having that condition among people with the variant. Reversing the conditioning changes the question.
For independent events, multiply probabilities: two independent events each with probability 0.5 both occur with probability 0.25. Independence is an assumption to justify. Measurements from relatives, repeated measurements from one person and technical replicates may be related.
Distributions describe patterns of variation
A distribution describes possible values and how frequently, or with what probability, they occur. A histogram displays the distribution of observed numerical values. Inspect its shape, spread, gaps and unusual observations before choosing a statistical method.
A normal distribution is symmetric and bell-shaped. Not all biological measurements follow it: counts are discrete, proportions are bounded, and concentrations may be strongly skewed. A binomial model describes the number of successes in a fixed number of independent trials with the same success probability. Its assumptions matter as much as its formula. NIST: binomial distribution.
The mean adds the values and divides by their number. The median is the middle ordered value, averaging the middle pair when necessary. The median is less affected by extreme observations. Neither summary replaces looking at the data. NIST: measures of location.
Worked example 1: what does “typical” mean?
Five fictional DNA preparations have concentrations of 8, 9, 10, 11 and 42 ng/µL. Their mean is (8 + 9 + 10 + 11 + 42) ÷ 5 = 16 ng/µL. Their median is 10 ng/µL. Four observations lie well below the mean because the fifth pulls it upwards.
The sample standard deviation (SD) measures spread around the sample mean, in the original units. Here it is √[(64 + 49 + 36 + 25 + 676) ÷ 4] ≈ 14.6 ng/µL. The denominator is n − 1 for this sample SD. NIST: measures of scale.
Self-check: should you delete the value 42?
No. Check the laboratory record, units and measurement process. It might be a genuine observation or a documented error. A value should not be removed merely because it changes the answer; explain any exclusion and consider whether conclusions depend on it.
Spread, precision and effect size
SD describes variation between observations. The standard error (SE) describes uncertainty in an estimated quantity across repeated samples. For a mean from independent, identically distributed observations with finite variance, the estimated SE is SD ÷ √n. Additional independent observations can improve precision without making individuals less variable.
A confidence interval expresses uncertainty using a specified procedure. Under the model assumptions, a method with 95% coverage would produce intervals containing the true parameter in about 95% of repeated samples. A particular frequentist interval is not a statement that the fixed parameter has a 95% probability of lying inside it. NIST: confidence limits.
Report the estimated effect size too: for example, a mean difference, a risk difference or a ratio. Include its units, comparison group and uncertainty. A precise but tiny difference can have limited practical importance. A wide interval may leave several materially different conclusions plausible.
Worked example 2: relative and absolute differences
In a fictional laboratory comparison, 20 of 200 preparations fail with method A and 10 of 200 fail with method B.
| Quantity | Calculation | Result |
|---|---|---|
| Failure proportion with A | 20 ÷ 200 | 10% |
| Failure proportion with B | 10 ÷ 200 | 5% |
| Difference, B minus A | 5% − 10% | −5 percentage points |
| Ratio, B divided by A | 0.05 ÷ 0.10 | 0.5 |
“Half the failure proportion” and “five percentage points lower” describe the same observations. They do not establish that method B caused the improvement. We also need uncertainty estimates and information about how samples were assigned.
Self-check: what if A was used on older, degraded samples?
Sample condition could confound the comparison: it is related to method assignment and could affect failure. A fairer design might randomly allocate comparable samples to methods. More observations alone would not repair the systematic imbalance.
Correlation and causation
Correlation summarises association; it does not identify its cause. Plot the observations to check for clusters, outliers and curved relationships. A strong relationship can arise through a shared cause, selection or measurement artefacts. A correlation near zero can conceal a curved relationship. The study design and biological context help distinguish explanations. NIST: scatter plots.
P-values and multiple testing
A p-value asks how unusual the observed result, or a more extreme one according to the chosen test, would be under a specified null model and its assumptions. It does not give the probability that the null hypothesis is true, measure effect size or prove biological importance. A large p-value does not demonstrate equivalence. American Statistical Association: p-value principles.
Testing many hypotheses creates many opportunities for false positives. If 1,000 null hypotheses are true and each valid test has a 5% false-positive probability, the expected number of false positives is 50. Independence is not required for this expectation. Bonferroni uses a threshold of α divided by the number of tests; at α = 0.05, that is 0.00005 for 1,000 tests. This controls the probability of at least one false positive across that defined family, subject to valid individual tests. NIST: Bonferroni's method.
False discovery rate (FDR) methods address a different aim: controlling the expected proportion of false positives among discoveries, counting this proportion as zero when there are none. Benjamini–Hochberg is a common approach under its dependence assumptions. FDR control does not assign a probability of being wrong to each individual finding. R documentation: multiple-comparison methods.
Self-check: is p = 0.03 automatically persuasive in a genome-wide study?
No. Ask how many hypotheses were examined, which analysis was planned, how multiple testing was handled, how large and precise the effect is, and whether independent evidence supports it.
Updated September 2026.