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A statistics interview tests whether you can connect a calculation to its assumptions and interpretation. These six questions use small worked examples. They focus on estimation and evidence; the separate probability guide covers event rules, and the A/B testing guide covers an experiment from design to decision.
1. When do the mean and median tell different stories?
For values 1, 2, 2, 3, and 12, the mean is 4 and the median is 2. The mean reflects the large value more strongly. That is not evidence that 12 should be deleted. Investigate whether it is an error, a valid extreme, or part of a different population.
from statistics import mean, median, variance
values = [1, 2, 2, 3, 12]
print("mean:", mean(values))
print("median:", median(values))
print("sample variance:", variance([2, 4, 6]))
The sample variance in the last line is 4: squared deviations from the sample mean are 4, 0, and 4, divided by n−1 = 2. Dividing by n instead gives the descriptive population variance of those three values, 8/3. State which quantity you need. Python’s statistics documentation distinguishes these functions.
2. How do standard deviation and standard error differ?
Standard deviation describes variation among observations. Standard error describes the sampling variation of an estimator. For independent, identically distributed observations with finite variance, the sample mean’s standard error is σ/√n when the population standard deviation σ is known, or estimated by s/√n when appropriate.
If σ = 2 and n = 100, the mean’s standard error is 0.2. Repeated measurements from the same user are not automatically 100 independent observations. Clustering or serial dependence changes the uncertainty calculation; increasing event count alone does not justify the independent-observation formula.
3. What does a 95% confidence interval mean?
Assume 100 independent observations from a normal population with known σ = 2 and observed mean 10. A two-sided normal interval is 10 ± z₀.₉₇₅ × 0.2, approximately [9.6080, 10.3920]. This known-variance example is deliberately simplified; an unknown variance generally requires a different procedure, such as a t interval under the corresponding assumptions.
from math import sqrt
from statistics import NormalDist
sample_mean, sigma, n = 10.0, 2.0, 100
critical = NormalDist().inv_cdf(0.975)
margin = critical * sigma / sqrt(n)
print(f"[{sample_mean-margin:.4f}, {sample_mean+margin:.4f}]")
The procedure covers the fixed population mean in 95% of repeated samples under its assumptions. It is not a statement that 95% of individual observations lie in this interval, nor a frequentist posterior probability that this particular fixed parameter lies in the realized interval.
4. What does a p-value establish?
A p-value measures how incompatible the observed statistic is with a specified null model, using the chosen test’s definition of results at least as extreme. It is not the probability that the null hypothesis is true, the probability the result will replicate, or the size of a useful effect.
Report the estimated effect, uncertainty, design, and decision threshold together. A small effect can be statistically detectable in a large sample yet unimportant in practice. A large estimated effect with a wide interval may still be too uncertain to act on. Failure to reject a difference is not proof of equivalence; equivalence requires an appropriate margin and analysis.
5. When is a paired analysis appropriate?
If the same four people have before scores [10, 20, 30, 40] and after scores [12, 21, 33, 42], their differences are [2, 1, 3, 2], with mean 2. A paired analysis works with within-person differences and their dependence structure. Treating the two sets as unrelated discards the pairing.
Pairing does not establish causality in an uncontrolled before/after study. Practice effects, time trends, or concurrent changes could explain improvement. Define the sampling unit and identify what the design can support before choosing a test.
6. What changes when you inspect many outcomes?
If you run 20 independent tests of true null hypotheses at α = 0.05, the probability of at least one false rejection is 1 − 0.95²⁰, approximately 64.15%. The independence assumption is part of that calculation; real outcomes can be correlated.
Predefine the primary question and a suitable multiplicity strategy for confirmatory claims. Label exploratory subgroup findings as exploratory and validate them on new evidence. Similarly, keep model tuning separate from final evaluation: repeatedly using the test set to select a model leaks evaluation information into the selection process.
For each interview answer, state the unit, estimator, assumptions, uncertainty, and practical meaning. A correct formula with the wrong design or denominator is still the wrong analysis.
