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Sampling and estimation
What is inferential statistics, and how does it differ from descriptive statistics?
Inferential statistics uses a random sample to approximate a population parameter, and the key issue is how well the sample statistic estimates it (NIST e-Handbook). Descriptive statistics only summarises the data in hand, with a mean or a chart. Inference adds a statement of uncertainty, such as an interval or a p-value, that says how far a conclusion can be trusted beyond the sample.
Why it is asked: it checks that you separate what you observed from what you claim. Read the full explanation: statistics interview questions, the complete guide.
What is the difference between a parameter and a statistic?
A parameter is a fixed number that describes a whole population, such as the true mean. A statistic is a number computed from one sample, such as the sample mean, and used to estimate the parameter. The parameter stays put between samples while the statistic changes, which is why a statistic has a distribution and a margin of uncertainty.
Why it is asked: every later answer depends on these two terms being kept apart. Read the full explanation: mean versus median.
What is the central limit theorem?
The central limit theorem says that in a population with a known or unknown distribution, if the sample size is sufficiently large, the distribution of the sample means is approximately normal, with its mean equal to the population mean (OpenStax). Only the average becomes normal; the raw data keep their shape. How large is large depends on the population, so avoid quoting one fixed sample size as a law.
Why it is asked: it is the reason normal-based intervals and tests work on non-normal data. Read the full explanation: the Gaussian distribution.
What is standard error, and how does it differ from standard deviation?
The standard error is the standard deviation of a statistic's sampling distribution (NIST glossary). For a sample mean it equals the population standard deviation divided by the square root of the sample size (OpenStax). Standard deviation describes how individual values spread; standard error describes how the estimate would spread across repeated samples. In OpenStax Example 7.1, a standard deviation of 15 and a sample size of 25 give a standard error of 3 (OpenStax).
Why it is asked: confusing the two is the most common slip in this topic. Read the full explanation: OpenStax, central limit theorem for sample means.
How does sample size affect the standard error?
The standard error shrinks with the square root of the sample size, so four times the data halves it. With a standard deviation of 15 (illustrative), a sample of 25 gives a standard error of 3, a sample of 100 gives 1.5 and a sample of 400 gives 0.75.1 The population standard deviation does not change; only the error of the estimate does, and each halving costs four times the data.
Text description of the diagram
- The standard error shrinks with the square root of the sample size, so four times the data halves it.
- With a standard deviation of 15, a sample of 25 gives a standard error of 3, a sample of 100 gives 1.5 and a sample of 400 gives 0.75.
Why it is asked: it tests whether you can reason about the cost of precision when planning a study. Read the full explanation: A/B testing interview questions.
Confidence intervals
What does a 95% confidence interval mean?
A confidence interval is a range of values likely to contain a population parameter. The 95% describes the procedure: if the same population is sampled many times and an interval is built each time, the intervals would bracket the true parameter in approximately 95% of cases (NIST e-Handbook). One computed interval either contains the parameter or does not. Saying "a 95% probability the true value is inside this interval" is the usual error.
Why it is asked: it separates a memorised phrase from understanding repeated sampling. Read the full explanation: what a t test is and how to run one.
Does a 95% confidence interval contain 95% of the data?
No. A confidence interval gives limits within which we expect a population parameter, such as the mean, to lie, while a tolerance interval gives limits within which we expect a stated proportion of the population to lie (NIST e-Handbook). Individual values spread far wider than the mean is uncertain, and the interval narrows as the sample grows while the data's spread does not.
Why it is asked: it is a classic misreading that interviewers use to probe depth. Read the full explanation: Chebyshev's inequality.
Why is a t interval wider than a z interval?
At the same confidence level, intervals are wider when the standard deviation is estimated from limited data, because that estimate adds uncertainty (NIST e-Handbook). The t distribution has heavier tails to allow for it. With a sample mean of 90, a sample standard deviation of 15 and 25 observations, the z interval is 84.1 to 95.9 and the t interval, with 24 degrees of freedom, is 83.8 to 96.2.1 The gap closes as the sample grows.
Why it is asked: it checks that you know when each is the right tool. Read the full explanation: the t test.
How are confidence intervals and hypothesis tests related?
For every test of a hypothesis there is an equivalent statement about whether the hypothesized value is included in a confidence interval (NIST e-Handbook, PDF copy). A two-sided test rejects a value exactly when the matching interval excludes it. With the t interval above, 83.8 to 96.2, a hypothesized mean of 80 falls outside and would be rejected, while 85 falls inside and would not.1
Why it is asked: it shows you can move between estimation and testing. Read the full explanation: the t test.
Knowing it is half the job. Saying it is the other half.
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Start Free Trial →Hypothesis tests and p-values
What is a p-value?
A p-value is the probability, under a specified statistical model, that a statistical summary of the data would be equal to or more extreme than its observed value (American Statistical Association (ASA) statement, PDF copy). The smaller it is, the greater the incompatibility of the data with the null hypothesis, if the assumptions used to calculate it hold (ASA statement).
Why it is asked: it is the most misused idea in statistics, so a precise definition stands out. Read the full explanation: the ASA statement on p-values.
Is the p-value the probability that the null hypothesis is true?
No. P-values do not measure the probability that the studied hypothesis is true, or the probability that the data were produced by random chance alone (ASA statement). A p-value assumes the model, null hypothesis included, and asks how surprising the data are. The probability of a hypothesis given the data needs a prior, which is the Bayesian route.
Why it is asked: it is the standard trap behind "what does p equal 0.03 mean". Read the full explanation: Bayes theorem interview questions.
What does a p-value of about 0.1 tell you in practice?
Take a test of a null mean of 85 with a sample mean of 90 and a standard error of 3 (illustrative): z is 1.667 and the two-sided p-value is about 0.096.1 That is above the common 0.05 threshold, yet it is not strong evidence of no difference, and the interval 84.1 to 95.9 sits almost entirely above 85. Conclusions should not rest only on whether a p-value passes a threshold (ASA statement).
Why it is asked: it tests judgement beyond "reject or not". Read the full explanation: the ASA statement on p-values.
Does a significant result mean the effect is important?
No. A p-value, or statistical significance, does not measure the size of an effect or the importance of a result (ASA statement, PDF copy). With enough data a tiny, useless difference can give a small p-value. Report the estimated effect and its interval next to the p-value, and ask whether that size matters for the decision.
Why it is asked: it checks whether you connect statistics to a business or product decision. Read the full explanation: A/B testing interview questions.
What does it mean to fail to reject the null hypothesis?
Failing to reject means the data did not give enough evidence against the null hypothesis. It is not quite the same as accepting it (Scribbr). The sample may be too small or too noisy to see a real effect. A careful statement is "we found no evidence of a difference", not "there is no difference".
Why it is asked: it separates careful inference from overclaiming. Read the full explanation: Scribbr, Type I and Type II errors.
Errors and power
What are Type I and Type II errors?
A Type I error means rejecting the null hypothesis when it is actually true, and its probability is the significance level, alpha. A Type II error means not rejecting the null hypothesis when it is actually false, usually with probability beta (Scribbr; NIST Technical Note 2106). In plain terms, a Type I error is a false alarm and a Type II error is a missed effect.
Why it is asked: it frames every testing trade-off. Read the full explanation: A/B testing interview questions.
What is statistical power, and what affects it?
Statistical power, one minus beta, is the probability of rejecting the null hypothesis when there is an actual difference (NIST Technical Note 2106). It rises with a larger sample, a larger true effect and lower variability. A power level of 80% or higher is usually considered acceptable (Scribbr). In the running example (illustrative), a true mean of 90 against a null of 85 gives power of about 0.38 at a sample of 25 and about 0.92 at 100.1
Why it is asked: it shows you can plan a test before running it. Read the full explanation: A/B testing interview questions.
What is the trade-off between Type I and Type II errors?
There is a trade-off between the two error types. A hypothesis test that never rejects the null hypothesis has zero Type I errors but no power to detect any real difference (NIST Technical Note 2106). Lowering alpha therefore raises beta unless the sample grows. Choose alpha by the cost of each error, not by habit.
Why it is asked: it tests whether lowering the threshold is understood as a cost, not a free fix. Read the full explanation: NIST Technical Note 2106.
Follow-up questions
What are the common inferential statistics interview questions?
The seventeen above cover the ones that recur: sampling, the central limit theorem, standard error, confidence intervals, p-values and the two error types with power. For counting and chance questions that sit next to them, see probability interview questions.
How do you explain a p-value to a non-technical person?
Say how surprising the data would be if nothing were going on. A small p-value means the data fit poorly with "nothing is going on", but it is not the chance that "nothing" is true.
When do you use a t test instead of a z test?
Use a t test when the standard deviation is estimated from the sample, which is the usual case, because the estimate widens the interval. The t test page covers the steps.
How are A/B tests related to inferential statistics?
An A/B test is a hypothesis test on a sample of users, so sample size, power and error rates apply directly. See the A/B testing questions. A related caution is that an observed difference alone does not prove a cause; see correlation versus causation.
What is the difference between frequentist and Bayesian inference?
Frequentist methods judge a procedure by its behaviour over repeated samples, as the confidence interval answer does. Bayesian methods combine a prior with the data to give a probability for the parameter itself. For a related estimation contrast, see maximum likelihood estimation (MLE) versus maximum a posteriori (MAP) estimation.
- Inference uses a sample to estimate a population parameter and states the uncertainty.
- Standard error shrinks with the square root of the sample size, so four times the data halves it.
- A confidence level describes the procedure over repeated samples, not one interval.
- A p-value is not the probability that the null hypothesis is true, and it does not measure effect size.
- Lowering alpha raises beta unless the sample grows, and power is the chance of detecting a real effect.
Sources
- 7.1.4. What are confidence intervals? (NIST/SEMATECH e-Handbook of Statistical Methods) — National Institute of Standards and Technology, 26 Oct 2026
- 7.2.6.3. Tolerance intervals for a normal distribution (NIST/SEMATECH e-Handbook) — National Institute of Standards and Technology, 26 Oct 2026
- NIST/SEMATECH e-Handbook glossary — National Institute of Standards and Technology, 26 Oct 2026
- Process or product monitoring and control (PDF copy of the e-Handbook chapter) — National Institute of Standards and Technology, 26 Oct 2026
- The ASA Statement on p-Values: Context, Process, and Purpose — The American Statistician (Wasserstein and Lazar, 2016), 26 Oct 2026
- The ASA Statement on p-Values, PDF copy with commentary — Yale University (hosted copy), 26 Oct 2026
- 7.1 The Central Limit Theorem for Sample Means (Averages), Introductory Statistics 2e — OpenStax, 26 Oct 2026
- Chapter 7 review, Introductory Statistics 2e — OpenStax, 26 Oct 2026
- NIST Technical Note 2106, Comparing Instruments — National Institute of Standards and Technology (Yen and Leber), 26 Oct 2026
- Type I and Type II Errors — Scribbr, 26 Oct 2026
