Statistics

The mode is the most frequent value. The median is the middle value of the sorted data — the 50th percentile, splitting it into two equal halves. The arithmetic mean is the sum of all values divided by their count, the balance point of the data.

NULL means the value is unknown or absent, NaN is an undefined result like 0/0, and zero is a real measured quantity. Treating them alike corrupts an average — counting NULLs as zeros drags the mean down and inflates the denominator, and a stray NaN poisons the sum.

A percentile is the value below which that share of observations falls: p95 is the latency 95% of requests beat. The mean hides the tail — slow calls inflate it and a bimodal mix averages to a value no request sees. p50 is the typical case, p95 the near-worst users feel.

A representative sample gives every member of the target population a known, non-zero chance of selection, so its composition mirrors the whole. A convenience sample of your most active users over-weights heavy usage, so metrics skew high and no extra data removes the bias.

The standard deviation measures spread in the raw data — a population property that does not shrink with n. The standard error is the SD of a statistic's sampling distribution, SD/√n for the mean, so it falls as n grows. The SD describes individuals; the SE, precision.

Variance is the mean squared deviation from the mean, so it lives in squared units. The standard deviation is its square root, back in the data's units and comparable to the mean. You report the SD because '±5 kg' is interpretable, while '25 kg²' is not.

First decide whether it is real or a data-entry error. A genuine ₽10M order is signal — keep it, and report the median beside the mean so one whale does not distort the typical value. Trimming or winsorizing biases real totals low; exclude only true errors.

The central limit theorem says the distribution of the sample mean of independent identically-distributed values with finite variance approaches a normal distribution as the sample size grows, whatever the original shape. It justifies normal-based confidence intervals and z/t-tests on means, and underpins A/B significance testing.

For a right-skewed distribution (long right tail, like incomes), the order is mode < median < mean. The long tail of large values pulls the mean up the most, the median less, and the mode sits at the dense low end. Left-skew reverses the order.

Common ones — normal (measurement errors), uniform (equally likely outcomes), Bernoulli/binomial (success counts, like conversions), Poisson (event counts per interval, like arrivals), exponential (waiting times), and log-normal (quantities like income). Each is discrete or continuous.

Four inputs set it: the metric's baseline variance, the minimum detectable effect (the 2% lift), the significance level α, and the power 1−β. n grows with variance and (z_α+z_β)² and falls with the square of the effect — so halving the lift quadruples n.

The law of large numbers says the sample mean converges to the true mean as n grows — it is about where the average lands. The central limit theorem says the distribution of that sample mean approaches a normal shape whatever the source — it is about the error around it.

Yes. The platform-wide average can fall though each category's average rose, because the category mix shifted toward the cheaper one. This is Simpson's paradox — a weighted aggregate depends on the weights, so it can't be judged from per-category trends alone.

The confounder is hot weather: summer heat drives both ice-cream sales and swimming, so the two move together, no causal link. Control for it by conditioning — stratify or regress on temperature or season. If the association vanishes once weather is held fixed, it was spurious.

Pearson's r measures only the strength of a linear relationship. A U-shaped or non-monotonic bond — like a dose effect that rises then falls — averages to r ≈ 0 even though the two are tightly linked. Plot the scatter, and use Spearman for non-linear dependence.

MCAR means missingness is unrelated to anything, so dropping rows only costs power. MAR means it depends on observed variables — safe to drop once you condition on them, biased otherwise. MNAR means it depends on the unseen value itself, so dropping always biases the estimate.

An A/A test has no real effect, so p = 0.036 is expected — at α = 0.05, 5% of clean A/A tests hit significance by chance. The tell is peeking: stopping at the first p < 0.05 before the planned horizon inflates the false-positive rate. A balanced split rules out selection bias.

Resampling uses the sample as a population proxy, so the statistic's spread approximates its sampling distribution — giving a standard error and CI with no formula. It works because the empirical distribution approaches the real one, and fails for tail-dependent statistics like the max and for tiny samples.