An estimate is a random variable (the key idea of inference)
This may be the most important lesson of the whole course — keep it in mind through all the following ones. All we have is a sample, and from it we ESTIMATE something: a mean, a share, a difference. The key insight: any such estimate is itself a random variable. Take another sample — you get another number. And this "wandering" has its own distribution, on which all of statistical inference rests.
Whichever estimate you pick — every sample has its own, i.e. it is itself random. Its spread falls as n grows. That is the sampling distribution — the object the CLT, intervals and hypothesis testing all stand on.
Let's separate two things that are easy to confuse. The truth (the estimand) is a number about the whole population: the true mean μ, the true share p. It is single and fixed, but unknown to us. The estimate (the estimator) is what we compute from a sample: x̄, p̂. It is known to us, but changes from sample to sample. The yellow line is the truth (one); the teal histogram below is how the estimate dances.
This frame instantly changes how you read any number. "Conversion is 12%" is not the truth but one realization of a random estimate; the right question is "how wide is its sampling distribution?". Intervals and significance testing grow naturally from here.
And this is also why sample size matters so much: it directly squeezes the estimate's sampling distribution, i.e. reduces its random wandering around the truth.
Polls, ratings, quality measurements, product metrics — everywhere you hold only a sample-based estimate. A professional automatically keeps its sampling distribution in mind and never confuses "the number we got" with "the truth".
That is why serious reports show not a bare point but an estimate with an interval — an honest admission that it is a random variable, not a verdict.
Definitions
The formula for the standard error, σ/√n (meet it properly in the next lesson), assumes independent observations. If rows are clustered (several visits by one person, cities, ad campaigns), the effective n is smaller than the row count — the real error is larger than the formula says.
The standard error describes only random scatter. Systematic error — an unrepresentative sample, a biased measurement — is not cured by √n: as n grows you converge ever more confidently to the wrong answer.
Deep dive: the math and the mechanism (optional)
Формально оценка — это функция от случайной выборки, поэтому она сама случайная величина. У неё есть математическое ожидание (куда она целится в среднем) и дисперсия (как сильно гуляет). Несмещённость — это «ожидание оценки = оцениваемое»; эффективность — «при той же выборке дисперсия меньше».
Дальше мы увидим, что для среднего это распределение имеет ширину σ/√n и стремится к нормальному (ЦПТ) — но сама идея «оценка случайна и имеет распределение» верна для любой статистики, даже когда формулы нет (тогда выручает бутстреп).