Sample (Statistics)
Statistics & ProbabilityA sample is a subset of a population selected for study in order to draw conclusions about the whole population.
Definition
A sample is a smaller group chosen from a larger population to represent it. Instead of studying everyone, we study the sample and use what we learn to make guesses about the whole group.
Example
To find out what snack the whole school prefers, you ask $50$ randomly chosen students. Those $50$ students are your sample.
Key Insight
A good sample is like a taste test: a small spoonful can tell you a lot about the whole pot, as long as you stir it first (mix it well, or randomize it).
Definition
A sample is a subset of a population that is selected and studied in order to make inferences about the population. The quality of a sample depends on how it is selected: a random sample gives every member an equal chance of being chosen and tends to be unbiased.
Example
A school has $1{,}200$ students. A researcher randomly selects $120$ students ($10\%$) and records their daily screen time. The mean screen time of the $120$ students is a sample statistic used to estimate the population mean.
Key Insight
Sample size matters: larger samples give more precise estimates. But a large biased sample can be worse than a small random one.
Definition
A sample $\{x_1, \ldots, x_n\}$ is an independent and identically distributed (i.i.d.) draw from population distribution $F(x; \theta)$. Sample statistics (e.g., $\bar{x}$, $s^2$) are functions of the sample used as estimators of population parameters. Key properties of estimators include unbiasedness, consistency, and efficiency.
Example
The sample mean $\bar{x} = \frac{1}{n}\sum x_i$ is an unbiased estimator of $\mu$: $E[\bar{x}] = \mu$. The sample variance $s^2 = \frac{1}{n-1}\sum (x_i - \bar{x})^2$ uses $n-1$ (Bessel's correction) to remain unbiased for $\sigma^2$.
Key Insight
The central limit theorem guarantees that $\bar{x}$ is approximately normally distributed for large $n$ regardless of the population distribution, enabling confidence intervals and hypothesis tests.