Statistics and Probability - Data collection and sampling techniques

Population and Sample: A population represents the entire group being studied (e.g., all students in a school), while a sample is a subset of that population (e.g., 50 students). Visually, imagine a large circle representing the population, with a smaller circle inside it representing the sample. The goal is for the sample to accurately reflect the characteristics of the larger circle.

Discrete vs. Continuous Data: Numerical data is classified into two types. Discrete data consists of distinct, countable values (e.g., the number of siblings, $x \in \{0, 1, 2, ...\}$). Visually, this is represented by isolated dots on a number line. Continuous data can take any value within a range (e.g., height, $150 \le h \le 190$ cm). Visually, this is represented by a solid, unbroken line or interval.

Random Sampling: In a simple random sample, every member of the population has an equal chance of being selected. This is often done using a random number generator or by pulling names from a hat. This method minimizes bias and ensures that the sample is not influenced by the researcher's preference.

Systematic Sampling: This technique involves selecting members at regular intervals from an ordered list. For example, selecting every $k^{th}$ person. Visually, if you have a line of individuals, you might pick the $3^{rd}$, $6^{th}$, $9^{th}$, and so on. The interval is calculated by dividing the population size by the desired sample size.

Stratified Sampling: The population is divided into distinct subgroups, or 'strata', based on shared characteristics (like age or gender). A random sample is then taken from each stratum in proportion to its size in the population. Visually, imagine a bar chart where each bar represents a group; the sample takes a 'slice' from each bar relative to its height.

Convenience and Quota Sampling: Convenience sampling involves choosing individuals who are easiest to reach (e.g., surveying friends). Quota sampling involves selecting a specific number of people from different groups until a target is met, but unlike stratified sampling, the selection is not random. These methods are often faster but more prone to bias.

Sampling Bias: Bias occurs when the sample does not accurately represent the population, leading to skewed results. Visually, this looks like a target where all the arrows are clustered in one corner far from the center. Common causes include small sample sizes, non-random selection, or excluding certain groups within the population.