Section 15.7: Comparing Populations

Property

A random sample is a subset of individuals (a sample) chosen from a larger set (a population). Each individual is chosen randomly and entirely by chance, such that each individual has the same probability of being chosen at any stage during the sampling process, and each subset of $k$ individuals has the same probability of being chosen for the sample as any other subset of $k$ individuals. A simple random sample is an unbiased surveying technique.

Examples

• To find the favorite sport of 500 students, a researcher puts all their names in a bowl and draws 50 names to survey.
• A quality inspector assigns a number to every one of the 1,000 toys produced and uses a random number generator to select 100 toys to test.
• To estimate the average number of pages in the library's books, a librarian randomly selects 30 books from the computer catalog to count their pages.

Explanation

Think of this as the fairest way to pick a small group to represent a big one. Everyone has an equal chance of being chosen, like drawing names from a hat. This helps make sure your sample truly reflects the whole population.

Property

When comparing populations, we use measures of center to summarize and contrast different groups. The mean and median each provide a single-number summary of a population's data. The choice between mean and median depends on the distribution shape:

Skewed Left: Extreme values pull the mean to the left of the median. Median is a better measure of center for comparison.

Property

When comparing populations, the choice of measure of center depends on the distribution shape. For symmetric distributions, the mean and median are approximately equal, so either can be used effectively. For skewed distributions, the median is typically preferred because it is less affected by outliers and extreme values, providing a better representation of the typical value in each population.

Examples