Unbiased Estimator of the Population Median

Alejandro took a random sample of five ping-pong balls from a drum containing balls numbered from 0 to 32, calculated their median, replaced them, and repeated the process for 50 trials. Based on his results, does the sample median appear to be a biased or unbiased estimator of the population median?

In this problem, Alejandro is conducting an experiment to understand if the sample median is a biased or unbiased estimator of the population median. This is a critical concept in statistics as it relates to the accuracy and reliability of statistical estimates drawn from sample data. The question focuses on evaluating whether the repeated calculation of the sample median from numerous trials provides close estimations to the actual median of the entire population, thereby being considered unbiased.

In statistical terminology, an estimator is said to be unbiased if its expected value equals the true parameter it estimates. Here, Alejandro's process involves repeatedly taking random samples and calculating their median, which is just one type of statistical estimator. By replacing each sample and repeating 50 trials, he ensures variability in the samples, which aids in reducing bias. The larger the number of trials, the more reliable and consistent the assessment of bias becomes, due to the Law of Large Numbers.

Understanding if the sample median is an unbiased estimator involves reviewing the central tendency of the sample medians as compared to the population median. If on average, the sample medians are equal to the population median, then the sample median is an unbiased estimator. This concept is particularly useful when analyzing small samples from a larger population, as many real-world situations require making inferences from limited data. This problem therefore provides an insight into the practicality of point estimation and sampling, core essentials in understanding statistical inference.