8. Sampling and Standard Error

The professor begins by discussing the importance of sampling in inferential statistics, emphasizing the use of random samples to make inferences about populations. The lecture introduces the concept of Monte Carlo simulation for estimating the value of pi and highlights the empirical rule for confidence intervals. The discussion then shifts to real-world sampling, such as political polls, and the difference between simulation-based sampling and actual data collection. Probability sampling is introduced, with a focus on simple random sampling to avoid bias.

The lecture delves into the specifics of simple random sampling, where every member of the population has an equal chance of being selected. It contrasts this with stratified sampling, which involves dividing the population into subgroups and sampling proportionally from each. The professor uses the example of surveying MIT students to illustrate the potential pitfalls of random sampling and the benefits of stratified sampling in ensuring representative subgroups.

The professor presents a practical example using temperature data from the US Centers for Environmental Information, covering a span of 55 years across 21 American cities. The dataset, comprising 422,000 examples, is used to demonstrate the process of data analysis in data science. The professor introduces numpy.std for standard deviation and random.sample for obtaining simple random samples, and discusses the concepts of sampling with and without replacement.

The lecture continues with the visualization of temperature data through histograms and the examination of its distribution. The professor discusses the difference between the population mean and sample mean, and the significance of standard deviation in understanding sample variability. The central limit theorem is introduced to explain the distribution of sample means and its approximation to a normal distribution.

The professor explores the concept of confidence intervals, demonstrating how they can be calculated using the standard deviation of sample means. The impact of increasing the number of samples and the size of each sample on the width of the confidence interval is discussed. The central limit theorem is again highlighted as the reason for the sample means' distribution's closeness to a normal distribution.

The lecture introduces error bars as a method for visualizing the variability of data and confidence intervals. Using an example of pulse rate versus exercise frequency, the professor explains how error bars can indicate trends and the significance of overlapping confidence intervals. The concept is then applied to the temperature dataset, demonstrating how error bars can provide a visual representation of the data's confidence intervals.

The professor revisits the central limit theorem, focusing on its third aspect related to the variance of sample means. The standard error of the mean is introduced, and its formula is explained. The lecture demonstrates how the standard error can be used to estimate the standard deviation of the population, given a single sample, and the importance of sample size in this estimation.

The lecture addresses the challenge of using the sample standard deviation as an estimator for the population standard deviation, given that the population standard deviation is typically unknown. The professor discusses the surprising accuracy of this approach and explores the relationship between sample size and the accuracy of the sample standard deviation as an estimator.

The professor investigates how different population distributions and sizes affect the accuracy of using the sample standard deviation as an estimator. Through experiments on uniform, normal, and exponential distributions, the lecture reveals that skewness significantly impacts the sample size needed for accurate estimation, while population size surprisingly does not.

The lecture concludes with a discussion on the practical application of sampling in political polls, emphasizing the importance of choosing an appropriate sample size based on the estimated skew of the population. The process of selecting a random sample, computing mean and standard deviation, estimating standard error, and generating confidence intervals is summarized, highlighting the role of the central limit theorem in these calculations.

In the final part of the lecture, the professor reflects on the importance of understanding standard error in statistical analysis. The concept is demonstrated through an experiment using temperature data to determine if 200 samples are sufficient for a representative estimate. The experiment results support the theory, showing that with the right sample size, accurate statistical inferences can be made.