Answer to Question #321580 in Statistics and Probability for Chacha

Question

Explain the usefulness of central limit theorem in solving problems involving sampling

Answer 1

The unexpected appearance of a normal distribution from a population distribution that is skewed (even quite heavily skewed) has some very important applications in statistical practice. Many practices in statistics, such as those involving hypothesis testing or confidence intervals, make some assumptions concerning the population that the data was obtained from. One assumption that is initially made in a statistics course is that the populations that we work with are normally distributed.

The assumption that data is from a normal distribution simplifies matters but seems a little unrealistic. Just a little work with some real-world data shows that outliers, ​skewness, multiple peaks and asymmetry show up quite routinely. We can get around the problem of data from a population that is not normal. The use of an appropriate sample size and the central limit theorem help us to get around the problem of data from populations that are not normal.

Thus, even though we might not know the shape of the distribution where our data comes from, the central limit theorem says that we can treat the sampling distribution as if it were normal. Of course, in order for the conclusions of the theorem to hold, we do need a sample size that is large enough. Exploratory data analysis can help us to determine how large of a sample is necessary for a given situation.

Central limit theorem is useful when analyzing large data sets. For example, investors can use central limit theorem to aggregate individual security performance data and generate distribution of sample means that represent a larger population distribution for security returns over a period of time.