Central limit theorem

The central limit theorem is a concept in statistics that states that the sampling distribution of sample means from repeated random samples approaches a normal distribution as the sample size gets larger. In other words, even if data obtained from independent random samples seems skewed to one particular side, when the sample size is increased, the distribution of sample means will move closer to a normal distribution. The central limit theorem is a fundamental pillar of statistics and allows researchers to arrive at conclusions about entire populations by examining data from smaller sample sizes. Statisticians disagree on what constitutes a large enough sample size for the central limit theorem to provide valid results. In general, sample sizes of thirty or more are considered sufficient, although some researchers believe samples should be more than forty or fifty.

Background

Statistics is a mathematical science that collects numerical data samples and analyzes those samples to determine the probability that they represent a larger whole. If a statistician wanted to discover the average height of every person in the United States, it would be a near-impossible task to measure millions of people. Therefore, a researcher would measure a smaller sample size of people chosen at random so as not to unintentionally influence the results. Taking the sum of all the heights in the example and dividing it by the number of people sampled would reveal the statistical mean, or average.

Because the statistician is only measuring a segment of the population, there are several variable factors that need to be taken into consideration. The variance measures the distance each number in a data set is from the mean. Mathematically, variance is determined by taking the distance each point is from the mean and squaring that number, or multiplying the number by itself. The variance is the average of those results. The standard deviation is a measure of the dispersion of the data set from the mean. This means the further the data points are from the mean, the higher the standard deviation will be. Standard deviation is measured as the square root of the variance. For example, if a sample size of four people reveals their heights to be 50 inches, 66 inches, 74 inches, and 45 inches, then the mean would be 58.75 inches. The variance in this case would be 137.69, and the standard deviation would be 11.73.

Normal distribution is a symmetric probability distribution, with most of the points situated around the mean. This can be illustrated in the common bell curve that is a graphic with a rounded peak in the center that tapers away at either end. In a graph representing a normal distribution, the mean is represented by the central peak of the curve, while the standard deviation determines the peak’s height.

Overview

Using height as an example is fairly straightforward, as most people in a population tend to be at or close to average. The central limit theorem comes into play when the data from a sample size does not fit the normal distribution and seems to misrepresent the statistical probability of the data. In an analysis of average height, recording the measurements of eight people may be able to yield reliable data that can lead to an accurate result. If a statistician is trying to measure wealth by recording the incomes of eight people, however, a disparity in one respondent’s income may skew the results. For example, seven of the eight people may earn between $30,000 and $100,000, but if the eighth person is a millionaire, then the statistical mean would calculate to far more than the second-wealthiest respondent. The central limit theorem holds that if the sample size is increased, then the distribution of sample means will move closer to a normal distribution and provide a more accurate depiction of average household wealth. Many statisticians say that a sample size of thirty or more is often used as a practical guideline, although the required sample size depends on the population distribution. In cases where the data points are unusually irregular, a statistician may need to utilize a larger sample size. The central limit theorem is also used in fields such as data science, machine learning, polling, and quality control to analyze large sets of data.

In practice, statisticians often take repeated random samples of the same size from the same population. These individual samples are then averaged together to get a data point. The process is repeated a number of times to arrive at a data set. If, in the wealth example, the sample size is three people, then the incomes of three people would be recorded, averaged to find the mean, and that figure would become a data point. Therefore, if a random sampling of three people asked for their income, and they responded with $21,000, $36,000, and $44,000, then their mean income would be $33,667.

For comparison’s sake, assume the average salary in the United States was $45,000. If the survey was done correctly, the normal distribution should be near this figure. A group of five data points that yielded income numbers of $33,000, $39,000, $44,000, $351,000, and $52,000 would result in a mean value of $103,800, more than double the national average. The numbers are obviously affected by the income in the fourth data point. If the sample size is increased to ten respondents, the likelihood increases that they are more representative of the true average salary in the United States. Assuming the other values stayed the same, if the fourth figure dropped to $118,000, then the mean would be $57,200, more in line with the normal distribution. Moving the sample size to fifteen, twenty, or thirty would bring the results increasingly closer to the normal distribution.

Bibliography

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