Central limit theorem

The central limit theorem is a key principle in statistics and probability. It finds use in areas like inferential statistics, quality control, financial analysis, biostatistics, and machine learning. It allows you to make educated guesses about population means based on a data sample, even with incomplete knowledge about the whole population.

The central limit theorem helps explain the behavior of sample means and sums of random variables, shedding light on the statistical properties of data. This forms the basis for many reliable statistical methods used in data analysis and inference.

The sampling distribution of the sample means

To understand the central limit theorem, you need to first comprehend data distribution and the sampling distribution of sample means.

Data Distribution

In statistics, the term 'distribution' describes the way data is spread out, signifying the pattern of values in a dataset. This pattern helps in activities like understanding your data or forecasting future data outcomes.

Moreover, multiple ways exist to display data distribution, including histograms, scatter plots, Box plots, and others. Of these, histograms are the most commonly used. The tallest column corresponds to the most common data point, while the shortest one indicates the least common point.
For instance, consider the following data set: $\{2, 3, 4, 5, 6, 6, 6, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11\}$
At first glance, spotting a pattern may be challenging. However, if you distribute these numbers using a histogram, you'll notice a clear pattern.

The number nine is the most frequent in the data set, and therefore, it occupies the tallest column in the histogram.

Sampling distribution of sample means

In this type of distribution, you draw several samples of the same size from a population. You calculate the mean for each sample, and these means make up your distribution dataset.

Imagine you want to measure the average monthly income of individuals in a city. You draw four samples, each containing $5$ individuals. For each sample, you ask everyone about their monthly income and then calculate the average for that sample.

The results are as follows:
First sample average $= \$2,000$
Second sample average $= \$2,500$
Third sample average $= \$1,900$
Fourth sample average $= \$2,100$

So our dataset for the sampling distribution of sample means is $\{2,000, 2,500, 1,900, 2,100\}$. This dataset indicates the distribution of sample means obtained from different population samples in the city.

Central limit theorem

The central limit theorem forms the foundation for many statistics topics. It indicates that regardless of the actual population's distribution, if you consistently take equal-sized samples from the population and find the mean of each sample ($\bar{x}$), the outcomes of these sample means will follow a normal distribution.

There are multiple reasons why you should value the normality of the sampling distribution; this notion aids in numerous statistical analyses. Examining the sampling distribution provides a substantial amount of information about the data population, solely from samples.

As the sample size grows, the distribution becomes increasingly similar to a normal distribution. If the sample size is $30$ or more, it will closely resemble the normal distribution.

Inferences of Population Parameters

The Central Limit Theorem (CLT) provides useful insights into the behavior of sample means and standard deviations. Here's what it says concerning sample means and standard deviations:

Sample Means:

The Central Limit Theorem asserts that if the sampling distribution resembles a normal distribution, the mean of the samples will also approximate the population mean. Simply put, the average of the sample means will be a near approximation to the population mean.

${{\mu}_{\bar{x}}}=\mu$

As you increase the sample size, the average sample means will draw closer to the population mean. To illustrate this, let's consider the average weight of the crop.

Imagine you have an apple crop and wish to estimate the average weight of apples across the whole crop. To calculate the population mean, you would need to weigh each apple in the harvest and compute the average. Instead of this, you collect some baskets, put 30 apples in each, and calculate the average weight per basket.

For simplifying calculations, you collect only five samples (baskets).

Then, you calculate the average weight of apples for each basket:

basket1, average weight $(\overline x_1)= 246$ grams; basket2, average weight $(\overline x_2)= 250$ grams; basket3, average weight $(\overline x_3)= 244$ grams; basket4, average weight $(\overline x_4)= 256$ grams; basket5, average weight $(\overline x_5)= 239$ grams

Now, calculate the average of these means.

You now have a reasonably accurate estimate of the average weight of the apples in the crop. Gathering more baskets will make this approximation even more accurate. In our example, we only sampled five baskets for simplicity.

Standard Deviation:

Looking at the shape of the sampled points' normal distribution, you'll find that as the sample size grows, the sampling distribution becomes narrower. This indicates a reduction in the standard deviation of the sampling distribution. Thus, you can infer a relationship between sample size and the distribution's standard deviation. As the sample size increases, the standard deviation decreases. A narrow distribution shows samples with a smaller standard deviation.

This relationship is known as the standard error. The standard error formula allows us to calculate the population's standard deviation based on the standard deviation of the sampling distribution and the quantity of samples.

Here, the standard deviation of the sampling distribution ( ${\sigma}_{\bar{x}}$) equals the population's standard deviation ($\sigma$ ) divided by the square root of the sample number.

For example, suppose you collected samples, each with a sample size of $30$. After constructing a sampling distribution for sample means, you found that this distribution’s standard deviation is $0.5$. Based on the standard error formula, you can estimate that the population standard deviation would be

.We've learned plenty about the central limit theorem. However, what are the common missteps when applying it? Let's explore that in the next section!

Central limit theorem: pitfalls

Despite being a powerful tool in statistics, the CLT can also be subject to some common mistakes and misunderstandings:

1. Misunderstanding of conditions. The CLT assumes all observations are independent and identically distributed. This means the outcome of one observation doesn't influence the outcome of another. In real-life applications, this assumption is often violated. For example, in time-series data, observations are often dependent on previous observations.

2. Misinterpretation of the sample size. The CLT doesn't specify a minimum sample size that ensures a nearly normal distribution. A frequently used rule of thumb is that the sample size should be at least $30$. However, this is not a strict rule. The required sample size can be smaller or larger, based on the original distribution of the data. Misinterpreting this rule can lead to an unwarranted confidence in results and incorrect conclusions.

3. Heavy-Tailed Distributions. The CLT might not work well with heavy-tailed distributions or distributions with extreme outliers. These distributions have larger variances, and the sample mean doesn't converge to the normal distribution as quickly. In some instances, such as the Cauchy distribution, the CLT doesn't apply at all because the variance is infinite.

4. Overconfidence. Misinterpreting the CLT can lead to overconfidence in the results. For example, if the sample size is not large enough or the data is heavily skewed or has extreme outliers, the sample mean might not follow a normal distribution. Still, if we indiscriminately apply the CLT, we might be overly confident in our conclusions, leading to potential errors.

What should you do if you come across these limitations? Here are several strategies:

• Increase the sample size. The larger the sample size, the closer the sample mean will be to a normal distribution. However, this might not always be possible due to time or cost constraints.

• Use non-parametric methods. Non-parametric methods don't assume a specific distribution and can be used when the assumptions of the CLT aren't met.

• Use a different theorem. For example, you could use the Law of Large Numbers instead of the CLT. This theorem says that as the sample size increases, the sample mean will get closer to the population mean.

• Transform the data. Sometimes, transforming the data (like a log transformation) can help the distribution become more symmetric and eliminate the effects of extreme values.

• Use Robust Statistics. These statistical methods aren't sensitive to outliers or violations of assumptions. They can provide valid results even when the data don't meet the assumptions of traditional methods.

Remember, while the Central Limit Theorem is a valuable tool, it's not a universal solution. Always consider your data's characteristics and the assumptions of your statistical methods.

Conclusion

• The central limit theorem is an essential concept in statistics and probability that enables us to make inferences about a population by studying samples from it.

• The central limit theorem states that the distribution of sample means tends to follow a normal distribution.

• When applying the central limit theorem, always check whether your problem meets the conditions for its application. Examine the result of its application attentively.