Central Limit Theorem in Real Life
Real time application of Central Limit Theorem
Are you curious about how pollsters conduct election polls? Have you ever wondered what makes poll results reliable? Or maybe you’re still scratching your head over what went wrong with the opinion polls of 2016 US Presidential Election Poll results?
Hello World !! Welcome back to my Statistical Symphony Series. Thanks to my lovely readers for the overwhelming response to my earlier article on the Central Limit Theorem. I’m back again with an interesting follow up article on the real life applications of Central Limit Theorem. In this article, we will discuss the Central Limit Theorem and its relevance in the real world. We will explore various examples that demonstrate the power of CLT in data analysis. Additionally let’s discuss concepts like the margin of error, confidence interval which go hand in hand with the CLT. This article is intended for anyone who’s interested in analyzing data & who wants to understand the significance of the Central Limit Theorem in real life & beyond. Buckle up and let’s go…!
A quick recap of the Central Limit Theorem
The Central Limit Theorem (CLT), a cornerstone of statistics, is a mind-boggling concept which states that regardless of the underlying distribution of the population, the sample mean of a sufficiently large sample size will follow a normal distribution. This theorem is a game-changer
“The Average of the Averages is the Average!”
This theorem is important because it allows us to use sample means to draw conclusions about a larger population mean. It is useful in real-world situations where it would be too time consuming or costly to collect data about each individual in a population.
Step by Step Implementation of Central Limit Theorem in a Real Life Business Problem
To implement the Central Limit Theorem (CLT) in a real-life business problem, follow these steps:
- Define the problem: Identify the business problem that needs to be solved. For example, a company may want to estimate the average weight of a product to ensure it meets certain standards.
- Determine the sample size: Decide on the sample size needed to obtain accurate results. The larger the sample size, the more accurate the results will be. Refer to my earlier article on CLT to understand how large is large enough?
- Collect data: Collect data from a random sample of products and record their weights.
- Calculate sample mean: Calculate the mean weight of the products in your sample.
- Repeat sampling: Repeat steps 3 and 4 multiple times with different random samples from the population (with replacement).
- Calculate mean of means: Calculate the mean of all means obtained from step 4.
- Determine standard deviation: Determine the standard deviation of all means obtained from step 4.
- Apply CLT formula: Use the CLT formula to calculate confidence intervals for your population mean based on your sample data, mean of means, standard deviation, margin of error.
9. Interpret results: Interpret your results and use them to make informed business decisions.
By following these steps, businesses can use CLT to solve real-life problems.
But wait.. How do we calculate the confidence interval and before that what is a confidence interval?
What is a Confidence Interval?
Confidence interval is an important concept in inferential statistics where the fundamental idea is to infer or draw conclusion about the population parameter based on a sample statistic. But the challenge is we might commit some error in this process and hence it is better to arrive at an interval instead of an estimate about the population.
A confidence interval is a range of values that is likely to contain the true value of a population parameter (such as a mean or proportion) based on a sample of data. The interval is calculated from the sample data and is used to estimate the range of values within which the population parameter is expected to fall.
The confidence interval is computed as
Confidence interval = Sample estimate +/- Margin of error
Well let’s understand all of these with a real world example — Election Polls.
Election Polls — Central Limit Theorem in Real World
Election (opinion) polls are an example of how the Central Limit Theorem applies in the real world. In opinion polls, pollsters use random sampling to select a subset of individuals from a larger population and ask them questions about their opinions on a particular topic. By using the CLT, pollsters can estimate what percentage of the larger population holds a particular opinion based on the responses of the randomly selected individuals. For example, suppose a pollster wants to know what percentage of voters in a state support a particular candidate. Instead of surveying every voter in the state, which would be time-consuming and expensive, they can use random sampling to select a smaller group of voters to survey. By using the CLT, they can estimate what percentage of all voters in the state support that candidate based on the responses of the randomly selected voters.
The agency can then use the Central Limit Theorem to calculate the margin of error and confidence interval for the poll results. This helps pollsters to determine the accuracy of their results and to make predictions about the behavior of a larger population.
What is Margin of Error, Standard Error and Confidence Interval and its connection with Central Limit Theorem?
Margin of error is an important concept in statistical inference that measures the degree of uncertainty in an estimate. It is the maximum amount by which the sample estimate can differ from the true population parameter, with a certain level of confidence.
In the case of election polls, margin of error refers to the amount of uncertainty associated with the percentage of votes a candidate is predicted to receive based on the poll results.
To compute the margin of error, a polling agency needs to consider two factors: sample size and confidence level.
- Sample size is the number of individuals surveyed in the poll.
- Confidence level is the degree of certainty required in the estimate. Commonly, the confidence level is set to 95% in election polls, meaning that if the poll were conducted 100 times, 95 times out of 100, the results would be within the margin of error.
The formula for margin of error is:
Margin of error = Critical value x Standard Error
The critical value is the number of standard deviations that corresponds to the desired level of confidence. For a 95% confidence level, the critical value is 1.96.
The standard error is the standard deviation of the sample mean, which is estimated using the formula:
Standard Error = Standard Deviation / √Sample Size
Standard deviation here refers to that of the population. The Standard deviation of the population is usually not known when conducting opinion polls. It is more common to use the standard deviation of the sample, called the standard error, as an estimate of the standard deviation of the population. This is because the sample standard deviation is a reasonable estimate of the population standard deviation when the sample size is large enough.
Once the margin of error is computed, the polling agency can construct a confidence interval, which is a range of values within which the true population parameter is expected to fall with a certain level of confidence. The confidence interval is computed as:
Confidence Interval = Sample Estimate +/- Margin of Error
Let’s connect all the “DOTS” with an example
For example, if a polling agency conducts a poll of 1000 individuals (sample size) and finds that 55% of them support a particular candidate, then the sample estimate is 0.55. If the confidence level is set to 95%, then the critical value is 1.96. If the standard deviation of the sample is known to be 0.05, then the standard error (SE) is:
SE = Sample Standard Deviation / √Sample Size
This estimate assumes that the sample is representative of the population and that the data is normally distributed.
Standard error (SE) = 0.05 / √1000 = 0.00158
Using the formula for margin of error, we have:
Margin of error = 1.96 x 0.00158 = 0.0031 or 0.31%
Thus, the margin of error for the poll results is 0.31%. The polling agency can then construct a confidence interval as:
Confidence interval = 0.55 +/- 0.0031 = [0.5469, 0.5531]
This means that we can be 95% confident that the true percentage of votes the candidate will receive in the upcoming election is between 54.69% and 55.31%.
Relationship between Margin or Error and Sample Size
The margin of error will depend on the sample size and the level of confidence desired. A larger sample size will result in a smaller margin of error, and a higher confidence level will result in a wider confidence interval
An interesting real-life example of the application of the central limit theorem in opinion polls is the 2016 US Presidential Election.
In the months leading up to the election, numerous opinion polls were conducted to gauge the public’s sentiment towards the candidates. However, many of these polls were criticized for their accuracy, as they failed to predict the outcome of the election. One reason for this discrepancy was the failure to account for the central limit theorem. Opinion polls typically sample a small subset of the population, and the results are extrapolated to represent the entire population.
However, if the sample size is too small, the results may not accurately reflect the population’s views.
In the case of the 2016 US Presidential Election, many polls had small sample sizes, which led to inaccurate predictions. Additionally, some polls may have been biased towards certain demographics, further skewing the results.
Overall, the application of the central limit theorem in opinion polls is crucial for ensuring accurate and representative results. By increasing the sample size and accounting for potential biases, pollsters can improve the reliability of their predictions and provide valuable insights into public opinion.
Other Real World Application of CLT
The CLT is used in many real-life situations such as economics, biology, agriculture and data analytics.
Central Limit Theorem in Manufacturing
In manufacturing, the CLT is quite popularly used in several ways. Quality control is a popular example of how the Central Limit Theorem applies in the real worldin manufacturing.
Suppose a manufacturing plant wants to ensure that the length of a product is within a specific range. The plant can take a sample of products and measure their length. The plant can then use the Central Limit Theorem to calculate the mean and standard deviation of the sample. The plant can set a tolerance limit based on the mean and standard deviation of the sample, which ensures that the length of the product is within the desired range.
Central Limit Theorem in Finance
The Central Limit Theorem (CLT) is used in financial analysis to estimate portfolio distributions and traits for returns, risk, and correlation. When analyzing large data sets such as securities funds, indexes or portfolios, CLT provides a shortcut to dealing with large data sets. It allows one to assume that means and standard deviations of the sampling distribution will be normally distributed which makes for easier statistical analysis. CLT is useful when analyzing a large collection of securities to predict the characteristics of a population more accurately. It is also helpful when examining financial data such as stock price history, returns, or correlations on a large number of securities.
Central Limit Theorem in Biology
One practical example of the Central Limit Theorem (CLT) in biology is its application in estimating the mean body weight of a population of animals or plants. Suppose a biologist is studying a population of python at the Amazon forest and wants to estimate their average weight. Collecting data on the weight of every single python in the forest would be impractical. Instead, the biologist can take a random sample of snakes from the population, record their weights, and calculate the sample mean weigh. This is very much applicable to any wild life study.
Central Limit Theorem in Traffic Analysis
In traffic analysis, the central limit theorem is used to estimate the distribution of travel times for different routes. This helps transportation planners to optimize traffic flow and to predict the impact of different road improvements like constructing fly-overs, layered roads, road expansions, traffic rerouting etc.
Central Limit Theorem in Medical Research
Medical research is an example of how the Central Limit Theorem applies in the real world. Suppose a medical researcher wants to estimate the average age of patients who have a specific medical condition. The researcher can randomly select a sample of patients with the medical condition and calculate the mean age. The researcher can then use the Central Limit Theorem to calculate the margin of error and confidence interval for the mean age.
The Central Limit Theorem is a powerful tool that helps any one analyzing data to make predictions and draw conclusions from a sample data. It is a cornerstone in statistics, and its relevance in the real world cannot be overemphasized. In this article, we have discussed the Central Limit Theorem and its relevance in the real world. We have explored various examples that demonstrate the power of CLT in real life.
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I’m Arun Prakash Asokan. Do check out my other articles on Statistical Symphony Series and stay tuned for the next one! See you soon !
