Central limit theorem plays a very important role when it comes to getting information on the sample mean. In simple words, it tells us that when we take a large number of samples from a population that is significantly large enough in terms of sample size (>30) and try to get the sample mean then it would be normally distributed.
Let’s break down why is the central limit theorem important with examples
Enabling Hypothesis Testing
When we do hypothesis testing the goal is to check whether the sample data we have comes from a population and meets a certain assumption like a specific mean or proportion.
The central limit theorem (CLT) helps in making these tests reliable by assuming sample mean regardless of the population distribution will follow a normal distribution as long as we have a sample size that is large enough.
For example,
Consider we are testing whether the average height of adult men in a country is 175 cm. We took a sample of 100 men and found their average height as 178 cm with a standard deviation of 10 cm.
If we want to test if the sample mean (178 cm) is significantly different from the assumed population mean of 175 cm.
As the sample size is big enough (100) so CLT tells us that the distribution of the sample mean will be approximately normal.
Now let’s calculate standard error and Z score. Z score is known as standard score which indicates how many standard deviations a data point is above or below the mean of the dataset.
Population Mean (μ): 175 cm. Sample Mean (X): 178 cm. Standard Deviation (σ): 10 cm. Sample Size (n): 100.
A z-score value of 3 indicates that the sample mean of 178 cm is 3 standard errors above the population mean of 175 cm. When we look in the standard normal distribution table, a z-score of 3 corresponds to a very small probability (p-value), telling us that the difference is statistically significant.
Confidence Intervals
The confidence interval provides a range of values within which we can be confident that the population parameter like population mean lies. CLT helps us to calculate reliable confidence intervals even when the population is not normally distributed.
For example
Let’s say we want to estimate the average income of workers in a city. We have randomly sampled 50 workers and found that the average income is $50,000, with a standard deviation of $8,000.
To calculate a 95% confidence interval for the population mean:
Sample Mean (X): $50,000 Standard Deviation (σ): $8,000 Sample Size (n): 50 Confidence Level: 95%
We first need to find the standard error which is given by the standard deviation divided by the square root of the sample size.
8000/sqrt(50) = 1131.37
From the standard normal distribution table let’s get the z value for 95 % confidence interval which comes to approximately 1.96.
To get the margin of error we have to multiply the z value of 95 % confidence and standard error.
ME=z×SE ME=1.96×1131.37 ME=2219.47
Finally, we get the confidence interval
50,000±2219.47=(47,780.53,52,219.47)
So we can say with 95% confidence that the true population mean income lies between $47,780.53 and $52,219.47.
From the above example, we got to know that CLT allows us to use the normal distribution to construct this interval even if we have a distribution of incomes that may be skewed or not be normally distributed
Predicting Outcomes
We have domains like economics, marketing, or healthcare where it is often impractical to collect data from the entire population. Instead, we can take a sample and use the CLT to make predictions about the larger population based on the sample data.
For example
Suppose you’re working in marketing, and you want to estimate the average amount of time customers spend on your website. Instead of surveying every visitor, you randomly sample 200 website sessions and find that the average time spent is 5 minutes, with a standard deviation of 2 minutes.
Sample Mean (X): 5 minutes Standard Deviation (σ): 2 minutes Sample Size (n): 200
Using the CLT, you can predict that the average time spent by all website visitors will be close to this sample mean (5 minutes), and you can construct a confidence interval around that estimate to get an idea of the true average.
Standard error (SE) = standard deviation/sqrt(sample size) SE=2/sqrt(200) SE=0.1414
Now let us get the z value from the standard normal distribution table for 95 % confidence interval. It comes to 1.96.
Margin Of Error (ME) = z value x Standard error ME = 1.96×0.1414 ME=0.277 Confidence interval: 5±0.277=(4.723,5.277)
So by these calculations, we can say that the true average time spent on the website is between 4.723 and 5.277 minutes. This prediction is valuable in decision making for optimizing website content.
Helping in Data-driven Decisions
The ability to make decisions based on sample data is important in business. Decisions have to be made quickly but accurately in today’s fast-paced world.
The Central Limit theorem helps us to get sample means which are reliable approximations of the population mean when we have a large population.
Conclusion
The central limit theorem is very important in statistics because it helps in
- Hypothesis testing
- Confidence intervals
- Outcome prediction
- Decision making
Even when the underlying data distribution is not normal. It allows data scientists to work with sample data to make accurate inferences about the entire population.
So knowing how the central limit theorem works and its importance is essential for anyone involved in data analysis.
You can also read our other blog on how to build dashboards in Python and how to get a data analytics internship in 2025.