The Central Limit Theorem (CLT)
The Central Limit Theorem (CLT) is a statistical theory that states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population. Furthermore, on taking multiple samples from the same population, the mean of the individual samples will form a normal distribution pattern, with all variances being approximately equal to the variance of the population divided by each sample's size.
Let’s understand the Central Limit Theorem stepwise:
- Take a random sample (S1) of size (n) from your data/population.
- Take the average of this sample (xˉ1).
- Take another sample (S2) of the same size and calculate its average (xˉ2).
- In this way calculate xˉ3, xˉ4, …. etc.
- Plot all the sample averages using a histogram i.e. plot xˉ1, xˉ2, xˉ3, ……, xˉn. The shape that you will observe will look like a normal distribution bell curve.
Example:
Consider a college of students with 4 divisions. We now randomly pick students from different divisions and make a sample. Each sample will include 20 students. Then, we will follow the following steps to solve it.
- Take all these samples and find the mean for each individual sample.
- Find the mean of the sample means.
- In this manner, we will get the approximate mean height of the students in the divisions.
- When the sample size gets larger, the sample means distribution will tend towards becoming a normal distribution as we keep on taking sufficient samples. In this way, we will get a bell-shaped curve if we will find the histogram of these sample mean heights.
CLT is all about “means which are normally distributed”
The mean values calculated from samples taken from a uniform distribution is the same as the mean values calculated from samples taken from either an exponential, beta or flat distributions and are normally distributed. It doesn’t matter what distribution you start with, if you collect samples from those samples, the means will be normally distributed as you keep increasing the sample size. The image below is a pictorial representation of what the Central Limit Theorem does.
Important points about Central Limit Theorem:
- The mean of a sample of data will be closer to the mean of the overall population in question as to the sample size increases, in spite of the actual distribution of the data, and whether it is normal or non-normal.
- More the sample size, more the distribution will tend towards being normal.
- It is said that the sample size must be at least 30 for CLT to show normal distribution. As the sample size increases to 10 and above, the distribution looks more and more like a normal distribution as can be seen in the following graphs.
Central Limit Theorem Formula:
The central limit theorem is applicable for a sufficiently large sample size (n≥30). For the random samples we take from the population, the mean of the sample means is calculated using;
The standard deviation of the sample means is calculated using;
Where,
μ = Population mean
σ = Population standard deviation
n = Sample size
Applications of Central Limit Theorem:
- It helps evaluate data from various distribution patterns.
- It is used in election polls to calculate the percentage of persons supporting a candidate which are seen on news as confidence intervals.
- It is also used to measure the average family income of a family in a particular region.
- By the 68-95-99 rule,
- 68% of the sample is within one standard deviation of the mean
- 95% is within 2 standard deviations
- 99.7% is within 3 standard deviations
- It can then be used to calculate something called a p-value which will help us in making inferences and forms the final step in hypothesis testing.
