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The central limit theorem is a concept of statistics that states that the sum of a large number of self-standing random variables is nearly normal. If we simplify this, we can say that the theorem means that if we keep drawing larger and larger samples and then calculate their means, then the sample means will form their normal distribution. The mean of the normal distribution is the same as the original distribution, and we divide the variance by n, which is the sample size. n denotes the number of values that are averaged together. With the increase in the number of samples, the graph will move toward a normal distribution.
Central limit theorem statement
The central limit theorem (CLT) says that as sample sizes grow higher, the distribution of sample means approaches a normal distribution, independent of the population’s distribution. For the CLT to hold, sample sizes of 30 or more are frequently regarded as sufficient.
The central limit theorem statement looks technical, but it can be comprehended if we think about it step by step. First, we can take a random sample where the number of individuals is n. Then it is easy to make a sample mean that will correspond to the mean of the measurement we want to make in our population.
Examples of the central limit theorem
1. The mean values of the sample collected from a larger sample are 11.8, 10.6, 11.7, 13.2, 12.3, 14.4, 14, 10, and 10.6. Find the mean of the population.
Solution: the given values are 11.8, 10.6, 11.7, 13.2, 12.3, 14.4, 14, 10, 10.6
μ = (11.8+ 10.6+11.7+13.2+12.3+14.4+14+10+10.6) / 9
= 108.6/9
= 12.07
2. There is an unidentified distribution whose mean is 45 and has a standard deviation of 8. The size of the sample is 30 which is drawn from the population randomly. Find out the probability of the sample mean to be between 42 and 50.
Solution: P(42<x<50)= ( 42, 50, 45, 8/ √30)
= 0.9797
3. The data weight of the male population follows a normal distribution. The mean is 75 kg and the standard deviation is 5 kg. Find out the mean and standard deviation of 40 males.
Solution: μ = 75, σ = 5 kg, n= 40
According to the central limit theorem, the sample mean and the population mean are equal
μ = 75 kg
= 5/√40
= 0.79
4. Cigarette smokers have a mean age of 30 years. If the standard deviation is 8 years and the sample size is 40. Find out the mean and standard deviation.
Solution: μ = 30, σ = 8, n = 40
As the central limit theorem states that sample mean and population mean is equal, then
μ = 30 years
= 8/√40
= 1.27
Central Limit Theorem’s importance
The central limit theorem is important in statistics for two reasons:
The normality assumption
The information that the sample distributions could approximate a normal distribution has some important applications. The normality assumption is essential for the parametric hypothesis test of the mean. However, it may look like these tests are invalid because the data is not distributed normally. If the sample size is large enough, then the central limit theorem will help make the sampling distributions that will approximate a normal distribution.
Precision of estimates
When we prepare graphs, we notice that when the sample size increases, the sampling distributions of the mean are grouped more strongly around the population mean. When we use a sample to estimate the mean of the whole population, this property becomes applicable. When we have a large sample size, then it is possible that the mean would be close to the real population, or we can say that we have a more precise estimate.
On the other hand, the sample distributions of the mean are wider for small samples. In the case of smaller sample sizes, it is normal for them to be away from the actual mean of the population, or it can be said that the obtained estimate would be less precise.
Conclusion
The central limit theorem says that the distribution of the sample means will be close to a normal distribution with the increase in the size of the sample irrespective of the population’s distribution. As we increase the number of samples, the graph will also move to a normal distribution. This theorem helps understand how population estimates behave when they are subjected to repeated sampling. However, certain conditions must be met to use the central limit theorem. The central theorem is important in statistics and has many applications as well.
