Central Limit Theorem

by Jaxton Ibrahim

Statement of Central Limit Theorem:

The theorem states that if sizeable random samples are taken with replacement from the population consisting of the mean (𝝁) and standard deviation (σ), then the distribution of the sample means is normal. The Central Limit Theorem is valid even in the cases where the population is normal or asymmetrical, the only condition is that the sample size (n) must be ≥ 30. The theorem works even for the samples smaller than 30 however the population should be normal. In fact, the theorem is applicable to the binomial distribution provided the minimum value of (n times p, n times q) to be greater than 5, where n represents the sample size, p being the probability of success.

Since the random samples are drawn from the population, computing the sample means and sample standard deviation is necessary.

In the case of the binomial distribution, the formula for the population mean and the standard deviation is given by

𝝁 = np

σ = √n (p) (1 – p)

A few examples of the Central Limit Theorem are

  • Consider the case of a coin being tossed a few times, the probability that the outcomes are heads tends to a normal distribution when the mean is the same as the sum of the number of times the coin is tossed. It gets equated to the normal distribution at the point where there exist an infinite number of tosses.
  • The sum of the distances of the morning or evening walks traversed by a person will tend to a normal distribution.
  • It gives an idea about the sample and population parameters in order to build better machine learning models.

The method used to compute the problem of the central limit theorem involving ‘greater than’ ‘less than’ or “between” is explained below:

1) A few values such as mean, population size, sample size, standard deviation along with their respective sign is to be identified in the problem.

2) Taking the mean as the central value, a graph is drawn.

3) The value of the z-score is calculated using the formula: .

4) The z-table is referred to obtain the z values.

5) 3 different cases arise consisting of different signs are:

a] Central Limit Theorem with “greater than”: 0.5 – [z-score value]

b] Central Limit Theorem with “less than”: 0.5 + [z-score value]

c] Central limit theorem involving “between”.

6) Using step 3, the z score is found by substituting the values.

7) In the final step, the number is converted from decimal to percentage.

Applications of Central Limit Theorem

1] The central limit theorem describes the display of the “bell curve” in density estimates that finds its application in real-world data.

2] With the increase in the samples, it is found that the sample mean deviation decreases which enables in finding the population mean more accurately.

3] Once the sample mean is computed, it helps in obtaining other values that include the mean of the population.

4] Application of central limit theorem can be found in election polls to determine the percentage of people who have casted their vote in favour of a particular candidate as confidence intervals are usually communicated by the news channels.

A few applications on the Central Limit Theorem are listed above for reference. For more information on theorems like Fundamental Theorem of Algebra, Lagrange’s Theorem, Pythagoras Theorem, Mean Value Theorem and many more, please visit BYJU’S. It gives detailed explanations about the concepts along with solved problems.

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