The Central Limit Theorem in Statistics states that as the sample size increases and its variance is finite, then the distribution of the sample mean approaches normal distribution irrespective of the shape of the population distribution. The query that how much the sample size should increase can be answered that if the sample size is greater than 30 then the statement of the Central Limit Theorem holds.

The central limit theorem posits that the distribution of sample means will invariably conform to a normal distribution provided the sample size is sufficiently large. This holds regardless of the underlying distribution of the population, be it normal, Poisson, binomial, or any alternative distribution.

In this article on the Central Limit Theorem, we will learn about the central limit theorem definition, examples, Formulas, proof of the Central Limit Theorem, and its applications.

What is the Central Limit Theorem?

The Central Limit Theorem explains that the sample distribution of the sample mean resembles the normal distribution irrespective of the fact that whether the variables themselves are distributed normally or not. The Central Limit Theorem is often called CLT in abbreviated form.

Central Limit Theorem Definition in Statistics

Central Limit Theorem states that:

When large samples usually greater than thirty are taken into consideration then the distribution of sample arithmetic mean approaches the normal distribution irrespective of the fact that random variables were originally distributed normally or not.

Central Limit Theorem Formula

Let us assume we have a random variable X. Let σ be its standard deviation and μ is the mean of the random variable. Now as per the Central Limit Theorem, the sample mean [Tex]\overline{X} [/Tex] will approximate to the normal distribution which is given as [Tex]\overline{X} [/Tex] ⁓ N(μ, σ/√n). The Z-Score of the random variable [Tex]\overline{X} [/Tex] is given as Z = [Tex]\dfrac{\overline x – \mu}{\frac{\sigma}{\sqrt n}} [/Tex]. Here [Tex]\overline x [/Tex] is the mean of [Tex]\overline X [/Tex]. The image of the formula is attached below.

Central Limit Theorem Proof

Let the independent random variables be X₁, X₂, X₃, . . . . , X_n which are identically distributed and where their mean is zero(μ = 0) and their variance is one(σ² = 1).

The Z score is given as, Z = [Tex]\dfrac{\overline X – \mu}{\frac{\sigma}{\sqrt n}} [/Tex]

Here, according to Central Limit Theorem, Z approximates to Normal Distribution as the value of n increases.

Let m(t) be the Moment Generating Function of Xi

⇒ M(0) = 1

⇒ M'(1) = E(Xi) = μ = 0

⇒ M”(0) = E(Xi²) = 1

The Moment Generating Function for Xi/√n is given as E[e^tXi/√n]

Since, X₁ X₂, X₃ . . . X_n are independent, hence the Moment Generating Function for (X₁ + X₂ + X₃ + . . . + X_n)/√n is given as [M(t/√n)]ⁿ

Let us assume as function

f(t) = log M(t)

⇒ f(0) = log M(0) = 0

⇒ f'(0) = M'(0)/M(0) = μ/1 = μ

⇒ f”(0) = (M(0).M”(0) – M'(0)2)/M'(0)2 = 1

Now, using L’ Hospital Rule we will find t/√n as t²/2

⇒ [M(t/√n)]² = [e^f(t/√n)]ⁿ

⇒ [e^nf(t/√n)] = e^(t²/2)

Thus the Central Limit Theorem has been proved by getting Moment Generating Function of a Standard Normal Distribution.

Central Limit Theorem Examples

Let’s say we have a large sample of observations and each sample is randomly produced and independent of other observations. Calculate the average of the observations, thus having a collection of averages of observations. Now as per the Central Limit Theorem, if the sample size is adequately large, then the probability distribution of these sample averages will approximate to a normal distribution.

Assumptions of the Central Limit Theorem

The Central Limit Theorem is valid for the following conditions:

• The drawing of the sample from the population should be random.
• The drawing of the sample should be independent of each other.
• The sample size should not exceed ten percent of the total population when sampling is done without replacement.
• Sample Size should be adequately large.
• CLT only holds for a population with finite variance.

Steps to Solve Problems on Central Limit Theorem

Problems of Central Limit Theorem that involves >, < or between can be solved by the following steps:

Step 1: First identify the >, < associated with sample size, population size, mean and variance in the problem. Also there can be ‘betwee; associated with range of two numbers.

Step 2: Draw a Graph with Mean as Centre

Step 3: Find the Z-Score using the formula

Step 4: Refer to the Z table to find the value of Z obtained in the previous step.

Step 5: If the problem involves ‘>’ subtract the Z score from 0.5; if the problem involves ‘<‘ add 0.5 to the Z score and if the problem involves ‘between’ then perform only step 3 and 4.

Step 6: The Z score value is found along [Tex]\overline X [/Tex]

Step 7: Convert the decimal value obtained in all three cases to decimal.

Central Limit Theorem Applications

Central Limit Theorem is generally used to predict the characteristics of a population from a set of sample. It can be applied in various fields. Some of the applications Central Limit Theorem are mentioned below:

• Central Limit Theorem is used by Economist and Data Scientist to draw conclusion about population to make a statistical model.
• Central Limit Theorem is used by Biologists to make accurate predictions about the characteristics of the population from set of sample.
• Manufacturing Industries use Central Limit Theorem to predict overall defective items produced by selecting random products from a sample.
• Central Limit Theorem is used in surveys to predict the characteristics of the population or to predict the average response of the population by analyzing a sample of obtained responses.
• CLT can be used in Machine Learning to make conclusion about the performance of the model.

Read More: Central Limit Theorem in Data Science & Machine Learning

Solved Examples on Central Limit Theorem

Example 1. The male population’s weight data follows a normal distribution. It has a mean of 70 kg and a standard deviation of 15 kg. What would the mean and standard deviation of a sample of 50 guys be if a researcher looked at their records?

Given: μ = 70 kg, σ = 15 kg, n = 50

As per the Central Limit Theorem, the sample mean is equal to the  population mean.

Hence, [Tex]\mu _{\overline{x}}    [/Tex] = μ = 70 kg

Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}}    [/Tex] = 15/√50

⇒ [Tex]\sigma _{\overline{x}}    [/Tex] ≈ 2.1 kg

Example 2. A distribution has a mean of 69 and a standard deviation of 420. Find the mean and standard deviation if a sample of 80 is drawn from the distribution.

Given: μ = 69, σ = 420, n = 80

As per the Central Limit Theorem, the sample mean is equal to the  population mean.

Hence, [Tex]\mu _{\overline{x}}    [/Tex] = μ = 69 

Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}} [/Tex]

⇒ [Tex]\sigma _{\overline{x}}    [/Tex]= 420/√80

⇒ [Tex]\sigma _{\overline{x}}    [/Tex] = 46.95 

Example 3. The mean age of people in a colony is 34 years. Suppose the standard deviation is 15 years. The sample of size is 50. Find the mean and standard deviation of the sample.

Given: μ = 34, σ = 15, n = 50

As per the Central Limit Theorem, the sample mean is equal to the  population mean.

Hence, [Tex]\mu _{\overline{x}}    [/Tex] = μ = 34 years

Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}} [/Tex]

⇒ [Tex]\sigma _{\overline{x}}    [/Tex]= 15/√50

⇒ [Tex]\sigma _{\overline{x}}    [/Tex] = 2.12 years

Example 4. The mean age of cigarette smokers is 35 years. Suppose the standard deviation is 10 years. The sample size is 39. Find the mean and standard deviation of the sample.

Given: μ = 35, σ = 10, n = 39

As per the Central Limit Theorem, the sample mean is equal to the  population mean.

Hence, [Tex]\mu _{\overline{x}}    [/Tex] = μ = 35 years

Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}}    [/Tex] = 10/√39

⇒ [Tex]\sigma _{\overline{x}}    [/Tex] = 1.601 years

Example 5. The mean time taken to read a newspaper is 8.2 minutes. Suppose the standard deviation is one minute. Take a sample of size 70. Find its mean and standard deviation.

Given: μ = 8.2, σ = 1, n = 70

As per the Central Limit Theorem, the sample mean is equal to the  population mean.

Hence, [Tex]\mu _{\overline{x}}    [/Tex] = μ = 8.2 minutes

Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}}    [/Tex] = 1/√70

⇒ [Tex]\sigma _{\overline{x}}    [/Tex] = 0.11 minutes

Example 6. A distribution has a mean of 12 and a standard deviation of 3. Find the mean and standard deviation if a sample of 36 is drawn from the distribution.

Given: μ = 12, σ = 3, n = 36

As per the Central Limit Theorem, the sample mean is equal to the  population mean.

Hence, [Tex]\mu _{\overline{x}}    [/Tex] = μ = 12

Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}}    [/Tex] = 3/√36

⇒ [Tex]\sigma _{\overline{x}}    [/Tex] = 0.5

Example 7. A distribution has a mean of 4 and a standard deviation of 5. Find the mean and standard deviation if a sample of 25 is drawn from the distribution.

Given: μ = 4, σ = 5, n = 25

As per the Central Limit Theorem, the sample mean is equal to the  population mean.

Hence, [Tex]\mu _{\overline{x}}    [/Tex] = μ = 4

Now, [Tex]\sigma _{\overline{x}}=\frac{\sigma }{\sqrt{n}}    [/Tex]

⇒ [Tex]\sigma _{\overline{x}}    [/Tex]= 5/√25

⇒ [Tex]\sigma _{\overline{x}}    [/Tex] = 1

Articles related to Central Limit Theorem:

• Probability Distribution Function
• Standard Deviation
• Central Tendency

Summary – Central Limit Theorem in Statistics

The Central Limit Theorem (CLT) is a cornerstone of statistical theory that establishes the conditions under which the mean of a large number of independent, identically distributed random variables, irrespective of the population’s distribution, will approximate a normal distribution.

Key assumptions for the CLT to hold include random sampling, independence among samples, a sample size that does not exceed 10% of the population for sampling without replacement, and a sample size generally greater than 30 to ensure the sample mean’s distribution approximates normality for populations with finite variance.

The theorem is mathematically represented by the formula X∼N (μ,σ/n​), where μ is the population mean, σ is the standard deviation, and n is the sample size. This principle is proved through the use of moment-generating functions, illustrating that as sample size increases, the distribution of the sample mean approaches that of a standard normal distribution.

The CLT’s applications span across various fields including economics, biology, manufacturing, and even machine learning, assisting in making predictions about population characteristics, statistical modeling, and understanding the performance of models through analysis of sample data.

Central Limit Theorem in Statistics – FAQs

What is the Central Limit Theorem?

The Central Limit Theorem states that with a sufficiently large sample size, the sampling distribution of the sample mean will approximate a normal distribution, regardless of the population’s original distribution. This holds true provided the sample size is generally considered to be 30 or more​.

What is Central Limit Theorem in Statistics?

Central Limit Theroem in statistics states that whenever we take a large sample size of a population then the distribution of sample mean approximates to the normal distribution.

When does Central Limit Theorem apply?

Central Limit theorem applies when the sample size is larger usually greater than 30.

Why is Central Limit Theorem important?

Central Limit Theorem is important as it helps to make accurate prediction about a population just by analyzing the sample.

How to solve Central Limit Theorem?

The Central Limit Theorem can be solved by finding Z score which is calculated by using the formula Z = [Tex]\dfrac{\overline x – \mu}{\frac{\sigma}{\sqrt n}} [/Tex]. The detailed process has been discussed under the heading “Steps to Solve Central Limit Theorem”.

What is Moment Generating Function?

Moment Generating Function is a function that encodes the moment of a random variable into a function. It is the expectation of a function of Random Variable. It acts as alternative to Probability Distribution Function and Cumulative Distribution Function for a random variable.

Why is the Central Limit Theorem important in statistics?

The CLT is crucial because it allows statisticians to make inferences about population parameters using sample statistics. It underpins many statistical methods, such as hypothesis testing and the creation of confidence intervals, by justifying the normality assumption of large sample statistics​.

Can the Central Limit Theorem be applied to any distribution?

Yes, one of the remarkable aspects of the CLT is that it does not require the population to be normally distributed. It applies to any distribution with a finite mean and variance, making it a powerful tool in statistical analysis and data science​.

What are the conditions necessary for the Central Limit Theorem to hold?

The theorem requires that the samples are independent and identically distributed, the sample size is sufficiently large (usually n ≥ 30), and the population from which samples are drawn has a finite variance. These conditions ensure the sampling distribution of the mean approximates normality​.

How does the Central Limit Theorem apply in real-world scenarios?

In practical terms, the CLT is used extensively across various fields such as finance, insurance, and economics for risk assessment, forecasting, and decision-making. For example, financial analysts use it to predict stock prices or manage portfolio risk by assuming that large sample means of financial returns are normally distributed​.