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Central Limit Theorem Calculator

Input the required statistical entities in their respective positions and the calculator will try to determine the mean and standard deviation by using the central limit theorem, with the steps shown.

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An online central limit theorem calculator enables you to determine the mean and standard deviation of a sample with its theorem. This free calculator uses the central limit theorem formula to provide step-by-step calculations. Let’s read to understand the different types and examples of this phrase.

What is the Central Limit Theorem?

The central limit theorem states that if the sample size is large enough, even if the population distribution is abnormal, the sample mean distribution will be approximately normal.

Central Limit Theorem Formula:

This formula points out that the distribution of the sample has the following central limit theorem conditions:
  • The mean of the sampling distribution is equal to the mean (μ) of population distribution:
$$x = μ$$
  • The deviation of the sampling distribution is similar to the deviation of the population distribution divided by the sample size:
$$s = σ / \sqrt{n}$$ This formula for sample size used by the central limit theorem calculator.

The Central Limit Theorem for Sample Mean:

The central limit theorem of the sample mean indicates that the sample you draw is getting larger and larger. When calculating its mean with the central limit theorem calculator, the sample mean forms its own normal distribution. The distribution has the mean as the original distribution, and the variation is equal to the variance divided by the sample size. The variable n is the average value summed together, not the number of times the experiment is run. When extracting a random sample from size n, the distribution of a random variable (x), which consists of the sample mean is called the sample distribution of the mean. The sample distribution of the mean is approximately normal as the sample size n increases. The variable X(bar) in one sample $$x =X(bar) -μ_x / σx/\sqrt {n}$$ μ_x is average of X and X(bar) $$σx(bar) = X(bar) - μ_x / σx/\sqrt {n}$$ standard deviation of X (bar) Which is called the standard error of mean. However, an Online Limit Calculator determine the positive or negative limits for a given function at any point.

The Central Limit Theorem for Sums:

The central limit theorem of summation assumes that A is a random variable whose distribution may be known or unknown (can be any distribution), μ = the mean of A σ = the standard deviation of A The central limit theorem of summation of the standard deviations of A points out that if you keep drawing more larger samples and take their sum. These sums form their own normal distribution (sample distribution), which is similar to the normal distribution.
  • As the number of samples increases, the mean of a normal distribution is the original mean multiplied by the sample size, and
  • The standard deviation is the square root of the original standard deviation multiplied by the sample size.
The variable ΣX has some z-score associated with it:
  • Σx is one sum.
  • z = Σx–(n)(μ)/(sqrt{n})(σ)
  • (n)(μ) = the mean of ΣX
  • (sqrt{n})(σ) = (σΣX) the deviation of ΣX
Example: During a writing test the mean was 35 where the standard deviation is 5. If a candidate scored 40, then what is the z−score? Solution: Z = x−μ/σ Z = 40−35/5 Z = 1 Hence, the central limit theorem example provides the z-score using sample mean and standard deviation. However, an Online Mean Value Theorem Calculator helps you to find the rate of change of the function using the mean value theorem.

How Central Limit Theorem Calculator Works?

A simple online central limit theorem probability calculator computes standard deviation and sample mean by following the given steps:


  • First of all, substitute the value of population mean.
  • Now, add population standard deviation and sample size.
  • Click the calculate button to continue the process.


  • The central limit theorem calculator gives the values of the sample mean and standard deviation.
  • It also displays the central limit theorem formula and step-wise calculation.


How many samples required for the central limit theorem?

At least 30 randomly selected across various sectors, stocks must be sampled, for the central limit theorem to hold.

Tell about the two crucial concepts of the Central Limit Theorem?

These are the two key points of the central limit theorem: the average of our sample is itself the average of the entire population. The standard deviation of the sample mean is the standard error of the population mean.

What is the relationship between the central limit theorem and the confidence interval?

Suppose we want to obtain an estimate of the 95% confidence interval of the unknown population mean. The central limit theorem states that for large samples, the distribution of the sample mean approximates the mean distribution, with the mean and standard deviation (also known as standard error).

Why large sample size is important?

The sample size is a vital aspect of research. Larger sample size can provide a more accurate average, identify outliers that can skew the data in a smaller sample, and provide less bias.


This free online central limit theorem calculator uses the population mean and sample size to calculate the sample mean and standard deviation of the sample. This means that the calculator will perform all calculations with an accuracy of 100, which is more beneficial for students and teachers.


From the source of Wikipedia: Central limit theorem, Independent sequences, Classical CLT, Lyapunov CLT, Lindeberg CLT, Multidimensional CLT,  Martingale difference CLT. From the source of Boston University School: Central Limit Theorem with a Normal Population, Central Limit Theorem with a Skewed Distribution, Standard error. From the source of Investopedia: What Is the Central Limit Theorem (CLT), distribution of sample, The Central Limit Theorem in Finance. From the source of Statistics by Jim: Distribution of the Variable in the Population, Sampling Distribution of the Mean, Central Limit Theorem and Approximating the Normal Distribution.  
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