What Is the Central Limit Theorem?

Key takeaways

• The central limit theorem (CLT) is a fundamental statistical concept that relates multiple, independent sample means to a population distribution [1].

• If you take the averages of enough samples and plot them, the CLT dictates that these averages will form a bell-shaped curve for most populations.

• When using the CLT, values must be independent, the population variance must be finite, and the sample size must be large enough.

You can use the CLT across industries to maximize resources while still developing insights into population data. Learn more about the central limit theorem by exploring its definition, real-world examples, how it compares to other theorems, and how you can use it professionally.

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What is the central limit theorem in simple terms?

The CLT tells you that when you take samples from almost any population, the averages of those samples will be normally distributed. Being “normally distributed” means it follows a bell curve shape.

For example, imagine you have a jar of candy in varying sizes. If you take multiple handfuls of candy and record the average size of each handful, these averages will eventually follow a bell-shaped curve. This principle helps you to take messy, unpredictable data and turn it into a predictable pattern.

What is the central limit theorem formula?

In statistical terms, the central limit theorem is the sampling distribution of the mean. It refers to the distribution of means that you get from repeatedly sampling a population. So if you have a population with mean μ and standard deviation σ, then for a sample size of n, the means will center on μ with a standard error (or spread) of σ/√n.

This implies that bigger samples tend to give you averages that are closer to the population “truth.” You often hear of a “z-score” in relation to the CLT. A z-score, calculated by the formula Z = (x − μ) / σ, rescales sample averages to show how many standard errors they sit from the population mean. Note that “x” represents the value being evaluated. This helps you understand how unusual a value may be compared to other population values.

What is the difference between CLT and LLN?

The law of large numbers (LLN) and CLT both relate to a growing sample size, but in different ways. The LLN tells you that as a sample size grows, the mean will get closer to the mean of the whole population, while the CLT tells you that for a sufficiently large sample size, the distribution of the sample means will approach a normal distribution. Put another way, the LLN is about how a single average becomes more accurate as sample size increases, while the CLT showcases the pattern of many averages. The LLN primarily validates expected theoretical values such as the mortality rate. The CLT, on the other hand, helps estimate confidence intervals or the probability of extreme events like bankruptcy [2].

Read more: 7 Statistical Analysis Methods Beginners Should Know

What are the three conditions of the central limit theorem?

The three conditions of the CLT are that observations must be random and independent, the population must have a finite variance, and the sample size must be large enough.

• Random and independent: If your sample is random and independent, one data point shouldn’t influence another. For example, if you flip a coin twice, the result of the first flip shouldn’t influence the second.

• Finite variance: Variance measures how spread out the data is. When a variance is finite, it means that the values may vary but don’t stretch toward extremes so aggressively that the variance becomes infinite. The CLT becomes less accurate if the variances become infinite.

• Large enough sample size: In order for the sample means to approach normality, the samples generally must have at least 30 data points in each.

Why is 30 a good sample size?

In statistics, professionals often reference 30 as a good target number for the CLT because it's small enough to practically collect but large enough that the averages tend to form the bell-shaped curve more nicely than small, more variable samples. However, if the distribution is heavily skewed, you may need a larger sample size.

What is an example of the central limit theorem in action?

Rolling a six-sided die repeatedly illustrates how averaging many sample sets produces a bell curve even when individual outcomes are uniformly distributed. If you rolled it 50 times and graphed the outcomes, you would see a flat, straight line. This is because each individual number, one through six, is equally likely. The average of all possibilities (one through six) is 3.5. So μ = 3.5, and the spread is σ ≈ 1.71.

This is where you can bring in the central limit theorem. Now consider that you don’t want to look at single rolls but instead want to look at averages across sets of rolls. You could take the following steps:

1. Roll the die 30 times and average all of the values. Suppose you get 3.7.

2. Now repeat another 30 times, and another. You might get 3.4, and then 3.6.

3. Repeat this hundreds of times, recording the average value each time.

4. Now plot all of the averages. They should form a bell-shaped curve centered on 3.5.

Even though one roll of one die is flat on a graph, combining the averages gives you a smooth bell curve. This means that you've turned your 'random' set of values into a tidy, predictable bell curve.

Where is the central limit theorem used in real life?

You can spot professionals across industries using the central limit theorem to make sense of information and predict population findings based on their sample values. This helps to optimize resources over time.

• Polling: You typically can’t poll millions of people, so a more practical solution is to find a few thousand instead and extrapolate the information. The CLT allows pollsters to take the values from a smaller set of people and make estimates about the larger population.

• Quality control: Instead of trying to inspect every item from a factory, quality control professionals can pull small samples and measure averages, using the CLT to flag when something seems off. The CLT allows you to make estimates across a broader population based on the outcome in your smaller samples.

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