Central Limit Theorem: The Four Conditions to Meet


The central limit theorem states that the sampling distribution of a sample mean is approximately normal if the sample size is large enough, even if the population distribution is not normal.

In order to apply the central limit theorem, there are four conditions that must be met:

1. Randomization: The data must be sampled randomly such that every member in a population has an equal probability of being selected to be in the sample.

2. Independence: The sample values must be independent of each other.

3. The 10% Condition: When the sample is drawn without replacement, the sample size should be no larger than 10% of the population.

4. Large Sample Condition: The sample size needs to be sufficiently large.

This tutorial provides a brief explanation of each condition.

Condition 1: Randomization

In order to apply the central limit theorem, the data that we use must be sampled randomly from the population by using a probability sampling method.

In statistics, there are two types of sampling methods:

1. Probability sampling methods: Sampling methods in which every member in a population has an equal probability of being selected to be in the sample. Examples include:

  • Simple random sample
  • Stratified random sample
  • Cluster random sample
  • Systematic random sample

2. Non-probability sampling methods: Sampling methods in which every member in a population does not have an equal probability of being selected to be in the sample. Examples include:

  • Convenience sample
  • Voluntary response sample
  • Snowball sample
  • Purposive sample

It’s important that a probability sampling method is used to obtain the sample because this maximizes the chances that we obtain a sample that is representative of the population.

Condition 2: Independence

In order to apply the central limit theorem, we must also assume that each of the sample values is independent of each other. That is, the occurrence of one event does not affect the occurrence of any other event.

This assumption is often met if we use a probability sampling method because these types of sampling methods choose observations to be included in the sample completely independently of each other. 

Condition 3: The 10% Condition

When the sample is drawn without replacement (which is almost always the case), the sample size must be no larger than 10% of the total population.

For example:

  • If our population size is 500, then our sample size should be no larger than 50.
  • If our population size is 1,000 then our sample size should be no larger than 100.
  • If our population size is 50,000, then our sample size should be no larger than 5,000.

And so on.

Condition 4: Large Sample Condition

Lastly, in order to apply the central limit theorem our sample size must be sufficiently large.

In general, we consider “sufficiently large” to be 30 or larger. However, this number can vary a bit based on the underlying shape of the population distribution.

In particular:

  • If the population distribution is symmetric, sometimes a sample size as small as 15 is sufficient.
  • If the population distribution is skewed, generally a sample size of at least 30 is needed.
  • If the population distribution is extremely skewed, then a sample size of 40 or higher may be necessary.

Depending on the shape of the population distribution, you may require more or less than a sample size of 30 in order for the Central Limit Theorem to apply.

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