# Central limit theorem

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## Definition: Central limit theorem

The Central limit theorem (CLT) states that if the population is normally distributed and samples are selected randomly from that population, then the sample means are also uniformly distributed. I.e. the distribution of means piles up at the centre. The size of the sample has to be sufficiently large, say above 30 for this to take place.

In short, if samples of size “n” (the sample size should be > 30) are drawn randomly from a population with mean “µ”, and a standard deviation of “σ”, the sample means are approximately normally distributed, regardless of the shape of the population.

Therefore,

1. Mean of population is equal to mean of sample means &
2. Standard deviation of sample means (standard error) is standard deviation of population means divided by square root of sample size

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