Central Limit Theorem

The Central Limit Theorem (CLT) forms the theoretical foundation for determining sample size. The theorem states that the sampling distribution of the mean x̄ (or the percentage value p̄) of a variable X, derived from a simple random sample will be normally distributed as the sample size increases, even if the population distribution is not normally distributed.

 

For example:

Universe size = N

Sample size = n

Mean value = μ

Sample mean values:

Sample 1: x̄1

Sample 2: x̄2

Sample 3: x̄3

Sample 4: x̄4 etc.

According to the CLT, the frequency distribution of the average values x̄1, x̄2, x̄3, x̄4 ... of a variable X, obtained from samples taken from the universe, follows the bell-shaped normal distribution curve shown in Exhibit 33.2.

While each sample yields a different mean for the variable being measured, based on the theorem, it can be expected that all samples of the same size and design will yield a result that is within a measured range around the true value.

The CLT further states that if repeated random samples of size n are drawn from a large population along some variable X, having a mean μ and variance S2, then the sampling distribution of sample mean will be a normal distribution having mean μ and variance s 2 = S2/n. The standard deviation s of the sampling distribution is referred to as the standard error of the mean.

For any parameter, while we do not know exactly how close the real value is to the measured value, based on the properties of the normal distribution, we can conclude with 95% confidence that it lies within ± two (1.96, to be precise) times the standard error.

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