# Central Limit Theorem

The Central Limit Theorem (CLT) forms the theoretical foundation for determining sample size. The theorem states that, as the sample size increases, the sampling distribution of the mean x̄ (or the percentage valuep̄) of a variable X, obtained from a simple random sample, will approach a normal distribution. This holds true regardless of the underlying distribution of the population.

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 34.2.

#### Exhibit 34.2  90% of the observations fall within a range of ±1.65 standard deviation from the mean.

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 lies 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 approximate a normal distribution with a mean μ and a 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, although we cannot determine the exact proximity of the true value to the measured value, we can rely on the properties of the normal distribution to infer with 95% confidence that it falls within ± two times the standard error (1.96, to be precise).

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