H2 Maths 6.3 - Normal Distribution: The Central Limit Theorem

Step 1 - The sample mean and its exact distribution

When we take a random sample of n observations from a normal distribution with mean mu and variance sigma squared, the sample mean X-bar has a predictable distribution.

Step 1 - The sample mean and its exact distribution

The mean of X-bar is mu, and the variance of X-bar is sigma squared over n - averaging n values reduces the spread by a factor of n.

Step 1 - The sample mean and its exact distribution

This holds exactly when the parent distribution is normal.

Step 2 - The Central Limit Theorem for non-normal populations

The remarkable result is that even if the parent distribution is not normal, the sample mean is approximately normal for large enough n.

Step 2 - The Central Limit Theorem for non-normal populations

This is the Central Limit Theorem. In practice, n greater than or equal to thirty is usually taken as sufficient.

Step 2 - The Central Limit Theorem for non-normal populations

The approximation improves as n increases, regardless of the shape of the parent distribution.

Step 3 - Set up the worked example

The mass X grams of an apple has mean one hundred and fifty and variance one hundred. A random sample of twenty-five apples is chosen.

Step 3 - Set up the worked example

The sample mean X-bar has mean one hundred and fifty, and variance one hundred over twenty-five, which is four.

Step 3 - Set up the worked example

Since the parent distribution is normal, X-bar follows exactly N of one fifty, four.

Step 4 - Find P(X̄ > 152)

We want the probability that the sample mean exceeds one hundred and fifty two.

Step 4 - Find P(X̄ > 152)

Z = (152 − 150) / 2 = 1.

Step 4 - Find P(X̄ > 152)

From the GC, P of Z greater than one equals zero point one five eight seven.

Step 5 - Effect of increasing the sample size

If we double the sample size to fifty, the variance of X-bar halves from four to two.

Step 5 - Effect of increasing the sample size

The standard deviation becomes the square root of two, approximately one point four one four.

Step 5 - Effect of increasing the sample size

The same threshold of one fifty two is now further out in the tail - the probability drops to about zero point zero seven eight six, showing that larger samples give more precise estimates of the mean.