The Central Limit Theorem for Sample Means (Average)
Suppose X is a random variable with a distribution that may be known or unknown (it can be any distribution).
Using a subscript that matches the random variable, suppose:
a. μX = the mean of X
b. σX = the standard deviation of X
If you draw random samples of size n, then as n increases, the random variable X which consists of sample means, tends to be normally distributed.
The central limit theorem for sample means says that if you keep drawing larger and larger samples (such as rolling one, two, five, and finally, ten dice) and calculating their means, the sample means to form their own normal distribution (the sampling distribution).
The normal distribution has the same mean as the original distribution and variance that equals the original variance divided by the sample size.
The variable n is the number of values that are averaged together, not the number of times the experiment is done.
To put it more formally, if you draw random samples of size n, the distribution of the random variable X, which consists of sample means, is called the sampling distribution of the mean.
The sampling distribution of the mean approaches a normal distribution as n, the sample size, increases.
The random variable X has a different z-score associated with it from that of the random variable X. The mean x is the value of X in one sample.
Using the Central Limit Theorem
It is important for you to understand when to use the central limit theorem. If you are being asked to find the probability of the mean, use the clt for the mean.
If you are being asked to find the probability of a sum or total, use the clt for sums.
This also applies to percentiles for means and sums.
The Empirical Rule
If X is a random variable and has a normal distribution with mean μ and standard deviation σ, then the Empirical Rule says the following:
• About 68% of the x values lie between –1σ and +1σ of the mean μ (within one standard deviation of the mean).
• About 95% of the x values lie between –2σ and +2σ of the mean μ (within two standard deviations of the mean).
• About 99.7% of the x values lie between –3σ and +3σ of the mean μ (within three standard deviations of the mean).
Notice that almost all the x values lie within three standard deviations of the mean.
• The z-scores for +1σ and –1σ are +1 and –1, respectively.
• The z-scores for +2σ and –2σ are +2 and –2, respectively.
• The z-scores for +3σ and –3σ are +3 and –3 respectively.
The empirical rule is also known as the 68-95-99.7 rule.