The Central Limit Theorem is a fundamental theorem of probability that allows researchers to run certain statistical tests on any data set that is assumed to be “normal,” meaning that the distribution of mean scores or values in the sample fits a symmetrical, bell-shaped curve, with most of the values centered around the mean.
According to this theorem, if we draw a random sample from a population and we plot all the possible means we might find in our sample, the distribution of these means will have the same mean and standard deviation as the population. Also, if we increase the size of our sample, the shape of our sampling distribution will become more normal.
The Central Limit theorem helps use determine how many cases or people we should sample based on the shape of the population distribution. If the population distribution is normal, as it is for IQ scores, then we do not need a large sample to be able to run accurate statistical tests analyzing that score. For data with a non-normal population distribution, such as income, we need a larger sample.