# Sampling Distribution

## What is a 'Sampling Distribution'

A sampling distribution is a probability distribution of a statistic obtained through a large number of samples drawn from a specific population. The sampling distribution of a given population is the distribution of frequencies of a range of different outcomes that could possibly occur for a statistic of a population.

## BREAKING DOWN 'Sampling Distribution'

A lot of data drawn and used by academicians, statisticians, researchers, marketers, analysts, etc. are actually samples, not population. A sample is a subset of a population. For example, a medical researcher that wanted to compare the average weight of all babies born in North America from 1995 to 2005 to those born in South America within the same time period cannot within a reasonable amount of time draw the data for the entire population of over a million childbirths that occurred over the ten-year time frame. He will instead only use the weight of, say 100 babies, in each continent to make a conclusion. The weight of 200 babies used is the sample and the average weight calculated is the sample mean.

Now suppose that instead of taking just one sample of 100 newborn weights from each continent, the medical researcher takes repeated random samples from the general population, and computes the sample mean for each sample group. So, for North America, he pulls up data for 100 newborn weights recorded in the US, Canada, and Mexico as follows: four 100 samples from select hospitals in the US, five 70 samples from Canada, and three 150 records from Mexico, for a total of 1200 weights of newborn babies grouped in 12 sets. He also collects a sample data of 100 birth weights from each of the 12 countries in South America. Each sample has its own sample mean and the distribution of the sample means is known as the sample distribution.

The average weight computed for each of the 2400 sample set is the sampling distribution of the mean. Not just the mean can be calculated from a sample. Other statistics, such as the standard deviation, variance, proportion, and range can be calculated from a sample data. The standard deviation and variance measure the variability of the sampling distribution. The number of observations in a population, number of observations in a sample, and the procedure used to draw the sample sets determine the variability of a sampling distribution. The standard deviation of a sampling distribution is called the standard error. While the mean of a sampling distribution is equal to the mean of the population, the standard error depends on the standard deviation of the population, the size of the population, and the size of the sample. Knowing how spread apart the mean of each of the sample sets are from each other and from the population mean will give an indication of how close the sample mean is to the population mean. The standard error of the sampling distribution decreases as the sample size increases.

## Normal Distribution and Shape

A population or one sample set of numbers will have a normal distribution. However, because a sampling distribution includes multiple sets of observations, it will not necessarily have a bell-curved shape. Following our example, the population average weight of babies in North America and in South America has a normal distribution because some babies will be underweight (below the mean) or overweight (above the mean), with most babies falling in between (around the mean). If the average weight of newborns in North America is seven pounds, the sample mean weight in each of the 12 sets of sample observations recorded for North America will be close to seven pounds as well. But if you graph each of the averages calculated in each of the 1200 sample groups, the resulting shape may result in a uniform distribution, but it is difficult to predict with certainty what the actual shape will turn out to be. The more samples the researcher uses from the population of over a million weight figures, the more the graph will start forming a normal distribution.