To this point, we have used regression analysis only to describe the relationship between two variables in a sample. However, in statistical analysis, we are not usually interested in the characteristics of a particular sample. More often, we are interested in estimating the characteristics of the population from which the sample was drawn. Whenever we wish to make statements about the characteristics of a population, based on the characteristics of a sample, we must rely on the logic of statistical inference. In particular, we must employ the concept of sampling distributions. Indeed, the logic of inferential statistics is based largely on the concept of sampling distributions. A sampling distribution is the theoretical distribution of a sample statistic that would be obtained from a large number of random samples of equal size from a population. Consequently, the sampling distribution serves as a statistical “bridge” between a known sample and the unknown population. Sample statistics, such as the sample mean and variance, are used to provide estimates of corresponding population parameters, such as the population mean and variance.
Keywords
- Sample Statistic
- Statistical Inference
- Central Limit Theorem
- Sampling Distribution
- Population Parameter
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
What Is a Sampling Distribution?
A sampling distribution is a probability distribution of a statistic obtained from a larger 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.
In statistics, a population is the entire pool from which a statistical sample is drawn. A population may refer to an entire group of people, objects, events, hospital visits, or measurements. A population can thus be said to be an aggregate observation of subjects grouped together by a common feature.
- A sampling distribution is a probability distribution of a statistic that is obtained through repeated sampling of a specific population.
- It describes a range of possible outcomes for a statistic, such as the mean or mode of some variable, of a population.
- The majority of data analyzed by researchers are actually samples, not populations.
Understanding Sampling Distribution
A lot of data drawn and used by academicians, statisticians, researchers, marketers, analysts, etc. are actually samples, not populations. 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 draw the data for the entire population of over a million childbirths that occurred over the ten-year time frame within a reasonable amount of time. They will instead only use the weight of, say, 100 babies, in each continent to make a conclusion. The weight of 100 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, they pull up data for 100 newborn weights recorded in the U.S., Canada, and Mexico as follows: four 100 samples from select hospitals in the U.S., five 70 samples from Canada, and three 150 records from Mexico, for a total of 1,200 weights of newborn babies grouped in 12 sets. They also collect 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 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 sample data. The standard deviation and variance measure the variability of the sampling distribution.
The number of observations in a population, the 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.
Special Considerations
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.
However, if you graph each of the averages calculated in each of the 1,200 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.