Sampling Distributions | Applied Biostatistics | BIO733_Topic068

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Sampling distributions are a fundamental concept in biostatistics that address the variability inherent when drawing multiple samples from the same population. Since it is often impossible to study an entire population, researchers rely on samples to estimate population parameters; however, different samples yield different estimates for the same statistic. The sampling distribution describes the probability of obtaining various values for a specific statistic across all possible random samples of a fixed size drawn from that population. This concept helps analysts understand how likely it is to observe a particular value and provides insight into the distribution of these statistics within the broader context of the population. The process of constructing a sampling distribution depends on whether sampling is done with or without replacement, which significantly affects the number of possible samples. In sampling with replacement, an individual unit is studied and then returned to the population, allowing it to be selected multiple times in different samples; mathematically, this results in $N^n$ possible combinations where $N$ is the population size and $n$ is the sample size. Conversely, sampling without replacement involves studying a unit and removing it from the pool, ensuring no individual is studied twice in the same sequence, which leads to $N$ choose $n$ distinct samples. Regardless of the method chosen, if multiple samples are taken, the resulting statistics will vary, and their collective behavior forms the sampling distribution. To empirically construct this distribution for a discrete and finite population, one follows a systematic three-step approach: first, draw all possible samples from the population; second, calculate the statistic of interest for each sample; and third, list these calculated values along with their frequencies to determine relative frequencies or probabilities. This data can be organized into a table showing distinct statistic values and how often they occur, where the sum of the frequencies equals the total number of possible samples. Alternatively, the distribution can be represented graphically by plotting the possible statistic values on the x-axis against their corresponding probabilities or relative frequencies on the y-axis, which often reveals a symmetric pattern, although symmetry is not guaranteed in every case. Understanding the sampling distribution allows researchers to focus on three critical characteristics: the mean, the variance, and the functional form of the distribution. While determining the exact functional form can be difficult for large or infinite populations, approximations are often used in such scenarios. Additionally, these distributions can sometimes be derived mathematically rather than just empirically. By analyzing these properties, statisticians can make informed inferences about population parameters based on sample data, acknowledging that while individual sample results may vary, the underlying distribution of those results follows predictable patterns that are essential for accurate statistical analysis.

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Sampling has a lot of advantages and disadvantages. We cannot study pop in the whole population in all the situations. That's why we rely on our sample. But one can take more than one sample from the same population. And every time you take a sample, one can get a different estimates coming from the sample. And these estimates could have different values. In this module, we will be talking about the sampling distribution. It talks about all those various possible values that can occur as a result of all possible samples. But when we draw a sample, sample could be drawn using two methodologies. One way is that we draw a sample with replacement. And the other method is that we draw a sample without replacement. Though in all the conditions, it's not always possible to draw sample with replacement or sometimes without replacement only. In with replacement sampling, we select a unit, after studying it, we place it back into the population. So in with replacement sampling, it is very likely that an individual could be selected twice or even more from the same population. Whereas in without replacement sampling, we select our individual, study it, and then we do not place them back. So that's the same individual that has already been studied will not be studied again. So, there there So when we are doing with replacement sampling, the number of all possible samples that could happen is n to the power n, where capital N denotes the population size and small n denotes the sample size. And likewise, for with replacement sampling, the all possible samples drawn could be equals to n choose n. Where capital N again represents the population size and small n represents the the sample size. Once we have decided to draw a sample with either with replacement sampling methodology or without replacement sampling methodology. So, one can take multiple samples from coming from the same population. When we take multiple sample and from each sample, we make we take the value of certain statistic. The distribution of all possible values that can be assumed by some statistic computing from the sample of the same size randomly drawn from the same population is called sampling distribution. When we are taking multiple samples from the same population, every time the sampling units and the sample units are going to be different. Hence, the statistics could be different in all the situation in all the various samples. The sampling distribution will help us to view that how what what's most likely for any specific value of a sample statistic to show up. Sa- sampling distribution describe how these different values are distributed in the population. Sampling distribution may be constructed empirically when sampling from a discrete and finite population. Hence, we follow these three steps. In the very first step, from a finite population of size n, we randomly draw all possible samples of size n. Once these all possible samples have been drawn, we compute the statistic of our interest fro- for each sample. And once we have calculated all the statistic of interest for each sample, we'll list them in one column, the different distinct observed values of the statistic. And in other column, we list the corresponding frequency of occurrence of each distinct observation value for the statistic. And eventually, we calculate the relative frequencies. Hence, it will form the complete sampling distribution. Let's look at this example. When we are saying that theta is an n n is a is a statistic, so the sampling distribution of this sample statistic theta could be given in this table. Where theta one, theta two, theta three, theta four, theta five are possible statistic which are obtained as a process of calculating the statistics from all possible samples. And the frequency n1, n2, n3, n4, n5 represents the number of times all these statistics estimator values were obtained. N1 represents the theta one occurred n1 number of times. And n5 represents that theta five occurred n5 number of times. And when you add all these frequencies up, this should be equals to all possible samples obtained. Then, we calculate the relative frequency. And here, we call it probability of theta. So, probability of theta one is equals to n1 divided by n. So, hence, it's overall, it it should add up to one because these are the relative frequencies or the probabilities and they all add up to one. There's another way to represent a sampling distribution, and that is graphical way. So, when we represent in graph, on x-axis, we can take the possible values of our statistic. And on y-axis, it could be probabilities or the relative frequencies. And if you look at it, in this picture, it shows quite symmetric symmetric distribution, but it's not necessarily true that we always get a symmetric distribution, but most of the time we do. There are a few important characteristics of the sampling distribution. We usually are interesting in knowing three important characteristics. The first one is mean, other is variance, and the functional form. For large or infinite population, sometimes it's not really easy to find out the functional form of the distribution. Hence, in those those situations, we try to approximate the sampling distribution. We can also obtain the sampling distribution mathematically. Thank you.