Types of Sampling: || Intro. to Sampling Techniques...:

Outline:

Introduction

Inferential statistics entails extrapolating results from a sample to the entire population. Determining how far sample data are likely to differ from each other and from the population parameter is an important component of inferential statistics. These conclusions are derived from sampling distributions.

Definition:Sampling Distribution

It is the probability distribution of a statistic derived from a larger number of samples gathered from a certain population. The sampling distribution of a population is the frequency distribution of a range of alternative outcomes that could occur for a population statistic.

The Sampling Distribution, also known as a finite-sample distribution, depicts the frequency distribution of how far apart distinct events will be for a given population. The underlying population distribution, the statistic being analyzed, the sampling process utilized, and the sample size used all determine the sampling distribution.

The numerical descriptive measures we calculate from a random sample from a population are referred to as statistics. These statistics alter or vary depending on the random sample we choose, hence they are random variables. As a result, statistical probabiity distributions are referred to as sampling distributions. This sampling distirbution provide two informations:

  • What values of the statistic can occur?
  • How often each value occurs?

    • Definition: The Sampling Distribution of a Statistic

    This is the probability distribution for the possible values of the statistic that results when random samples of size n are repeatedly drawn from the population.

    Example 1:

    Consider a population that has a mean μ and standard deviation σ. Assume we gather samples of a given size from this population on a regular basis and compute the arithmetic mean for each sample. The sample mean is the term coined to this statistic. Each sample has its own average value, and the “sampling distribution of the sample mean” is the distribution of these averages. Although sampling distributions are frequently near to normal even when the population distribution is not, this distribution is normal because the underlying population is normal.

    Example 2:

    Assume you wish to know the average height of students from each Region at a specific University X. You select 100 students at random from each region and compute the mean for each sample group.
    In Region II, for example, you select data regarding student heights at random and calculate the mean for 100 of them. You also determine the mean height for 100 students using data from Region IX at random.
    You may compute the mean of the sampling distribution by obtaining the mean of all the average heights of each sample group as you continue to find average heights for each sample group of students from each location.

    Sampling

    In order to make statistical inferences and estimate population characteristics, sampling is a way of selecting individuals or a subset of the population. Several sampling procedures are widely used in research so that researchers do not have to survey the entire community in order to obtain useful information.

    To estimate population characteristics, samples are used. The mean of a sample, for example, is used to determine the population mean. However, because the sample is a subset of the population, the sample mean is unlikely to be exactly identical to the general mean. Similarly, the sample standard deviation is unlikely to be identical to the population standard deviation. As a result, we can expect a disparity between a sample statistic and the population parameter that corresponds to it.

    SAMPLING ERROR - The difference between a sample statistic and its corresponding population parameter.

    There are three ways to find the sampling distribution of a statistic:

  • Derive the distribution mathematically using the laws of probability.
  • Use a simulation to approximate the distribution. That is, draw a large number of samples of size n, calculating the value of the statistic for each sample, and tabulate the results in a relative frequency histogram. when the number of samples is large, the histogram will be very close to the theoretical sampling distribution.
  • Use statistical theorems to derive exact or approximate sampling distribution.

    • It was not too difficult to derive these sampling distribution in the number of elements in the population was very small. When this is not the case, you may need to use on e of these methods:

  • Use a simulation to approximate the sampling distribution emmpirically.
  • Rely on statistical theorems and theoretical results.

    • THE CENTRAL LIMIT THEOREM

    The Central Limit Theorem says that sums and averages of random samples of measures chosen from a population tend to have an approximately normal distribution under fairly generic assumptions. The Central Limit Theorem asserts that as the sample size grows higher, the sampling distribution of the sample means approaches a normal distribution, regardless of the form of the population distribution. This is especially true for sample sizes greater than 30.

    How Large the Sample Size is Large Enough?

  • If the sampled population is normal, then the sampling distribution of sample mean will also be normal as well, regardless of the sample size you choose.
  • When the sampled population is approximately symmetric, the sampling distribution of the sample mean becomes approximately normal for relatively small values of n.
  • When the sampled population is skewed, the sample size n must be larger, with n at least 30 before the sampling distribution of the sample mean becomes approximately normal.

    • Thank you for Reading!

    I'd love to hear your thoughts about the Statistics and Sampling Distribution. Feel free to leave your comment section below.

    References:

  • https://online.stat.psu.edu/stat500/lesson/4
  • https://www.investopedia.com/terms/s/sampling-distribution.asp
  • https://www.khanacademy.org/math/statistics-probability/sampling-distributions-library
  • https://www.geneseo.edu/~pyan/bs/c07.pdf
  • https://online.stat.psu.edu/stat500/lesson/4

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