sampling distribution

(noun)

The probability distribution of a given statistic based on a random sample.

Related Terms

Examples of sampling distribution in the following topics:

  • Creating a Sampling Distribution

    • Learn to create a sampling distribution from a discrete set of data.
    • Specifically, it is the sampling distribution of the mean for a sample size of 2 ($N=2$).
    • The more samples, the closer the relative frequency distribution will come to the sampling distribution shown in the above figure.
    • As the number of samples approaches infinity , the frequency distribution will approach the sampling distribution.
    • This means that you can conceive of a sampling distribution as being a frequency distribution based on a very large number of samples.

  • Properties of Sampling Distributions

    • Knowledge of the sampling distribution can be very useful in making inferences about the overall population.
    • Sampling distributions are important for inferential statistics.
    • Fortunately, this information is directly available from a sampling distribution.
    • The most common measure of how much sample means differ from each other is the standard deviation of the sampling distribution of the mean.
    • Finally, the variability of a statistic is described by the spread of its sampling distribution.

  • What Is a Sampling Distribution?

    • The sampling distribution of a statistic is the distribution of the statistic for all possible samples from the same population of a given size.
    • These determinations are based on sampling distributions.
    • Sampling distributions allow analytical considerations to be based on the sampling distribution of a statistic rather than on the joint probability distribution of all the individual sample values.
    • The sampling distribution depends on: the underlying distribution of the population, the statistic being considered, the sampling procedure employed, and the sample size used.
    • Each sample has its own average value, and the distribution of these averages is called the "sampling distribution of the sample mean. " This distribution is normal since the underlying population is normal, although sampling distributions may also often be close to normal even when the population distribution is not.

  • Sampling Distribution of p

    • Compute the mean and standard deviation of the sampling distribution of p
    • State the relationship between the sampling distribution of p and the normal distribution
    • The sampling distribution of p is a special case of the sampling distribution of the mean.
    • Therefore, the mean of the sampling distribution of p is 0.60.
    • The sampling distribution of p is a discrete rather than a continuous distribution.

  • Mean of All Sample Means (μ x)

    • The mean of the distribution of differences between sample means is equal to the difference between population means.
    • The distribution of the differences between means is the sampling distribution of the difference between means.
    • The variance sum law states that the variance of the sampling distribution of the difference between means is equal to the variance of the sampling distribution of the mean for Population 1 plus the variance of the sampling distribution of the mean for Population 2.
    • Recall that the standard error of a sampling distribution is the standard deviation of the sampling distribution, which is the square root of the above variance.
    • The resulting sampling distribution as diagramed in , is normally distributed with a mean of 10 and a standard deviation of 3.317.

  • Sampling Distributions and Statistic of a Sampling Distribution

    • You can think of a sampling distribution as a relative frequency distribution with a great many samples.
    • (See Sampling and Data for a review of relative frequency).
    • If you let the number of samples get very large (say, 300 million or more), the relative frequency table becomes a relative frequency distribution.
    • A statistic is a number calculated from a sample.
    • The sample mean $\bar{x}$ is an example of a statistic which estimates the population mean $\mu$.

  • Introduction to Sampling Distributions

    • The distribution shown in Figure 2 is called the sampling distribution of the mean.
    • For this simple example, the distribution of pool balls and the sampling distribution are both discrete distributions.
    • The more samples, the closer the relative frequency distribution will come to the sampling distribution shown in Figure 2.
    • As the number of samples approaches infinity, the relative frequency distribution will approach the sampling distribution.
    • In later sections we will be discussing the sampling distribution of the variance, the sampling distribution of the difference between means, and the sampling distribution of Pearson's correlation, among others.

  • Shapes of Sampling Distributions

    • The overall shape of a sampling distribution is expected to be symmetric and approximately normal.
    • As previously mentioned, the overall shape of a sampling distribution is expected to be symmetric and approximately normal.
    • When calculated from the same population, it has a different sampling distribution to that of the mean and is generally not normal; although, it may be close for large sample sizes.
    • Sample distributions, when the sampling statistic is the mean, are generally expected to display a normal distribution.
    • Give examples of the various shapes a sampling distribution can take on

  • Sampling Distributions and the Central Limit Theorem

    • The central limit theorem for sample means states that as larger samples are drawn, the sample means form their own normal distribution.
    • The central limit theorem for sample means specifically says that if you keep drawing larger and larger samples (like rolling 1, 2, 5, and, finally, 10 dice) and calculating their means the sample means form their own normal distribution (the sampling distribution).
    • The upshot is that the sampling distribution of the mean approaches a normal distribution as $n$, the sample size, increases.
    • The usefulness of the theorem is that the sampling distribution approaches normality regardless of the shape of the population distribution.
    • Illustrate that as the sample size gets larger, the sampling distribution approaches normality

  • Introduction to one-sample means with the t distribution

    • First, a large sample ensures that the sampling distribution of $\bar{x}$ is nearly normal.
    • The second motivation for a large sample was that we get a better estimate of the standard error when using a large sample.
    • The standard error estimate will not generally be accurate for smaller sample sizes, and this motivates the introduction of the t distribution, which we introduce in Section 5.3.2.
    • We will see that the t distribution is a helpful substitute for the normal distribution when we model a sample mean $\bar{x}$ that comes from a small sample.
    • While we emphasize the use of the t distribution for small samples, this distribution may also be used for means from large samples.