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The Distribution of Sample Means - Coggle Diagram
The Distribution of Sample Means
Samples, Populations, and the Distribution of Sample Means
distribution of sample means: the collection of sample means for all the possible random samples of a particular size (n) that can be obtained from a population.
sampling distribution: a distribution of statistics obtained by selecting all the possible samples of a specific size from a population.
Sampling error: the natural discrepancy, or amount of error, between a sample statistic and its corresponding population parameter.
Characteristics of the Distribution of Sample Means
The distribution of sample means is approximately normal, with most values near the population mean and fewer as they move farther away.
Larger samples produce means closer to the population mean, while smaller samples produce more scattered results.
Sample means should pile up around the population mean, representing the population even if not perfect.
Shape, Central Tendency, and Variability for the Distribution of Sample Means
The Mean of the Distribution of Sample Means: The Expected Value of M
expected value of M: the mean of the distribution of sample means is equal to the mean of the population of scores, and is called the expected value of M
The Standard Error of M
law of large numbers: states that the larger the sample size (n), the more probable it is that the sample mean will be close to the population mean.
standard error of M: the standard deviation of the distribution of sample means is called the standard error of M. The standard error provides a measure of how much distance is expected on average between a sample mean (M) and the population mean (μ)
The standard error describes the distribution of sample means. It provides a measure of how much difference is expected from one sample to another.
Standard error measures how well an individual sample mean represents the entire distribution. Specifically, it provides a measure of how much distance is reasonable to expect between a sample mean and the overall mean for the distribution of sample means.
The Shape of the Distribution of Sample Means
Distributions are almost perfectly normal of either of the following two conditions are satisfied:
The population from which the samples are selected is a normal distribution.
The number of scores (n) in each sample is relatively large, around 30 or more.
Three Different Distributions
sample selected from population
the distribution of sample means.
the original population of scores.
The Central Limit Theorem
For any population with mean and standard deviation, the distribution of sample means for sample size n will have a mean of μ and a standard deviation of σ/√n and will approach a normal distribution as n approaches infinity.
z-Scores and Probability for Sample Means
a z-score identifies the location with a signed number so that
the sign tells whether the location is above (+) or below (−) the mean.
the number tells the distance between the location and the mean in terms of the number of standard deviations.
the z-score for a sample mean can be defined as a signed number that identifies the location of the sample mean in the distribution of sample means so that
the sign tells whether the sample mean is located above (+) or below (−) the mean for the distribution (which is the population mean)
the number tells the distance between the sample mean and population mean in terms of the number of standard errors.
More about Standard Error
Sampling error: The general concept of sampling error is that a sample typically will not provide a perfectly accurate representation of its population. More specifically, there typically is some discrepancy (or error) between a statistic computed for a sample and the corresponding parameter for the population.
Standard error: When most of the sample means are relatively close to the population mean (those in the center of the distribution). These samples provide a fairly accurate representation of the population. On the other hand, some samples produce means that are out in the tails of the distribution, relatively far from the population mean. These extreme sample means do not accurately represent the population.
Looking Ahead to Inferential Statistics
Specifically, there is always a margin of error that must be considered whenever a researcher uses a sample mean as the basis for drawing a conclusion about a population mean.
The sample mean is not perfect. The distribution of sample means and the standard error will be critical elements in the inferential process.
The natural differences that exist between samples and populations introduce a degree of uncertainty and error into all inferential processes.